Introduction

Regenerative chatter is a regular and complicated phenomenon that occurs frequently during machining operations. Chatter is not directly dependent on the machining parameters rather relies on the combination of these parameters. Therefore, it is quite pertinent that a proper combination of input parameters should be selected so that machining at these combinations will result in minimum chatter and maximum MRR.

In the last few decades, researchers are using process-based approach to identify chatter. In this regard, first step is to acquire the signal to identify the machining state. To achieve this, researchers have used various sensors like dynamometers [1,2,3], accelerometers [4,5,6], microphone [7,8,9,10,11] and Motor current [12, 13] for acquiring the signal. Delio et al. [7] compared acceleration sensors, displacement sensors with microphone, and deduced that chatter detection using acoustic signal are more effective and less costly.

The analysis of the experimentally recorded signals is the next step in tool chatter recognition. To examine the signal, some researchers employed time–frequency analysis such as Short-Time Fourier Transform (STFT) [14, 15], Continuous Wavelet Transform (CWT) [16, 17], Wavelet Transform (WT) [18, 19] and Wigner-Ville Distribution (WVD) [20]. However, due to the constraints of uncertainty principle, time–frequency methods are not so accurate. Flaws related to time–frequency analysis have been also reported in the previous literature [21].

To overcome such issues, researches have proposed various novel Self-Adaptive techniques such as Empirical Mode Decomposition (EMD) [22] and Ensemble Empirical Mode Decomposition (EEMD) [5, 6], Empirical Wavelet Transform (EWT) [23] and Variational Mode Decomposition (VMD) [24]. However, all these Self-Adaptive signal processing techniques have their own limitations such as EMD and EEMD is known for limitations like end effect, mode aliasing, sensitivity to noise and sampling. In VMD method, mode numbers must be indicated in advance [25]. EWT can lead to an improper segmentation in the frequency domain [26, 27]. To address the difficulty of EMD, a new self-adaptive signal processing techniques i.e., LMD was proposed by smith [28]. To smooth the original local mean function and envelope estimate function, the LMD process employs the moving averaging algorithm. However, LMD has also some inherent defects [29]. Recently, Mishra et al. [11] used Spline-Based Local Mean Decomposition (SB-LMD) technique to detect tool chatter in milling process using statistical indicators.

After signal processing and feature extraction, it is quite essential to develop a chatter severity model which can ensure better surface finish. In the present competitive and time-driven scenario, industries need a reliable cutting parameter for having minimum chatter. In the recent years, neural network approach has been applied to monitor machining processes [30,31,32]. Lamraoui et al. [33] adopted neural network approaches for chatter detection in milling. Kumar and Singh [34] employed an Artificial Neural Network (ANN) technique to forecast the safe cutting zone and metal removal rate in turning. Main concern in ANN modelling is to select the training function and optimized number of neurons in the hidden layer, to prevent it from overfitting and under fitting. However, several researchers have made an attempt to optimize the neurons in the hidden layer [35, 36].

Still, a clear concept is still missing regarding the selection of optimal neurons and the training function for applying ANN approach to build chatter severity model in terms of input milling parameters. In the present work, an attempt has been made to overcome these issues. Moreover, audio signals obtained during milling have been processed using SB-LMD technique. The remaining paper is organized as follows: Sect. 2 enumerates the outline of the presented work. Section 3 presents chatter signal simulation model. In Sect. 4, mathematical background SB-LMD is presented. Section 5 comprises the experimental details on extraction of tool chatter features. ANN modelling of chatter severity and details related to selection of optimized neuron in hidden layer and training and activation function is presented in Sect. 6. Result and discussion is presented in Sect. 7. At last, the conclusions drawn are summed up in Sect. 8.

Outline of the Presented Work

Figure 1 presents the outline of the presented work. The presented work has been divided into four stages for easy comprehension. The machine, tool, workpiece and cutting parameters have been selected in the first stage. Audio signals have been acquired using the microphone. In the second stage, viability of proposed signal processing technique i.e., SB-LMD has been tested on acquired audio signals. Normalized energy ratio (NER) and Co-relation coefficient (CC) are employed to choose the most prominent product function (PF) having higher information of tool chatter. Moreover, a new Chatter Indicator (CI) has been developed to differentiate the state of machining condition. In the third stage, An ANN model has been developed for CI using optimized neuron in hidden layer and training function.

Fig. 1
figure 1

Outline of the presented work

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Chatter Signal Simulation Model

In the present section, the frequency domain has been used to simulate real time working using Zeroth-Order Approximation (ZOA) [37] as shown in Fig. 2. ‘N’ number of teeth has been taken at milling cutter with a zero-helix angle at angular speed (Ω) (rad/s) and having cutter rotation period as ‘T’. Any arbitrary tooth number (let say k) has been selected for developing the model of end milling cutter displacement. In this context, the variation of tooth engagement angle is represented by \(\theta_{k} (t) = \Omega t\) where (Ω) (rad/s), is the angular speed of spindle. Resultant chip thickness is given by

$$h\left( {\theta_{k} } \right) = \left[ {f_{t} {\text{ sin}}\theta_{k} + \left( {z_{k,0} - z_{k} } \right)} \right]u\left( {\theta_{k} } \right).$$
(1)
Fig. 2
figure 2

Regenerative chatter in milling system

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Resultant chip thickness is a combination of the static chip thickness \((f_{t} \sin \theta_{k} )\) and dynamic chip thickness. Where, feed rate per tooth is represented by \(f_{t}\) and displacement of the cutter at the previous and present tooth periods is given by \((z_{k,0} ,z_{k} )\). Because static chip thickness does not contribute much in regenerative phenomenon, it has been omitted from the calculation. Now, modified chip thickness in terms of ‘x’ and ‘y’ direction displacement can be written as:

$$h(\theta_{k} ) = \left[ {\Delta x\sin \theta_{k} + \Delta y\cos \theta_{k} } \right] \, u(\theta_{k} ).$$
(2)

The mechanistic cutting forces are directly proportional to the dynamic chip thickness and are defined in matrix form as:

$$\left\{ {\begin{array}{*{20}c} {F_{x} } \\ {F_{y} } \\ \end{array} } \right\} = \frac{1}{2}bK_{t} \left[ {\begin{array}{*{20}c} {d_{xx} } & {d_{xy} } \\ {d_{yx} } & {d_{yy} } \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {\Delta x} \\ {\Delta y} \\ \end{array} } \right\},$$
(3)

where b, Kt and Kr are axial depth of cut and the cutting coefficients, respectively. Time-varying directional dynamic milling force coefficients are gives as

$$\begin{gathered} d_{xx} = \sum\limits_{k = 0}^{N - 1} {\left[ {K_{r} (1 - \cos 2\theta_{k} ) + \sin 2\theta_{k} } \right]} ( - u_{k} ); \, d_{xy} = \sum\limits_{k = 0}^{N - 1} {\left[ {K_{r} \sin 2\theta_{k} + (1 + \cos 2\theta_{k} )} \right]} ( - u_{k} ) \hfill \\ d_{yx} = \sum\limits_{k = 0}^{N - 1} {\left[ { - K_{r} \sin 2\theta_{k} + (1 + \cos 2\theta_{k} )} \right]} (u_{k} ); \, d_{yy} = \sum\limits_{k = 0}^{N - 1} {\left[ { - K_{r} (1 + \cos 2\theta_{k} ) + \sin 2\theta_{k} } \right]} (u_{k} ). \hfill \\ \end{gathered}$$

Equation (3) can be expressed in the time domain matrix, given by;

$$\left\{ {F(t)} \right\} = \frac{1}{2}bK_{t} [D(t)]\}$$
(4)

Applying Fourier transform, Eq. (4) can be expanded into the Fourier series as follows:

$$\left. \begin{gathered} \left[ {D(\omega )} \right] = {\mathcal{F}}[D(t)] = \sum\limits_{r = - \infty }^{ + \infty } {[D_{r} ]\delta (\omega - r\omega_{T} ) = \sum\limits_{r = - \infty }^{ + \infty } {[D_{r} ]{\text{e}}^{{ir\omega_{T} t}} } } \hfill \\ \, [D_{r} ] = \frac{1}{T}\int_{0}^{T} {[D(t)]{\text{e}}^{{ - ir\omega_{T} t}} dt} \hfill \\ \end{gathered} \right\},$$
(5)

where Ƒ and δ are the Fourier and Dirac delta function, respectively. Tooth engagement angle is given by \(\theta_{k} = \theta + k\theta_{p}\) and cutter pitch angle is \(\theta_{p} = {\raise0.7ex\hbox{${2\pi }$} \!\mathord{\left/ {\vphantom {{2\pi } N}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$N$}}\).

After substituting \(\omega_{T} t = N\theta\), the directional matrix is given by

$$\begin{gathered} [D_{r} ] = \frac{1}{\Omega T}\sum\limits_{k = 0}^{N - 1} {\int_{{k\theta_{p} }}^{{(k + 1)\theta_{p} }} {\left[ {\begin{array}{*{20}c} {d_{xx,k} } & {d_{xy,k} } \\ {d_{yx,k} } & {d_{yy,k} } \\ \end{array} } \right]{\text{e}}^{ - irN\theta } } } \hfill \\ { = }\frac{1}{{\theta_{p} }}\left( {\int_{0}^{{\theta_{p} }} {\left[ {\begin{array}{*{20}c} {d_{xx,0} } & {d_{xy,0} } \\ {d_{yx,0} } & {d_{yy,0} } \\ \end{array} } \right]} \, e^{ - irN\theta } d\theta + \int_{{\theta_{p} }}^{{2\theta_{p} }} {\left[ {\begin{array}{*{20}c} {d_{xx,1} } & {d_{xy,1} } \\ {d_{yx,1} } & {d_{yy,1} } \\ \end{array} } \right]} {\text{ e}}^{ - irN\theta } d\theta + ....} \right) \hfill \\ { = }\frac{N}{2\pi }\int\limits_{0}^{2\pi } {\left[ {\begin{array}{*{20}c} {d_{xx} } & {d_{xy} } \\ {d_{yx} } & {d_{yy} } \\ \end{array} } \right]} {\text{ e}}^{ - irN\theta } d\theta . \, \hfill \\ \end{gathered}$$
(6)

When number of harmonics,\(r = 0, \pm 1,\) then directional coefficient matrix becomes;

$$\left[ {D(t)} \right] = \sum\limits_{r = - 1}^{ + 1} {\left[ {D_{r} } \right]{\text{ e}}^{{ir\omega_{T} t}} }$$
(7)

White Gaussian noise has been added in the simulated model in order to realize the real working condition signal as represented in Fig. 3. This real working signal has been developed taking N (number of teeth) = 4, cutting coefficients Kt and Kr are 796 N/mm2and 0.212, respectively.

Fig. 3
figure 3

Simulated signal with white Gaussian noise in time domain

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Proposed Signal Processing Techniques

In the present work, an effort has been made to overcome the limitations of conventional LMD (C-LMD) by using spline interpolation. In this proposed signal processing technique, cubic spline interpolation has been used instead of moving average interpolation. Details of cubic spline interpolation has been given in ensuing section:

Cubic Spline Interpolation

Consider the problem of interpolating between the data points \((x_{0} ,y_{0} ),(x_{1} ,y_{1} )...(x_{n} ,y_{n} )\) by means of spline fitting.

Then, the cubic spline f (x) must follow listed conditions:

  1. i.

    \(f\left( x \right)\) is a linear polynomial outside the interval \(\left( {x_{0} ,x_{n} } \right)\),

  2. ii.

    \(f\left( x \right)\) is a cubic polynomial in each of the subintervals,

  3. iii.

    \(f^{\prime}\left( x \right)\) and \(f^{\prime\prime} \left( x \right)\) are continuous at each point.

Since \(f\left( x \right)\) is cubic in each of the subintervals then \(f^{\prime\prime} \left( x \right)\) shall be linear.

Taking equally spaced values of x so that \(x_{i + 1} - x_{i} = h\), we can write

$$f^{^{\prime\prime}} \left( x \right) = \frac{1}{h}\left[ {\left( {x_{i + 1} - x} \right)f^{^{\prime\prime}} \left( x \right) + \left( {x - x_{i} } \right)f^{^{\prime\prime}} \left( {x_{i + 1} } \right)} \right]$$
(8)

Integrating twice, we have

$$f(x) = \frac{1}{h}\left[ {\frac{{\left( {x_{i + 1} - x} \right)}}{3!}f^{^{\prime\prime}} \left( x \right) + \frac{{\left( {x - x_{i} } \right)}}{3!}f^{^{\prime\prime}} \left( {x_{i + 1} } \right)} \right]a_{i} \left( {x_{i + 1} - x} \right) + b_{i} \left( {x - x_{i} } \right)$$
(9)

The constants of integration ai, bi are determined by substituting the values of \(y = f\left( x \right)\) at xi and xi+1. Thus,

$$a_{i} = \frac{1}{h}\left[ {y_{i} - \frac{{h^{2} }}{3!}f^{^{\prime\prime}} \left( {x_{i} } \right)} \right],b_{i} = \frac{1}{h}\left[ {y_{i} - \frac{{h^{2} }}{3!}f^{^{\prime\prime}} \left( {x_{i + 1} } \right)} \right]$$

Substituting the values of ai, bi and writing \(f^{^{\prime\prime}} \left( {x_{i} } \right) = M_{i}\), Eq. 9 takes the form

$$f(x) = \frac{{(x_{i + 1} - x)^{3} }}{6h}M_{i} + \frac{{(x - x_{i} )^{3} }}{6h}M_{i + 1} + \frac{{x_{i + 1} - x}}{h}\left( {y_{i} - \frac{{h^{2} }}{6}M_{i} } \right) + \frac{{x - x_{i} }}{h}\left( {y_{i + 1} - \frac{{h^{2} }}{6}M_{i + 1} } \right)$$
(10)
$$\therefore\, f^{^{\prime}} (x) = - \frac{{(x_{i + 1} - x)^{2} }}{2h}M_{i} + \frac{{(x - x_{i} )^{2} }}{6h}M_{i + 1} - \frac{h}{6}(M_{i + 1} - M_{i} ) + \frac{1}{h}(y_{i + 1} - y_{i} )$$

To impose the condition of continuity of \(f^{\prime} \left( x \right)\), we get.

\(f^{^{\prime}} (x - \varepsilon ) = f^{^{\prime}} (x + \varepsilon )\) as \(\varepsilon \to 0\)

$$\therefore\, \frac{h}{6}\left( {2M_{i} + M_{i - 1} } \right) + \frac{1}{h}\left( {y_{i} - y_{i - 1} } \right) = - \frac{h}{6}\left( {2M_{i} + M_{i + 1} } \right) + \frac{1}{h}\left( {y_{i + 1} - y_{i} } \right)$$
$$M_{i - 1} + 4M_{i} + M_{i + 1} = \frac{6}{{h^{2} }}\left( {y_{i - 1} - 2y_{i} + y_{i + 1} } \right),i = 1.....(n - 1)$$

There are only \(\left( {n - 1} \right)\) terms but the unknown parameters are (n + 1). To get the remaining terms, the first derivative values at endpoints has been used, which are known constants. Substituting the value of Mi in Eq. 10, gives the concerned cubic spline.

Spline-Based Local Mean Decomposition (SB-LMD)

The aforementioned cubic spline interpolation has been invoked to overcome the limitation of coventional LMD (C-LMD) method. Details of step followed in the SB-LMD are presented in Fig. 4.

Fig. 4
figure 4

Flowchart of SB-LMD

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Simulated Chatter Signal Processing via C-LMD

To test the feasibility of the proposed signal processing technique, first, a simulated chatter signal was developed (details given in Sect. 3) and has been processed with conventional LMD (C-LMD). The frequencies of the obtained simulated signal are extracted using C-LMD. The simulated chatter signal is decomposed by C-LMD into a sequence of PFs, as shown in Fig. 5a, and the corresponding Fast Frequency Transform (FFT) of PFs is shown in Fig. 5b. It is evident from these figures that the original signal's frequency peaks are not obtained.

Fig. 5
figure 5

a C-LMD processed simulated chatter signal, and b FFT of first three PF’s

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Simulated Chatter Signal Processing via SB-LMD

It is evident from the above discussion that the conventional LMD method is not always appropriate for processing a non-stationary and non-linear signal. To address the disadvantage of conventional LMD, SB-LMD is used. The PF’s of the simulated chatter signal has been decomposed as shown in Fig. 6a and the corresponding FFT of first three PF’s is shown in Fig. 6b. Chatter frequencies peaks are evidently marked in Fig. 6b. As a result, it can be concluded that the SB-LMD procedure is ideal for processing a non-stationary and non-linear signal signals.

Fig. 6
figure 6

a SB-LMD processed simulated chatter signal, and b FFT of first three PF’s

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Experimental Setup

End milling experiments (slotted) is performed after validating the proposed signal processing technique (SB-LMD) for simulated chatter signal. For performing the experiments, four fluted HSS milling cutter and Al 6061-T6 as a work piece material has been selected. Experimental set-up is shown in Fig. 7. Microphone (Model No: AHUJA AGN-480) having 600 Ω impedance and 2 mV/Pa sensitivity, has been used for gathering the generated sound signals (real working signal) during machining. Total 27 experiments have been performed with full factorial combinations of the input parameters, as presented in Table 1. Figure 8 shows a sample audio signal generated during a milling operation with an axial depth of cut of 1 mm, a feed rate of 50 mm/min, and a spindle speed of 1000 rpm.

Fig. 7
figure 7

End milling experimental set-up

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Table 1 Experimental setting (3 parameters at 3 levels)
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Fig. 8
figure 8

Sound signals for test run 1

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SB-LMD in Real Working Signals

For extracting the valuable components and removing the time series and associated spatial noise components from sound signal, SB-LMD technique is applied. Extracted Product Functions (PF’s) of the signal are shown in Fig. 9. Two major indicators are introduced to pick the important PF’s, containing the principle data of processing state. Relative energy ratio (NER) of the foremost noticeable PF shows higher energy content. When cutting is steady, energy dispersion is uniform and relative energy ratio is small. However, when chatter starts, relative energy ratio increases. Therefore, relative energy ratio is a good indicator to select the crucial PF’s. Second indicator, co-relation coefficients (CC) is also used.

Fig. 9
figure 9

SB-LMD processed real chatter signal

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The first three PFs have a higher relative energy ratio as well as higher correlation coefficient, as shown in Fig. 10. Therefore, first three PF’s are used to constitute the main signal information.

Fig. 10
figure 10

a CC and b NER of PF’s

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Figure 11 displays the newly developed signal's frequency spectrum. It is evident from Fig. 11 that SB-LMD technique is able to extract all three basic important frequency components, i.e., chatter frequency, tooth passing frequency (ωT = 66.6 Hz) and multiples of tooth passing frequency (ωk).

Fig. 11
figure 11

FFT of reconstructed signal

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Extraction of Tool Chatter Features

For ANN modelling, tool features are required. Therefore, in this section, SB-LMD processed audio signals are further explored by employing a new statistical indicator, Chatter Indicator (CI). Chatter Indictor is based on coefficient of variation, i.e., the ratio of mean to the standard deviation given by following relation:

$$CI\,{ = }\,\frac{{{\text{Mean (}}\overline{x} {)}}}{{{\text{Standard deviation (}}\sigma {)}}} \, {.}$$
(12)

Coefficient of variation is employed, because it reflects the degree of variability in proportion to the mean of the sound data point. Therefore, the value of CI is directly proportional to signal’s chatter components. CI is calculated for each of the 27 experimental runs and presented in Fig. 12.

Fig. 12
figure 12

Coefficient of variation (CI) v/s experimental number

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For all 27 experimental runs, the variation of CI has been calculated and marked in Fig. 12. Upper (red) and lower (green) threshold limits are defined using 3σ criterion to classify the three phases of chatter severity (stable, moderate and unstable), as demonstrated in Fig. 12. Green line represents the stable machining, while CI points above the red line represents unstable machining. The points in the middle (between green and red) represent a transition from stable to unsafe machining.

Surface topography of the machined surface was carefully studied at experimental runs 7 and 27, as shown in Fig. 13a, b, respectively, to understand the effect of feed rate. When looking at the surface topology of machined surfaces, it is obvious that clear and distinct chatter marks can be seen.

Fig. 13
figure 13

Surface topology for (a) experiment 7 and (b) Experiment 27

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Artificial Neural Network

Furthermore, a suitable prediction model has been created to investigate the relative dependency of chatter severity in terms of input milling parameters. Thus, to meet this objective, ANN technique has been invoked as presented in the present section.

Overview and Architecture of Artificial Neural Network

The human brain and biological nervous system are the cause of motivation behind the development of artificial neural network. Artificial neurons in ANN are the simplified models of biological neurons that are used for computational or processing the data. In the present work, architectures of ANN considered consists of three layers viz. Input [having 3 neurons, ‘Spindle Speed (SS)’, ‘Feed Rate (F)’ and ‘Axial Depth of Cut (D)’], hidden (having optimized number of neurons) and output layer [one neuron, i.e. ‘Chatter Indicator (CI)’] as shown in Fig. 14. Input layer receives machining data from the user end. Hidden layers perform most of the complex processing for extracting the features. An output layer presents the final output.

Fig. 14
figure 14

ANN architecture

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In ANN modelling, various training functions are available and can be used for training the data. Viability of training function is problem specific. In this work, different variants of BP algorithms have been invoked for training the model to improve the efficiency of ANN for producing better prediction models for chatter identification. To enhance the efficiency of back propagation in terms of its convergence, several algorithms have been developed known as variants of Back propagation. The variants of BP algorithms are as follows:

  1. a)

    Resilient Propagation (RP).

  2. b)

    Conjugate Gradient-based Algorithms (CGP).

  3. c)

    Levenberg–Marquardt Algorithm (LM).

To improve the prediction capability of ANN, different activation function has been used. List of activation functions are as follows:

  1. a)

    Hyperbolic tangent sigmoid transfer function (TANSIG) = \(\frac{2}{{1 + {\text{e}}^{ - 2z} }} - 1\)

  2. b)

    Log sigmoid transfer function (LOGSIG) = \(\frac{1}{{1 + {\text{e}}^{ - z} }}\)

  3. c)

    Linear transfer function (PURELIN) = z

In this study, PURELIN function has been discarded due to linear property which does not meet with the required condition.

To compare the above listed training (RP, CGP and LM) and activation function (TANSIG and LOGSIG) has been taken in consideration for better prediction of chatter and their details are given in sub sequent sections. In this study, an effort has been done to get the optimized neurons for the selected training and activation function to avoid the case of under or over fitting of the prediction model. Later, these optimized neurons for hidden layer are used to train the data. Details are given in subsequent section.

Selection of Number of Neurons in the Hidden Layer

Optimized Neuron in Hidden Layer for TANSIG

To get optimized neuron in the hidden layer for selected training function (RP, CGP and LM) and TANSIG as an activation function, Mean squared error (MSE) has been taken as a statistical indicator to select the number of neurons. To achieve the optimized neurons, data have been trained for all selected training algorithm taking TANSIG as an activation function, by varying the number of neurons in the hidden layer from 1 to 30. Mean squared error have been calculated for all these combinations. Calculated mean squared errors (MSE) have been plotted against number of neurons as shown in Fig. 15.

Fig. 15
figure 15

MSE v/s number of neurons for selected training algorithms using TANSIG

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Form Fig. 15, it has been observed that, optimized number of neurons are different for different algorithm. Optimized numbers of neurons in hidden layer for algorithms have been decided based on minimum difference between the train data and the validation data. From Fig. 15, it is observed that the optimum number of neurons for RP, CGP and LM are 8, 9 and 5 neurons, respectively.

Optimized Neuron in Hidden Layer for LOGSIG

Similarly, optimized neurons in the hidden layer have been evaluated for selected training function (RP, CGP and LM) and LOGSIG used as an activation function as presented in above section. Calculated mean squared errors (MSE) have been plotted against number of neurons as shown in Fig. 16.

Fig. 16
figure 16

MSE v/s number of neurons for selected training algorithms using LOGSIG

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Form Fig. 16, it has been observed that, optimized number of neurons are different for different algorithm. Optimized numbers of neurons in hidden layer for algorithms have been decided based on minimum difference between the train data and the validation data. From Fig. 16, it is observed that the optimum number of neurons for RP, CGP and LM are 10, 9 and 6 neurons, respectively.

Selection of Training function

To select the best training algorithm for the prediction of chatter severity taking TANSIG and LOGSIG as an activation function, ANN model is trained individually using RP, CGP, LM algorithms. Three input parameters [i.e. Spindle speed (SS), Feed rate (F) and Axial depth of cut (D)] are used as the input neurons layer. To anticipate the chatter, the output layer has one neuron i.e., Chatter Indicator (CI). Optimized number of neurons has been for further analysis.

When compared to the depth of cut (0.5–1.5 mm) and feed rate (0.1–0.2 m/min), the spindle speed range (1000–3000 rpm) is numerically too broad. As a result, the available data set is standardised to facilitate training and testing. The normalised values are in the range of 0 to 1 in this case. When the disparity between each parameter data is too great, normalised values are employed. This allows us to compare the sizes on a common scale to identify which component has the most impact on the response. This also aids in the preservation of orthogonality. The transformation of non-normalized variables into normalized form of input and output variable has been done by using the following relations:

$$SS_{n} = \frac{SS}{{SS_{\max } }}; \, F_{n} = \frac{F}{{F_{\max } }}; \, D_{n} = \frac{D}{{D_{\max } }}; \, CI_{n} = \frac{CI}{{CI_{\max } }}$$

Finally, the final trained network weights are those that result in the least amount of inaccuracy. To evaluate the prediction capability of the algorithms, Absolute Percentage Deviation (APD) has been calculated and presented in Table 2. Later, final Average Absolute Percentage Deviation (AAPD) has been calculated for every algorithm.

Table 2 AAPD for TANSIG and LOGSIG
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When TANSIG is used as an activation function, the Average Absolute Percentage Deviation (AAPD) for RP, CGP and LM are 7.49, 12.66 and 3.11, respectively. In LOGSIG activation function, average absolute percentage deviations for RP, CGP and LM are 7.92, 9.69 and 3.84, respectively. After comparing all possible combinations of selected training algorithm and activation functions, LM with TANSIG is found to be the most suitable combination for predicting the output (Chatter Index).

Prediction Model for CI

After selecting best training and activation function and optimized number of neurons in the previous sections, prediction model has been developed for CI (Chatter indicator). Proposed ANN model for CI is shown in Fig. 17. Three neurons (SSn, Fn, Dn) in the input layer, five neurons in the hidden layer, and one neuron (CIn) in the output layer are used to create a CI prediction model. For ANN modelling, LM algorithm as a training function along with TANSIG activation function (between the input and hidden layer) and TANSIG (between the hidden and output layer) has been used. Table 3 shows the optimal weight and bias for the hidden layer from the input layer. Table 4 shows the optimal weight and bias for the output layer from the hidden layer. For better understanding, superscript is used to represent the number of layer and subscript is used to represent the position of neuron in that layer.

Fig. 17
figure 17

ANN model

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Table 3 Weight to hidden layer from input layer
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Table 4 Weight to output layer from hidden layer
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Development of Mathematical Expression

To develop the mathematical expression for ANN model, θ expression has been used to represent the weight, whereas 0 is used to present the position of bias in the mathematical expression.

Mathematical expression for hidden layer is expressed as:

$$\begin{gathered} z_{1}^{2} \, = \, \theta_{11}^{1} \times a_{1}^{1} \, + \, \theta_{12}^{1} \times a_{2}^{1} \, + \, \theta_{13}^{1} \times a_{3}^{1} \, + \, \theta_{10}^{1} \hfill \\ z_{2}^{2} \, = \, \theta_{21}^{1} \times a_{1}^{1} \, + \, \theta_{22}^{1} \times a_{2}^{1} \, + \, \theta_{23}^{1} \times a_{3}^{1} \, + \, \theta_{20}^{1} \hfill \\ z_{3}^{2} \, = \, \theta_{31}^{1} \times a_{1}^{1} \, + \, \theta_{32}^{1} \times a_{2}^{1} \, + \, \theta_{33}^{1} \times a_{3}^{1} \, + \, \theta_{30}^{1} \hfill \\ z_{4}^{2} \, = \, \theta_{41}^{1} \times a_{1}^{1} \, + \, \theta_{42}^{1} \times a_{2}^{1} \, + \, \theta_{43}^{1} \times a_{3}^{1} \, + \, \theta_{40}^{1} \hfill \\ z_{5}^{2} \, = \, \theta_{51}^{1} \times a_{1}^{1} \, + \, \theta_{52}^{1} \times a_{2}^{1} \, + \, \theta_{53}^{1} \times a_{3}^{1} \, + \, \theta_{50}^{1} \hfill \\ \end{gathered}$$

where θ (Tables 3 and 4) is the weight, θ superscript represents the connection between the layers (input to hidden layer), first number in θ subscript is representing the position of neuron in the second layer. Second number in θ subscript is the position of neuron in the first layer. To get the value of neurons in the hidden layer, z values need to be fed in to the activation function (TANSIG).

Now, neurons in the hidden layer are calculated using the expressions;

$$\begin{gathered} a_{1}^{2} \, = \, \frac{2}{{1 + {\text{e}}^{{ - 2 \times z_{1}^{2} }} }} - 1; \, a_{2}^{2} \, = \, \frac{2}{{1 + {\text{e}}^{{ - 2 \times z_{2}^{2} }} }} - 1; \, a_{3}^{2} \, = \, \frac{2}{{1 + {\text{e}}^{{ - 2 \times z_{3}^{2} }} }} - 1 \hfill \\ a_{4}^{2} \, = \, \frac{2}{{1 + {\text{e}}^{{ - 2 \times z_{4}^{2} }} }} - 1; \, a_{5}^{2} \, = \, \frac{2}{{1 + {\text{e}}^{{ - 2 \times z_{5}^{2} }} }} - 1 \hfill \\ \end{gathered}$$

Mathematical expression for output layer is expressed as

$$z_{1}^{3} \, = \, \theta_{11}^{2} \times a_{1}^{2} \, + \, \theta_{12}^{2} \times a_{2}^{2} \, + \, \theta_{13}^{2} \times a_{3}^{2} \, + \, \theta_{14}^{2} \times a_{4}^{2} \, + \, \theta_{15}^{2} \times a_{5}^{2} + \, \theta_{10}^{2}$$

To get the value of neuron at the output layer, TANSIG function is used. Therefore, neuron value in the output is determined using the following expression

$$a_{1}^{3} {\text{(CI}}_{n} {) } = \, \frac{2}{{1 + {\text{e}}^{{ - 2z_{1}^{3} }} }} - 1$$

This developed mathematical expression for CI has been used to explore the effect of individual parameter on chatter severity. Outcome of this study has been presented in the subsequent section, i.e., result and discussion.

Result and Discussion

Effect of Individual Parameters on CI

Effect of individual input milling parameters on Chatter Indicator (CI) has been studied by plotting the outputs with respect to the input parameters viz. Spindle Speed (SS), Feed Rate (F) and Axial Depth of Cut (D) as shown in Fig. 18. On analyzing Fig. 18a, it can be interpreted that Chatter Indicator (CI) increases with the increase in spindle speed. By critically analyzing blue, red and green line in Fig. 18a, it is observed that the percentage change in Chatter Indicator (CI) is 81.91%, 49.35% and 30.47%, respectively. Percentage change in blue line which is drawn at low feed and depth of cut (F = 50 mm/min and D = 1 mm) is higher as compared to other lines. Thus, it can be concluded that the relative effect of Spindle Speed (SS) is dominating at lower values of Feed rate (F) and Axial Depth of Cut (D).

Fig. 18
figure 18

Effect of milling parameters on CI

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From Fig. 18b, it can be inferred that Chatter Indicator (CI) is not so much affected by the variation in Feed Rate (F) as already discussed in Fig. 12. Moreover, the percentage change in CI with Feed Rate (F) at respective Spindle Speed; SS (1000, 2000 and 3000 rpm) and Axial Depth of Cut; D (1, 1.5 and 2 mm) are 33.16%, 5.46% and 7.65%, respectively. However, percentage change in CI w.r.t feed rate at low spindle speed and axial depth of cut (SS = 1000 rpm and D = 1 mm) is higher as compared to the rest two states.

In Fig. 18c, effect of Axial Depth of Cut (D) at constant Spindle Speed (SS) (1000, 2000 and 3000 rpm) and Feed Rate (F) (50, 75 and 100 mm/min) on Chatter Indicator (CI) has been shown in three colours viz. blue, red and green. By observing Fig. 18c, it can be deduced that Chatter Indicator (CI) increases with the increase in Axial Depth of Cut (D). Percentage variations Chatter Indicator (CI) for blue, red and green lines are 76.36%, 51.67% and 41.77%, respectively. The net differences in the value of Chatter Indicator (CI) for blue, red and green lines are 1.129, 0.9705 and 0.962, respectively. Thus, on comparing Fig. 18a–c, it is obvious that the effects of Spindle Speed (SS) and Axial Depth of Cut (D) on Chatter Indicator (CI) are more severe as compared to Feed Rate (F). A similar result has also been presented earlier by analyzing Fig. 12.

Effect of interaction parameters on CI and MRR

Occurrence of chatter in milling results in poor surface finish and degraded tool life. Therefore, to maximize productivity along with better surface finish and longer tool life, it is necessary to minimize Chatter Indicator (CI), which ultimately depends on the interaction of input milling parameters. Therefore, the effects of interaction parameters on Chatter Index have been discussed in the present section with the help of contour and surface plots as presented in Fig. 19.

Fig. 19
figure 19

Contour and surface plots for CI

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Figure 19a represents the variation of Chatter Indicator (CI) with respect to spindle speed and axial depth of cut at constant feed rate (75 mm/min). Figure 19b represents the variation of Chatter Index with respect to feed rate and axial depth of cut at constant spindle speed (2000 rpm). Figure 19c represents the variation of Chatter Index with respect to spindle speed and feed rate at constant axial depth of cut (1.5 mm). These variations in Chatter Index have been presented with different colours (green, dark green, blue, yellow, tangerine and red). As colour level changes from green to red, chatter level increases. Minimum and maximum chatter within the considered range of parameters is represented by green and red colours, respectively. From these figures, it is evident that combined range of axial depth of cut, spindle speed and feed rate should be selected in such a way that it will result in minimum chatter (represented by green colour).

Finally, on collectively analyzing all these figures, favourable output range of CI (green colour) has been evaluated as shown in Table 5.

Table 5 Chatter severity regimes
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Safe Milling Zone

After analyzing the severity of individual parameters on chatter severity, range of parameters pertaining to stable milling have been ascertained by extracting common range of values from Fig. 19 and are presented in Table 6.

Table 6 Stability range
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To validate the obtained range, further experiment has been performed. This experiment has been performed considering the developed stable range and is presented in Table 7. Result of validation experiment shows that at this milling range CI value is below the threshold stability value as shown in Fig. 12. Moreover, surface texture of the machined surface is also presented in Table 7. On examining this surface texture, it is also evident that the developed range is correct. Thus, the proposed methodology is quite suitable for ascertaining the stable milling parameters that will result in higher productivity along with better surface finish.

Table 7 Experimental validation
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Conclusion

In the present work, a methodology has been proposed to ascertain tool chatter severity regimes in milling operation. A new modified LMD approach based on cubic spline interpolation and ANN modelling has been used to achieve the aforementioned objective. Moreover, to explore the influence of input milling parameters on chatter severity, ANN modelling has been invoked. Modelling has been done using various training functions (RP, CGP and LM), activation functions (LOGSIG and TANSIG) and also optimal number of neurons in the hidden layer. Mean squared errors have been evaluated to estimate the optimized number of neurons in the hidden layer. LM training function with optimal five neurons in hidden layer and TANSIG activation function is found to be the most suitable one for the prediction of CI with an average deviation of 3.11%. Finally, developed mathematical expression has been studied critically to explore the effect of milling parameters on chatter severity. It has been inferred that effects of spindle speed and axial depth of cut on chatter severity is more pronounced as compared to feed rate.