Use of artificial neural network and multi-objective genetic algorithm approach to predict and ascertain stable cutting zone in conventional turning process

Abstract

This article is focused on the investigation of stable cutting zone in turning operation. Experiments have been conducted to acquire raw chatter signals. Generally, raw chatter signals are contaminated with ambient noise. Wavelet transform has been used for pre-processing and denoising these signals. In order to predict the severity of tool chatter, a new parameter denoted as chatter index has been evaluated considering the aforesaid denoised signals. In the present work, mathematical models have been developed for chatter index and metal removal rate using feedforward backpropagation–based artificial neural network considering three activation functions: TANSIG, LOGSIG and PURELIN. Furthermore, multi-objective genetic algorithm technique has been applied to evaluate stable cutting zones with maximized metal removal rate. TANSIG activation function found to be best option to achieve the aforesaid objectives. Good correlation between the artificial neural network predicted results and experimental ones validate the developed technique.

Keywords

Wavelet denoising chatter index metal removal rate artificial neural network genetic algorithm

Introduction

In machining process, direct contact between tool and work piece results in violent vibration known as tool chatter. Tool chatter adversely affects the product quality resulting in wastage of work-piece material and loss of production time. Tool chatter is an ineluctable phenomenon frequently encountered in turning process. Chatter cannot be fully eliminated but it can be minimized. Chatter identification and suppression has been one of the challenging tasks for the researchers.^1,2 In the last few decades, many researchers have adopted several approaches to suppress chatter by using parametric excitation,³ passive adaptor,⁴ piezo-electronic actuator,⁵ electrostatic actuators⁶ and active magnetic bearings.⁷ Recently, spindle speed variation (SSV) approach has been adopted by researchers for chatter suppression.^8–10 In order to suppress tool chatter, Yang et al.¹¹ have proposed the active use of multiple tuned mass dampers (TMD). They observed that TMD enhances the chatter resistance of machine components. Apart from the aforesaid methods, some researchers have presented tool chatter mathematically considering single degree of freedom (SDoF),¹⁰ second degree of freedom¹² and third degree of freedom.¹³ Furthermore, researchers have also found that depth of cut, feed rate, cutting speed and spindle speed are most influencing cutting parameters that govern tool chatter in turning process. Clancy and Shin¹⁴ observed that, by reducing the depth of cut and feed rate, chatter can be minimized to a great extent. They also observed that the cutting stability is greatly increased at low cutting speed. Moreover, Tangjitsitcharoen¹⁵ has suggested that cutting force components also affect tool chatter and are responsible for continuous and broken chip formation. Altintas and Weck¹⁶ observed that the dynamic cutting force depends not only on chip thickness but also on the shear angle oscillation and tool flank wavy surface contact mechanism.

Moreover, some researchers have tried to explore the mechanism of tool chatter by processing the acquired raw tool chatter signals. Recently, Fourier transforms (FT)^17,18 and short-time Fourier transforms (STFT)¹⁹ have been used by researchers to investigate the chatter phenomenon. Furthermore, due to certain shortcomings in aforesaid techniques, wavelet transform (WT) technique^20,21 came into picture. In comparison with FT and STFT, WT possesses certain advantages such as performing local analysis,²² handling both stationary and non-stationary signals and also providing efficient time-frequency analysis. In the present work, WT has been adopted to pre-process the raw chatter signal by denoising it.

Moreover, improving the product quality and simultaneously minimizing the production cost is the foremost priority of every machining industry. Therefore, in order to improve the product quality, it is necessary to reduce surface roughness as well as chatter severity and enhance metal removal rate (MRR), which directly affects the production cost. Many researchers have considered various statistical and computational approaches for predicting the aforesaid responses. Davim et al.,²³ Kohli and Dixit²⁴ and Natarajan et al.²⁵ have designed artificial neural network (ANN) models for predicting surface roughness in turning. They observed that ANN is well suited for predicting surface roughness. Moreover, prediction of both surface roughness and MRR simultaneously is a multi-optimization problem. In this concern, some researchers have proposed multi-objective optimization techniques to minimize surface roughness and maximize MRR. Kumar et al.²⁶ have proposed a multi-response optimization technique based on utility concept and Taguchi approach for surface roughness and MRR prediction in turning process. They deduced that depth of cut, cutting speed and feed rate are the governing parameters. Paiva et al.²⁷ adopted multi-objective optimization technique to examine the cumulative effect of various cutting parameters on MRR including other response. However, till date no work has been reported on determining chatter-free stable cutting zone with maximized MRR in turning process. In the present work, ANN has been considered for predicting the integrated effect of cutting parameters on tool chatter and MRR. Thereafter, multi-objective genetic algorithm (MOGA) has been used to find out the stable cutting zone in turning process.

In the present work, raw chatter signals have been acquired experimentally and a new parameter denoted as chatter index (CI) has been evaluated by denoising of these signals. Signals have been denoised using WT technique. Moreover, MRR has been evaluated by weighting the work-piece sample before and after turning process. Thereafter, mathematical models have been developed for CI and MRR using feedforward backpropagation–based ANN considering three activation functions: TANSIG, LOGSIG and PURELIN. After deciding the best suitable activation function MOGA technique has been adopted to evaluate the stable cutting zones with maximized MRR.

Theoretical analysis

In the present work, an SDoF turning operation with a flexible tool and rigid work-piece has been considered as shown in Figure 1. Mathematical model for turning operation has been represented by an SDoF equation as given by^2,28

m \frac{d^{2} q (t)}{d t^{2}} + c \frac{d q (t)}{d t} + k q (t) = F_{f} (t)

(1)

where $F_{f} (t)$ is the dynamic cutting force in feed direction, $m$ is the mass of tool, c represents the damping coefficient and $k$ is the stiffness.

Figure 1.

(a) Mechanism of regeneration and (b) SDoF model for turning.

Equation (1) has been further written as follows

\frac{d^{2} q (t)}{d t^{2}} + \frac{c}{m} \frac{d q (t)}{d t} + \frac{k}{m} q (t) = \frac{1}{m} F_{f} (t)

(2)

$F_{f} (t)$ is given by

\begin{array}{l} F_{f} (t) = k_{f} \times b \times d {(t)}^{3 / 4} = \\ k_{f} b [d_{0} + q (t - τ) - q {(t)}^{3 / 4}] \end{array}

(3)

where $b$ is the chip width, $k_{f}$ is the cutting force coefficient, $d_{0}$ is nominal chip thickness and $τ$ is the time delay between current and previous time. Here,τ = 60 / N, $N$ is the spindle speed in r/min.

In the above equation $d (t)$ is the dynamic chip thickness due to tool vibration and is equal to $[d_{0} + q (t - τ) - q (t)]$ .

For apparent understanding, a framework of the proposed chatter detection has been shown in Figure 2.

Figure 2.

Flow chart of proposed methodology.

Experimentation

Turning has been performed on ASTM A36 mild steel by using high-speed precision lathe NH22 (Hindustan Machine Tool Ltd.) with different set of cutting parameters. Schematic diagram of the experimental setup has been presented in Figure 3. Figure 4 shows the experimental setup. A single point cutting carbide tool has been used for turning of mild steel bars.

Figure 3.

Schematic diagram of experimental set-up.

Figure 4.

Experimental set-up.

An accelerometer (Make: PCB PIEZOTRONICS and Model: 356A16) and data acquisition system (Make: OROS group and Model: OR35-multi analyser) have been used to acquire the raw chatter vibration signals. Furthermore, MRR has been obtained by weighing of work piece sample before and after turning process. Chemical composition of work piece has been listed in Table 1. For each set of turning, a new bar of ASTM A36 has been used. Work piece of 200 mm length and 40 mm diameter has been. Here, 140 mm length of work piece has been used for turning, while remaining 60 mm length was used for holding purpose in chuck.

Table 1.

Chemical composition of work material (A36 mild steel bar).

C	Cu	Fe	Mn	P	Si	S
0.25%–0.290%	0.20%	98.0%	1.03%	0.040%	0.280%	0.050%

A total of 27 runs of turning have been performed considering the cutting parameters: depth of cut (d), feed rate (f) and spindle speed (N). Here, experimental runs have been designed using full factorial design of experiment. The various levels of cutting parameters considered have been shown in Table 2.

Table 2.

Control factors and their level used in experimentation.

S.no.	Symbol	Factor	Unit	Level 1	Level 2	Level 3
1	d	Depth of cut	mm	1	1.5	2
2	f	Feed rate	mm/rev	0.1	0.15	0.2
3	N	Spindle speed	r/min	850	1000	1150

Experiments have been performed in order to acquire the raw chatter signals and some of the recorded signals have been shown in Figures 5 –7. These recorded raw signals have severe noise inclusions. In the present study, wavelet denoising has been done using MATLAB software and hybrid thresholding rule has been adopted to acquire much smoother results.

Figure 5.

Recorded noisy signal at d = 2 mm, f = 0.1 mm/rev and N = 1000 r/min.

Figure 6.

Recorded noisy signal at d = 1.5 mm, f = 0.1 mm/rev and N = 1150 r/min.

Figure 7.

Recorded noisy signal at d = 2 mm, f = 0.15 mm/rev and N = 1150 r/min.

Wavelet-based denoising technique

The inclusion of noise in the signal interrupts the identification of exact chatter. In this study, wavelet denoising technique has been done. In wavelet denoising technique, the noisy signal is first decomposed using WT, where the level of decomposition depends upon the length of the signal. After decomposition, the thresholding of the coefficients has been done. If the wavelet coefficient is smaller than the threshold level, it is then set as zero and if the coefficient is larger than threshold level, it is either adapted or kept as it is.

Wavelet decomposition

The aforesaid denoising technique has been implemented in two steps, that is, wavelet decomposition and wavelet thresholding. In wavelet decomposition technique, the acquired signal of finite energy is passed through low pass and high-pass filter. Low-pass filter will result in approximate coefficient, while high-pass filter will yield detailed coefficient. The level of decomposition depends on the length of the signal. In Figures 8 –10, d₁, d₂, d₃ and d₄ are stationary detailed coefficients acquired at decomposition levels 1, 2, 3 and 4, respectively. A detailed coefficient has been obtained when raw signal is passed through high-pass filter, while approximated coefficient ‘a₄’ has been obtained at lower frequency.

Figure 8.

Denoising of signal at d = 2 mm, f = 0.1 mm/rev and N = 1000 r/min.

Figure 9.

Denoising of signal at d = 1.5 mm, f = 0.1 mm/rev and N = 1150 r/min.

Figure 10.

Denoising of signal at d = 2 mm, f = 0.15 mm/rev and N = 1150 r/min.

In this study, Daubechies 5 (db5) wavelet with decomposition level 4 has been selected. Selection of decomposition level depends on the length of the signal (n). In the present work, decomposition level ‘4’ has been considered to reduce computing time and enhance chatter responsiveness. The wavelet which supremely matches the signal has been detected by several trials, and it has been found that db5 is the best option.

Wavelet thresholding rule

Generally, hard or soft thresholding rules have been adopted for denoising purpose. If the wavelet coefficient values are greater than the given threshold level, then hard thresholding function will retain all wavelet coefficients above threshold level and rest are set as zero.²⁹ Hard thresholding is defined as follows

W_{T_H} = {\begin{cases} W | W | \geq T \\ 0 | W | < T \end{cases}

(4)

where W is the noisy wavelet coefficient and T is the threshold.

In soft thresholding, if the wavelet coefficient values are greater than given threshold, then soft thresholding function shrinks the wavelet coefficient and the rest are set as zero.²⁹ Soft thresholding is defined as follows

W_{T_S} = {\begin{cases} W - T W \geq T \\ 0 | W | < T \\ W + T W \leq - T \end{cases}

(5)

However, hard thresholding is based on keep or remove approach; hence, it classifies a true signal as noise and vice versa. However, soft thresholding shrinks the wavelet coefficients. Due to these difficulties, both the rule are unable to denoise the signal accurately and may leave the chatter undetected. In order to overcome the limitations of the aforesaid thresholding methods, an adaptive hybrid thresholding approach has been developed recently²² and is given by

W_{T} = {\begin{cases} W - sgn (W) (1 - ζ) \times T | W | \geq T \\ 0 | W | < T \end{cases}

(6)

Here, $ζ$ is a parameter in the range of (0, 1) and T is the thresholding level given by

T = σ \sqrt{2 \log (n)}

(7)

where n is the length of signal, and $σ$ is the standard deviation of noise.

In the present study, hybrid thresholding rule has been adopted to acquire much smoother results. The detailed coefficients and denoised signals have been shown in Figures 8 –10. Furthermore, a new parameter denoted as CI has been evaluated to quantify the chatter severity.

Calculation of CI and MRR

In the present work, chatter severity has been explored by evaluating a new parameter called CI considering aforesaid denoised signals. Higher the value of CI, more will be the resultant chatter. Thus, CI helps in identifying the severity of chatter. CI is evaluated using the given relation

C I = \sqrt{\frac{1}{n} \sum_{i = 1}^{N} {(x_{i} - μ)}^{2}}

(8)

where CI is the chatter index, n is the length of signal and µ is the mean.

MRR has been obtained by using the following relation

M R R = \frac{W_{i} - W_{f}}{T}

(9)

where W_i is the initial mass of work piece before turning, W_f is the final mass of work piece after turning and T is the time of one turning pass.

The absolute values of CI and MRR obtained at different set of cutting parameters have been listed in Table 3.

Table 3.

Chatter index (CI) at different cutting conditions.

S.no.	Depth of cut (d), mm	Feed rate (f), mm/rev	Spindle speed (N), r/min	CI	MRR, g/s
1	1	0.1	850	1.612	1.36
2	1	0.1	1000	0.412	2.00
3	1	0.1	1150	1.548	1.84
4	1	0.15	850	0.926	2.04
5	1	0.15	1000	0.383	2.4
6	1	0.15	1150	1.365	2.76
7	1	0.2	850	1.241	2.72
8	1	0.2	1000	0.369	3.20
9	1	0.2	1150	1.213	3.69
10	1.5	0.1	850	1.445	2.02
11	1.5	0.1	1000	0.948	2.37
12	1.5	0.1	1150	1.846	2.73
13	1.5	0.15	850	1.263	3.02
14	1.5	0.15	1000	0.779	3.56
15	1.5	0.15	1150	1.613	4.09
16	1.5	0.2	850	1.135	4.03
17	1.5	0.2	1000	0.686	4.74
18	1.5	0.2	1150	1.458	5.46
19	2	0.1	850	1.76	2.65
20	2	0.1	1000	1.965	3.12
21	2	0.1	1150	1.448	3.59
22	2	0.15	850	1.642	3.98
23	2	0.15	1000	1.776	4.68
24	2	0.15	1150	1.758	5.39
25	2	0.2	850	1.511	5.31
26	2	0.2	1000	1.677	5.55
27	2	0.2	1150	1.142	7.18

MRR: metal removal rate.

ANN

Overview and methodology

ANN is a soft computing technique based on the behaviour of neurons in human brain. Architecture of ANN has been shown in Figure 11. ANN consists of three layers: input, hidden and output layers. Each input layer has connecting links (synapse) and are characterized by their own weights. Input and output layers of ANN are defined as node, while the relation between input and output is provided by hidden layer.³⁰

Figure 11.

ANN architecture.

When the input data are given to input layer (I_i), this will be fed to nodes (j) through weights function (W_ji) in hidden layer.³¹ The net input to node ‘j’ is given as follows

{(N E T)}_{j} = \sum_{i} W_{i j} I_{i} - B_{j}

(10)

where B_j is the bias over node j.

The output of the node j is given as follows

{(o / p)}_{j} = \frac{2}{1 + e^{- 2 \times {(N E T)}_{j}}} - 1 for' TANSIG'

(11)

{(o / p)}_{j} = \frac{1}{1 + e^{- {(N E T)}_{j}}} for' LOGSIG'

(12)

{(o / p)}_{j} = {(N E T)}_{j} for' PURELIN'

(13)

Further in feedforward stage, the hidden layer output is fed into nodes in output layer. After that output (O_pn) has been calculated in the output layer, which will be different as target value (T_pn). For that the average system error is calculated by

E_{s} = \frac{1}{2 p} \sum_{p} \sum_{n} {(T_{p n} - O_{p n})}^{2}

(14)

where n is the number of output neurons and p is the output training pattern

This error is then back propagated to the hidden layer from output layer by using gradient search method. After that weights are updated by using the relation

W_{j i} (s + 1) = W_{j i} (s) + γ δ_{p j} O_{p i} + β Δ W_{j i} (s)

(15)

where $s$ is the learning step, $γ$ is the learning rate, $β$ is the momentum constant and $δ_{p j}$ pj is the error term.

In this study, ANN structure as shown in Figure 12 has been used for modelling and predicting CI and MRR in turning process.

Figure 12.

Proposed 3-5-2 ANN architecture.

For ANN training and testing purpose full factorial 27 experimental runs has been considered as shown in Table 3. The numerical value of spindle speed range (850–1150) is too large as compared with depth of cut (1–2) and feed rate (0.1–0.2). Hence, for smooth training and testing purpose, input parameters have been converted into coded form as presented in Table 4. Here, the coded values are in the range of −1 to +1. The transformation of un-coded input variables into coded form has been done by using the relation

\begin{array}{l} d (c o d e d) = \frac{d - 1.5}{0.5}, f (c o d e d) = \\ \frac{f - 0.15}{0.05}, N (c o d e d) = \frac{N - 1000}{150} \end{array}

(16)

Table 4.

Experimental data for ANN training and testing (coded form).

S.no.	Depth of cut (d), mm	Feed rate (f), mm/rev	Spindle speed (N), r/min	Experimental chatter index (CI)	Experimental MRR, g/s
1	−1	−1	−1	1.612	1.36
2	−1	−1	0	0.412	2.00
3	−1	−1	1	1.548	1.84
4	−1	0	−1	0.926	2.04
5	−1	0	0	0.383	2.4
6	−1	0	1	1.365	2.76
7	−1	1	−1	1.241	2.72
8	−1	1	0	0.369	3.20
9	−1	1	1	1.213	3.69
10	0	−1	−1	1.445	2.02
11	0	−1	0	0.948	2.37
12	0	−1	1	1.846	2.73
13	0	0	−1	1.263	3.02
14	0	0	0	0.779	3.56
15	0	0	1	1.613	4.09
16	0	1	−1	1.135	4.03
17	0	1	0	0.686	4.74
18	0	1	1	1.458	5.46
19	1	−1	−1	1.76	2.65
20	1	−1	0	1.965	3.12
21	1	−1	1	1.448	3.59
22	1	0	−1	1.642	3.98
23	1	0	0	1.776	4.68
24	1	0	1	1.758	5.39
25	1	1	−1	1.511	5.31
26	1	1	0	1.677	5.55
27	1	1	1	1.142	7.18

MRR: metal removal rate.

ANN training and testing

ANN training has been done using MATLAB software for 27 experimental values. The training function ‘TRAINLM’ and ‘LEARNGDM’ learning function has been used. Furthermore, three activation functions, ‘TANSIG’, ‘LOGSIG’ and ‘PURELIN’, have been invoked to obtain the desired output. These aforesaid activation functions provide three different sets of desired output values for each response. In ANN testing, these 27 output values corresponding to each activation function has been compared with the experimental outputs and the error in prediction is evaluated by using the equation

e_{a i} = (\frac{V_{E} - V_{P}}{V_{E}}) \times 100 %

(17)

where e_ai represents average individual error, and V_E and V_P show the experimental and predicted values, respectively.

Selection of suitable activation function

After completion of training and testing, the ANN model has been developed corresponding to these three different activation functions. Furthermore, a comparative study has been done to decide the best suited activation function for the prediction of chatter and MRR.

ANN ‘TANSIG’ activation function

Here, the training and testing samples are 23 (for CI and MRR) and 4 (for CI and MRR), respectively. Training data are different from that of the testing data. In the network considering TANSIG activation function, predicted values and error in output have been shown in Table 5.

Table 5.

Error of ANN ‘TANSIG’ activation functions.

S.no.	d	F	N	Experimental CI	Predicted CI	Error %	Experimental MRR	Predicted MRR	Error %
1	−1	−1	−1	1.612	1.578	2.1	1.36	1.40	−2.9
2	−1	−1	0	0.412	0.424	−3.0	2.00	1.99	0.5
3	−1	−1	1	1.548	1.599	−3.3	1.84	1.90	−3.3
4	−1	0	−1	0.926	0.956	−3.3	2.04	2.01	1.6
5	−1	0	0	0.383	0.363	5.3	2.4	2.35	2.1
6	−1	0	1	1.365	1.409	−3.3	2.76	2.65	3.9
7	−1	1	−1	1.241	1.169	5.8	2.72	2.77	−1.9
8	−1	1	0	0.369	0.358	3.1	3.20	3.33	−3.9
9	−1	1	1	1.213	1.171	3.5	3.69	3.70	−0.2
10	0	−1	−1	1.445	1.390	3.8	2.02	2.11	−4.2
11	0	−1	0	0.948	0.898	5.3	2.37	2.40	−1.3
12	0	−1	1	1.846	1.779	3.6	2.73	2.64	3.4
13	0	0	−1	1.263	1.322	−4.6	3.02	2.93	2.9
14	0	0	0	0.779	0.752	3.4	3.56	3.60	−1.1
15	0	0	1	1.613	1.570	2.6	4.09	4.15	−1.5
16	0	1	−1	1.135	1.170	−3.0	4.03	4.02	0.3
17	0	1	0	0.686	0.710	−3.5	4.74	4.62	2.5
18	0	1	1	1.458	1.409	3.4	5.46	5.55	−1.6
19	1	−1	−1	1.76	1.700	3.4	2.65	2.70	−1.9
20	1	−1	0	1.965	1.898	3.4	3.12	3.17	−1.6
21	1	−1	1	1.448	1.500	−3.6	3.59	3.64	−1.3
22	1	0	−1	1.642	1.577	3.9	3.98	4.01	−0.7
23	1	0	0	1.776	1.708	3.8	4.68	4.71	−0.7
24	1	0	1	1.758	1.691	3.8	5.39	5.25	2.6
25	1	1	−1	1.511	1.468	2.8	5.31	5.12	3.6
26	1	1	0	1.677	1.620	3.4	5.55	5.63	−1.5
27	1	1	1	1.142	1.171	−2.5	7.18	6.95	3.2

ANN: artificial neural network; CI: chatter index; MRR: metal removal rate.

Regression plot for CI has been shown in Figure 13, which indicates the performance of developed model. Here, the ‘R’ value of training data is 0.99693 and for testing data is 0.9576. Here, the ‘R’ value of validation is 0.99302. ‘R’ values represent the proximity of the predicted values with the target values.

Figure 13.

Regression plot considering TANSIG activation function.

ANN ‘LOGSIG’ activation function

Here, LOGSIG activation function has been considered for training the network to get the desired output, weight functions and regression plots for CI and MRR. The predicted values and error in output have been shown in Table 6.

Table 6.

Error of ANN ‘LOGSIG’ activation functions.

S.no.	d	F	N	Experimental CI	Predicted CI	Error %	Experimental MRR	Predicted MRR	Error %
1	−1	−1	−1	1.612	1.555	3.5	1.36	1.40	−3.2
2	−1	−1	0	0.412	0.431	−4.6	2.00	2.00	0.2
3	−1	−1	1	1.548	1.484	4.1	1.84	1.73	5.7
4	−1	0	−1	0.926	0.970	−4.8	2.04	1.99	2.5
5	−1	0	0	0.383	0.398	−3.9	2.4	2.46	−2.5
6	−1	0	1	1.365	1.333	2.4	2.76	2.75	0.2
7	−1	1	−1	1.241	1.136	8.5	2.72	2.76	−1.4
8	−1	1	0	0.369	0.382	−3.5	3.20	3.29	−2.8
9	−1	1	1	1.213	1.216	−0.3	3.69	3.65	1.0
10	0	−1	−1	1.445	1.387	4.0	2.02	1.98	2.0
11	0	−1	0	0.948	0.989	−4.3	2.37	2.43	−2.5
12	0	−1	1	1.846	1.854	−0.4	2.73	2.77	−1.6
13	0	0	−1	1.263	1.212	4.0	3.02	3.05	−1.0
14	0	0	0	0.779	0.812	−4.2	3.56	3.62	−1.8
15	0	0	1	1.613	1.630	−1.1	4.09	4.04	1.2
16	0	1	−1	1.135	1.052	7.3	4.03	3.96	1.7
17	0	1	0	0.686	0.711	−3.6	4.74	4.59	3.2
18	0	1	1	1.458	1.342	8.0	5.46	5.30	2.9
19	1	−1	−1	1.76	1.839	−4.5	2.65	2.67	−0.9
20	1	−1	0	1.965	1.809	7.9	3.12	3.06	1.8
21	1	−1	1	1.448	1.489	−2.8	3.59	3.61	−0.4
22	1	0	−1	1.642	1.671	−1.8	3.98	3.97	0.3
23	1	0	0	1.776	1.761	0.8	4.68	4.63	1.0
24	1	0	1	1.758	1.705	3.0	5.39	5.59	−3.7
25	1	1	−1	1.511	1.458	3.5	5.31	5.12	3.6
26	1	1	0	1.677	1.610	4.0	5.55	5.69	−2.5
27	1	1	1	1.142	1.091	4.4	7.18	7.05	1.8

ANN: artificial neural network; CI: chatter index; MRR: metal removal rate.

Figure 14 shows that the ‘R’ value of training data is 0.99608 and for testing data is 0.97194. Here, the ‘R’ value of validation is 0.9857 and is slightly closer to the corresponding values obtained using ‘TANSIG’ activation function.

Figure 14.

Regression plot considering LOGSIG activation function.

ANN ‘PURELIN’ activation function

Here, third activation function, PURELIN, has been considered for training the network with 27 experimental runs as shown in Table 4. Predicted values and error in output are shown in Table 7.

Table 7.

Error of ANN ‘PURELIN’ activation functions.

S.no.	d	F	N	Experimental CI	Predicted CI	Error %	Experimental MRR	Predicted MRR	Error %
1	−1	−1	−1	1.612	0.866	46.3	1.36	1.86	−36.6
2	−1	−1	0	0.412	0.598	−45.1	2.00	1.98	1.0
3	−1	−1	1	1.548	1.140	26.3	1.84	2.13	−15.5
4	−1	0	−1	0.926	0.805	13.0	2.04	2.31	−13.3
5	−1	0	0	0.383	0.530	−38.3	2.4	2.52	−4.9
6	−1	0	1	1.365	1.068	21.8	2.76	2.76	0.2
7	−1	1	−1	1.241	0.750	39.6	2.72	3.04	−11.9
8	−1	1	0	0.369	0.565	−53.1	3.20	3.34	−4.5
9	−1	1	1	1.213	0.997	17.8	3.69	3.67	0.6
10	0	−1	−1	1.445	1.232	14.7	2.02	2.57	−27.4
11	0	−1	0	0.948	1.373	−44.8	2.37	2.82	−19.0
12	0	−1	1	1.846	1.501	18.7	2.73	3.10	−13.5
13	0	0	−1	1.263	1.159	8.2	3.02	3.42	−13.4
14	0	0	0	0.779	1.303	−67.3	3.56	3.75	−5.4
15	0	0	1	1.613	1.438	10.8	4.09	4.10	−0.2
16	0	1	−1	1.135	1.087	4.3	4.03	4.47	−10.8
17	0	1	0	0.686	1.231	−79.5	4.74	4.81	−1.5
18	0	1	1	1.458	1.372	5.9	5.46	5.14	5.9
19	1	−1	−1	1.76	1.573	10.6	2.65	3.84	−44.9
20	1	−1	0	1.965	1.670	15.0	3.12	4.19	−34.2
21	1	−1	1	1.448	1.748	−20.7	3.59	4.53	−26.3
22	1	0	−1	1.642	1.516	7.7	3.98	4.90	−23.0
23	1	0	0	1.776	1.624	8.6	4.68	5.22	−11.5
24	1	0	1	1.758	1.711	2.7	5.39	5.52	−2.4
25	1	1	−1	1.511	1.455	3.7	5.31	5.80	−9.3
26	1	1	0	1.677	1.572	6.2	5.55	6.04	−8.8
27	1	1	1	1.142	1.670	−46.2	7.18	6.24	13.0

ANN: artificial neural network; CI: chatter index; MRR: metal removal rate.

From Figure 15, it is evident that ‘R’ value of training data is 0.9827 and for testing data is 0.93751. Here, ‘R’ value of entire data has been found to be 0.98024 which is less as compared with the other two activation functions.

Figure 15.

Regression plot considering PURELIN activation function.

Comparative study of ANN activation functions

A comparison of these three activation function has been presented in Table 8. From Table 8, it is evident that the percentage error considering ‘TANSIG’ function is very less as compared with the other two. Hence, ‘TANSIG’ function is best suited for the prediction of chatter and MRR simultaneously.

Table 8.

Comparison of three activation functions.

S.no.	d	f	N	TANSIG (%Error)		LOGSIG (%Error)		PURELIN (%Error)
				CI	MRR	CI	MRR	CI	MRR
1	−1	−1	−1	2.1	−2.9	3.5	−3.2	46.3	−36.6
2	−1	−1	0	−3.0	0.5	−4.6	0.2	−45.1	1.0
3	−1	−1	1	−3.3	−3.3	4.1	5.7	26.3	−15.5
4	−1	0	−1	−3.3	1.6	−4.8	2.5	13.0	−13.3
5	−1	0	0	5.3	2.1	−3.9	−2.5	−38.3	−4.9
6	−1	0	1	−3.3	3.9	2.4	0.2	21.8	0.2
7	−1	1	−1	5.8	−1.9	8.5	−1.4	39.6	−11.9
8	−1	1	0	3.1	−3.9	−3.5	−2.8	−53.1	−4.5
9	−1	1	1	3.5	−0.2	−0.3	1.0	17.8	0.6
10	0	−1	−1	3.8	−4.2	4.0	2.0	14.7	−27.4
11	0	−1	0	5.3	−1.3	−4.3	−2.5	−44.8	−19.0
12	0	−1	1	3.6	3.4	−0.4	−1.6	18.7	−13.5
13	0	0	−1	−4.6	2.9	4.0	−1.0	8.2	−13.4
14	0	0	0	3.4	−1.1	−4.2	−1.8	−67.3	−5.4
15	0	0	1	2.6	−1.5	−1.1	1.2	10.8	−0.2
16	0	1	−1	−3.0	0.3	7.3	1.7	4.3	−10.8
17	0	1	0	−3.5	2.5	−3.6	3.2	−79.5	−1.5
18	0	1	1	3.4	−1.6	8.0	2.9	5.9	5.9
19	1	−1	−1	3.4	−1.9	−4.5	−0.9	10.6	−44.9
20	1	−1	0	3.4	−1.6	7.9	1.8	15.0	−34.2
21	1	−1	1	−3.6	−1.3	−2.8	−0.4	−20.7	−26.3
22	1	0	−1	3.9	−0.7	−1.8	0.3	7.7	−23.0
23	1	0	0	3.8	−0.7	0.8	1.0	8.6	−11.5
24	1	0	1	3.8	2.6	3.0	−3.7	2.7	−2.4
25	1	1	−1	2.8	3.6	3.5	3.6	3.7	−9.3
26	1	1	0	3.4	−1.5	4.0	−2.5	6.2	−8.8
27	1	1	1	−2.5	3.2	4.4	1.8	−46.2	13.0

CI: chatter index; MRR: metal removal rate.

The average percentage error for three activation functions has been shown in Table 9. From Table 9, it quite clear that ‘TANSIG’ is the best suitable activation function.

Table 9.

Average percentage error of different activation function.

S.no.	TANSIG (%Error)		LOGSIG (%Error)		PURELIN (%Error)
S.no.	CI	MRR	CI	MRR	CI	MRR
1	3.6	2.1	3.9	2.0	25.1	13.3

CI: chatter index; MRR: metal removal rate.

From the above analysis, it is inferred that the proposed ANN model could be successfully used to predict CI and MRR in turning process.

Furthermore, in order to ascertain the relative influence of cutting parameters on chatter severity, three-dimensional (3D) and contour plots have been drawn as shown in Figures 16 –18. Figure 16 shows the effect of depth of cut and feed rate on CI. Here, very little variation in feed rate line shows the value of CI is almost same at all three range of feed rate, while variation in depth of cut line is high and has more effect on CI.

Figure 16.

Effect of d and f on CI (a) surface plot and (b) contour plot.

Figure 17.

Effect of d and N on CI (a) surface plot and (b) contour plot.

Figure 18.

Effect of f and N on CI (a) surface plot and (b) contour plot.

Furthermore, Figure 17 presents the effect of depth of cut and spindle speed on CI. Here, variation in spindle speed line is higher but comparatively low as compared with depth of cut line; it means proper combination of both parameters gives better results for CI.

Figure 18 shows the effect of feed rate and spindle speed on CI at holding depth of cut in middle range. Here, CI is higher at the lower value of feed rate, while lower value of CI is obtained at middle range of both feed rate and spindle speed.

From the above discussion, it has been inferred that the depth of cut is most influencing parameter and the reason is, with the increase in depth of cut keeping other parameters constant, the associated radial force increases prominently as compared with the other cutting forces. This increase in radial force results in increased waviness and unevenness along the surface of the work piece This waviness leads to further time delay between the two corresponding turning passes along the job surface. So, ultimately increase in depth of cut results in pronounced tool chatter as compared with the rest of the cutting parameters.

Moreover, 3D and contour graphs have been plotted to determine the influence of cutting parameters on MRR. Figures 19 –21 show that the minimum MRR is obtained at lower values of all three parameters and vice versa. Furthermore, variation of spindle speed line is higher than feed rate line, while variation of depth of cut line is more as compared with other two and has more effect on CI.

Figure 19.

Effect of d and f on MRR (a) surface plot and (b) contour plot.

Figure 20.

Effect of d and N on MRR (a) surface plot and (b) contour plot.

Figure 21.

Effect of f and N on MRR (a) surface plot and (b) contour plot.

Moreover, above discussion indicates the monotonic nature of MRR. Above discussion also concludes that the depth of cut and feed rate are the most influencing parameter in MRR prediction. The reason is, with the increase in depth of cut keeping other parameters constant, the cutting tool penetrates the work material more resulting in increased material removal. Similarly, with the increase in feed rate, the removal of material in each revolution is increased. So, ultimately increase in depth of cut and feed rate results in enhanced MRR.

Optimal CI and MRR using MOGA

Genetic algorithm (GA) is a robust optimization technique based on evolutionary technique and natural selection. It is used to find out the best solution for optimization problem and can be applicable for both linear and non-linear optimization problem. Based on evolutionary biology, GA involves selection, crossover and mutation operators. First, it is required to select randomly two couples from the set of various parents to generate the new population. In each couple, either both are equally good or one feasible candidate is better than one. Second, these two best couples are crossed by two point crossover operators. Two point crossover operators provide better diversity preservation. Finally, mutation is used to find some random changes in code string and introduce new feature in population.

In turning process, CI reflects the surface quality, while MRR improves the productivity. Therefore, it is required to minimize CI for better surface quality and maximize MRR to increase productivity. Thus, concerned problem is a multi-objective optimization problem. Moreover, single response optimization technique shows the unacceptable result with respect to other response.

From the previous analysis, it is concluded that ‘TANSIG’ activation function is best for training the ANN model for CI and MRR. So, ANN modelling has been done using the aforesaid activation function. The resulting mathematical model has been considered for determining the stable cutting zone using MOGA technique. Algorithm for the development of mathematical model has been outlined as follows: final optimal weights obtained between the input and hidden layers; hidden and output layers have been listed in Tables 10 and 11, respectively.

Table 10.

The weight values between input and hidden layer for ‘TANSIG’.

No. of neurons	Wdk	Wfk	WNk	WBok
1	0.49625	−1.6533	1.3474	−3.5345
2	0.51406	−0.56509	3.3339	−2.0965
3	−2.0988	−0.40426	3.0672	0.27722
4	0.017474	0.47449	−1.3053	0.96377
5	0.35629	0.32362	2.6124	2.5541

Table 11.

The weights values between hidden layer and output layer for ‘TANSIG’.

No. of neurons	Wck	Wmk	WBo1	WBo2
1	−0.80349	0.61283	0.091558	−1.0931
2	0.029973	2.3237
3	−1.189	−0.39007
4	−1.8094	3.2819
5	0.19818	1.1548

The mathematical model for normalized CI and MRR has been obtained as follows

y (C I) = \frac{2}{[1 + \exp (- 2 \times n C I]} - 1

(18)

y (M R R) = \frac{2}{[1 + \exp (- 2 \times n M R R]} - 1

(19)

where nCI and nMRR are the neuron outputs of the output layer and has been evaluated using the weights between the hidden and output layer as listed in Table 11 and the following equations

\begin{array}{l} n C I = - W_{C 1} \times A + W_{C 2} \times B - W_{C 3} \times \\ C - W_{C 4} \times D + W_{C 5} \times E + W B_{01} \end{array}

(20a)

\begin{array}{l} n M R R = W_{m 1} \times A + W_{m 2} \times B - W_{m 3} \times \\ C + W_{m 4} \times D + W_{m 5} \times E - W B_{02} \end{array}

(20b)

where A, B, C, D and E are the outputs of the hidden layer and are given by

\begin{array}{l} A = \frac{2}{[1 + \exp (- 2 \times n a)]} - 1, \\ B = \frac{2}{[1 + \exp (- 2 \times n b)]} - 1, \\ C = \frac{2}{[1 + \exp (- 2 \times n c)]} - 1, \\ D = \frac{2}{[1 + \exp (- 2 \times n d)]} - 1 \\ E = \frac{2}{[1 + \exp (- 2 \times n e)]} - 1 \end{array}

(21)

In equation (21), na, nb, nc, nd and ne are the neuron outputs of the hidden layer and has been evaluated using the weights between the input and hidden layer as listed in Table 10 and the following equations

n a = W d_{1} - W f_{1} + W N_{1} - W B_{01}

(22a)

n b = W d_{2} - W f_{2} + W N_{2} - W B_{02}

(22b)

n c = - W d_{3} - W f_{3} + W N_{3} - W B_{03}

(22c)

n d = W d_{4} + W f_{4} - W N_{4} + W B_{04}

(22d)

n e = W d_{5} + W f_{5} + W N_{5} + W B_{05}

(22e)

Here, equations (18) and (19) have been used as an objective function. The set of optimization parameters considered is as follows: population type, double vector; population size, 250; no. of iteration, 600; crossover probability, 0.8; and mutation probability, 0.8. In MOGA, it is not possible to achieve a single optimal solution. After execution of 600 generations, the MOGA resulted in 89 optimal Pareto solutions. Some of the selected Pareto-optimal solution has been listed in Table 12.

Table 12.

Selected MOGA optimal solutions.

S.no.	Control factors			Responses
S.no.	d	f	N	CI	MRR
1	1.0	0.20	980	0.596	3.25
2	2.0	0.20	1139	1.538	6.90
3	2.0	0.20	1087	1.300	6.50
4	1.9	0.20	1130	1.441	6.63
5	1.6	0.20	1091	1.047	5.36
6	2.0	0.20	1126	1.473	6.81
7	1.2	0.20	996	0.656	3.75
8	1.4	0.20	1033	0.824	4.50
9	1.4	0.20	1010	0.739	4.28
10	1.2	0.20	1055	0.749	4.00
11	1.6	0.20	1027	0.914	5.15
12	1.0	0.20	1035	0.644	3.37
13	1.9	0.20	1086	1.207	6.15
14	1.7	0.20	1091	1.119	5.64
15	1.5	0.20	1067	0.919	4.98
16	2.0	0.20	1131	1.489	6.78
17	1.7	0.20	1057	1.008	5.50
18	1.7	0.20	1056	0.980	5.37
19	2.0	0.20	1133	1.510	6.86
20	1.9	0.20	1121	1.384	6.52
21	1.2	0.20	1041	0.733	4.06
22	1.9	0.20	1138	1.496	6.74
23	2.0	0.20	1139	1.532	6.86
24	2.0	0.20	1064	1.252	6.41
25	2.0	0.20	1121	1.418	6.66
26	1.7	0.20	1040	1.014	5.57
27	2.0	0.20	1094	1.338	6.60
28	1.5	0.20	1083	0.980	5.11
29	1.0	0.20	1075	0.759	3.50
30	1.5	0.19	1019	0.840	4.63
31	1.4	0.20	1004	0.761	4.40
32	2.0	0.20	1150	1.599	6.97
33	2.0	0.20	1141	1.548	6.91
34	1.2	0.20	1029	0.702	3.97
35	1.9	0.20	1108	1.348	6.49
36	1.3	0.20	989	0.725	4.06
37	1.8	0.20	1027	1.070	5.77
38	2.0	0.20	1103	1.379	6.68
39	1.6	0.20	1044	0.946	5.27
40	1.5	0.20	1024	0.816	4.69

MOGA: multi-objective genetic algorithm; CI: chatter index; MRR: metal removal rate.

However, depending on the priority of the response (CI), best solution can be decided, for example, at d = 1 mm, f = 0.20 mm/rev and N = 980 r/min; the value of CI is 0.596 and MRR is 3.25 g/s as shown in Table 12 (S.no. 1). But our aim is to minimize CI and maximize MRR simultaneously. Pareto optimal fronts given by MOGA with detailed spread have been shown in Figures 22 and 23 in coded form. Figure 22 represents the value of CI plotted against MRR for the same value of independent cutting variables. Here, a small average distance between individuals in Pareto front shows that the solutions are evenly distributed.

Figure 22.

Pareto front.

Figure 23.

Average distance between individuals.

Figure 23 shows the average distance between individuals and is used to measure the diversity of population. Here, the averages distance between individual shows the good measure of the diversity of population.

Prediction of stable cutting zones using MOGA

The aim of present multi-objective optimization is to be finding the optimum cutting parameters for which MRR is maximized along with the minimum chatter of turning tool. MOGA results for both CI and MRR with respect to the cutting parameters is plotted on a contour and 3D surface graphs.

It has been observed that CI is less for the values of depth of cut and feed rate as shown in Figure 24 in orange colour for which CI is less than 0.60. Similarly, stable cutting zone has been evaluated for other parameters as shown in Figures 25 and 26.

Figure 24.

Stable cutting zone with respect to depth of cut and feed rate.

Figure 25.

Stable cutting zone with respect to spindle speed and depth of cut.

Figure 26.

Stable cutting zone with respect to spindle speed and feed rate.

Furthermore, same procedure has been adopted to find the optimal values of cutting parameters for which MRR is maximized as shown in Figures 27 –29. So, aforesaid developed technique is quite suitable in predicting the stable cutting zones during turning operation with maximized MRR.

Figure 27.

Stable cutting with maximized MRR with respect to depth of cut and feed rate.

Figure 28.

Stable cutting with maximized MRR with respect to spindle speed and depth of cut.

Figure 29.

Stable cutting with maximized MRR with respect to spindle speed and feed rate.

Conclusion

This article presents a signal processing technique, ANN model and MOGA analysis for predicting chatter and MRR in turning process. A full factorial experimental runs as listed in Table 4 have been considered to acquire the raw chatter signals and wavelet denoising technique has been adopted to denoise it. A new parameter called ‘CI’ has been evaluated and MRR has been obtained. Moreover, developed ANN model has been used as an objective function for MOGA analysis. From the above analysis, following conclusions have been drawn:

Raw chatter signals contaminated with noise are well denoised by wavelet denoising technique.

ANN model is quite suitable to predict chatter and MRR in turning process.

As compared with other two, ‘TANSIG’ activation functions results in least percentage deviation between the experimental and predicted values (3.6% for CI and 2.1% for MRR, simultaneously).

From the above analysis, it is evident that the depth of cut mostly influences the CI and MRR.

3D graphs show that the dependence of CI on cutting parameters is non-monotonic, while dependence of MRR on cutting parameters is monotonic in nature.

Multi-objective genetic algorithm is well suited for the prediction of stable cutting zones along with maximized metal removal rate.

In the present work, a methodology has been suggested to select the suitable range of cutting parameters that will result in maximum MRR along with minimum chatter. This technique can serve as guidelines for the engineers in machining industries to select the suitable cutting parameters in turning process. It will result in increased productivity and better surface finished products.

Footnotes

Acknowledgements

The authors gratefully acknowledge the Mechanical Engineering Department, IIT Indore, India, for their help in conducting experiments.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship and/or publication of this article.

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