1 Introduction

Sustainable manufacturing can bring economic and environmental benefits to manufacturing industry as well as human society [1]. To promote sustainable manufacturing, it is a complex and emerging problem to reduce the energy consumption (EC) for machining operations of machine tools [2]. Many effective techniques were employed to solve this problem [3], including energy-efficient design, process planning, and scheduling [4]. Among these techniques, cutting parameters optimisation is selected in this research for its effectiveness [5]. The investigation showed that 6–40% of EC was effectively reduced when selecting the appropriate schemes of cutting parameters at the process planning stage [6].

To realise the energy-efficient cutting parameters optimisation, an important prerequisite is to accurately develop the EC model for machining operations of machine tools [7]. Specifically, Zhou et al. [8] reviewed more than 20 EC models and suggested that the accuracy of existing models could be improved by the correlation analysis of machining conditions. To cope with uncertainties in machining conditions, Bhinge et al. [9] developed a data-driven EC model using Gaussian process regression and machine learning [10], and the accuracy of the proposed model was over 94%. To be more accurate, Shi et al. [11] assumed the additional power of the machine tool caused by metal cutting to be proportional to cutting power at the tool-tip, and the accuracy achieved over 97%. In these references, the optimisation techniques such as mathematic algorithms were not provided to minimise the EC. To bridge this knowledge gap, many related optimisation studies have been conducted for various machining operations [12]. For the external turning operations, the turning parameters were optimised to minimise the EC for cutting the 7075 aluminium alloy based on the response surface methodology [13], and the EC was reduced by 13.55%. With the data mining techniques, a two-stage knowledge-driven method was proposed to realise the energy-related optimisation of turning parameters for multiple machining configurations [14]. For the face milling operations, the effect of milling parameters on the EC was analysed, and an experiment approach was provided for minimising the EC for AISI 1050 carbon steel [15]. To further improve the energy efficiency of milling operations, the embodied EC of materials was considered [12], and the specific EC was reduced by 19.59% for a case. Besides, the grinding, drilling, and threading parameters were optimised to reduce the EC for grinding [16], drilling [17], and threading [18] operations, respectively.

The aforementioned research only focuses on the machining operations whose material removal rate (MRR) is constant [19]. Taking external turning operations as example, all cutting parameters including cutting speed, feed rate, and cutting depth remain unchanged during material-cutting [20]. Therefore, the MRR and cutting power are constant. In actual production, the machining operations whose MRR is variable are widely used, such as end face turning, grooving, and chamfering. As shown in Figs. 1 and 2, taking end face turning as example, the cutting speed and depth change with time during material-cutting, thereby leading to the dynamic change of MRR and cutting power. When the MRR is variable, the power characteristic is complicated. To the best of the authors’ knowledge, the cutting parameters optimisation has not been conducted for the EC with variable MRR. An investigation suggested that the end face turning consumes considerable energy when machining a part [21]. In this study, single-pass end face turning is focused. The aim is to minimise the end face turning energy consumption (EFTEC) by optimising the cutting parameters.

Fig. 1
figure 1

End face turning operations for part A

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Fig. 2
figure 2

Power curves of the machine tool based on two schemes: a 720 rpm and 0.25 mm/r; b 1080 rpm and 0.25 mm/r. The forward slashes areas denote the EC for spindle speed changes, the blue grids areas denote the EC for feeding activities, and the red nets areas denote the material-cutting EC

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The first step towards solving the optimisation problem is to develop the EFTEC model. The EFTEC is divided to three portions: the material-cutting EC, the EC for feeding activities, and the EC for spindle speed changes [22]. The material-cutting EC represents the additional EC of the machine tool for removing the materials to get the feature of end face [23]. The EC for feeding activities represents the EC of the machine tool for completing the tool path required by the end face turning without material removal [24]. The EC for spindle speed changes represents the EC of the machine tool for accelerating the spindle from 0 to required spindle rotation speed (SRS) and decelerating the spindle from required SRS to 0 [25]. Generally, the EC for spindle speed changes accounts for 5% of the total machining EC [25], but it has not received attention in the existing cutting parameters optimisation. The material-cutting EC model for end face turning was developed by Jia et al. [21] with fully reflecting the dynamic changing cutting power characteristics. The experiment was conducted to acquire the fitting coefficients and exponents [21]. The case study showed that the accuracy of the proposed model is 95.0%. The accurate EC models for feeding activities and spindle speed changes were built by Hu et al. [25] to solve the operation sequencing problem.

Although the aforementioned three models can be referred to integrate the EFTEC model, the following important gaps and insufficiencies have motivated this study. (1) The operation sequencing problem is discrete. However, our cutting parameters optimisation problem is continuous at intervals. Thus, the existing EC models for feeding activities and spindle speed changes should be modified to be suitable for our problem. (2) The algorithm for optimising the material-cutting EC of end face turning has not been provided by Jia et al. [21], which restricts the energy reduction. (3) The optimisation of material-cutting EC can result in the increase of EC for feeding activities and spindle speed changes [26], and the total EFTEC may increase. Thus, the trade-off among three portions of EC should be made to optimise the total EFTEC. (4) The effect of the design parameter (part diameter) on the selection of cutting parameters for minimising the EFTEC has not been considered. In sum, it is novel to optimise the EFTEC with variable MRR considering the spindle speed changes, and the proposed solutions are the original contributions.

This paper aims to analyse the conflicts among the material-cutting EC, the EC for feeding activities, and the EC for spindle speed changes, and to develop the integrated EFTEC model based on the cutting parameters. Especially, the design parameter of the diameter is included in the EFTEC model. The required cutting depth is a constant for single-pass end face turning because the initial and finished lengths of the part have been given. Further, the optimisation method is used to search for the optimal cutting parameters that lead to the minimum EFTEC. The type of our EFTEC optimisation problem is continuous at intervals. Yusup et al. [27] reviewed five popular heuristic algorithms including simulated annealing (SA), genetic algorithm [28], particle swarm optimisation, ant colony optimisation, and artificial bee colony for solving this type of problem from 2007 to 2011, and validated that SA performed better than other algorithms in terms of the solution quality [29]. Based on experiments and tests, Hu et al. [30] verified that SA can obtain the global optimum in a short computation time when the SRS and feed rate are continuous at intervals. Thus, SA is selected and is compared with the enumeration method to verify its performance about solution quality and computation speed for solving the EFTEC optimisation problem.

This paper is further organised as follows. Section 2 investigates the effect of cutting parameters on the EFTEC. In Sect. 3, the EFTEC model is developed. Section 4 describes the application of SA for optimising the EFTEC. Section 5 shows a case study to demonstrate the EFTEC model and optimisation methods. Section 6 analyses and discusses the optimisation results. In Sect. 7, the conclusions and future research are given.

2 Problem Description

The effect of the change of cutting parameters on the EFTEC is investigated. For example, part A is machined by a lathe, as shown in Fig. 1. This explains the material-cutting EC, the EC for feeding activities, and the EC for spindle speed changes, and the conflict among them when changing the cutting parameters. The required cutting depth is \(d\). To finish part A, two schemes of cutting parameters including SRS and feed rate can be employed: (a) 720 rpm and 0.25 mm/r; (b) 1080 rpm and 0.25 mm/r. Dry cutting is adopted. In Fig. 1a, the red arrowed lines represent the tool paths in end face turning. The acceleration and deceleration of the spindle are represented by and , respectively. According to aforementioned two schemes, power curves of the lathe are illustrated in Fig. 2. The power curves are drawn based on the simulation module developed by He et al. [31].

Firstly, the machining activities are identified and classified. In single-pass end face turning, there are one material-cutting activity, three feeding activities (\(A_{1}\), \(A_{2}\), and \(A_{3}\)), and two spindle speed changes (\(C_{1}\) and \(C_{2}\)), as shown in Fig. 1a. The feeding modes for the first, second, and third activities are rapid, normal, and rapid, respectively. The SRS for them is the same as that of material-cutting activity. The cutting parameters can affect the EC for all of these activities [32]. For example, the EC for both \(C_{1}\) and \(C_{2}\) will increase when increasing the SRS from 720 rpm to 1080 rpm. In Fig. 2, the forward slashes areas denote the EC for spindle speed changes, and the blue grids areas denote the EC for feeding activities, and the red nets areas denote the material-cutting EC. The sizes of these areas between Fig. 2a and b are compared. It shows that changing schemes of cutting parameters can lead to different values of the material-cutting EC, the EC for feeding activities, and the EC for spindle speed changes.

The possible conflicts among three portions of EC are explained. The SRS of the first scheme (720 rpm) is slower than that of the second scheme (1080 rpm). Thus, the material-cutting power for the first scheme is smaller while the material-cutting time is longer, as shown in Fig. 2. The material-cutting EC model is required to calculate and compare the values of material-cutting EC between two schemes. As shown in Fig. 1b, the material-cutting process is divided into cutter entering stage (①⟶②) and fully cutting stage (②⟶③) with variable MRR, which increases the material-cutting EC modelling complexity. Since the low SRS requires the less spindle rotation power, the EC of \(A_{1}\) and \(A_{3}\) for the first scheme is smaller than that of the second scheme, as shown in Fig. 2. The EC of \(C_{1}\) and \(C_{2}\) for the first scheme is also smaller because the low SRS consumes the less EC [33]. However, the EC of \(A_{2}\) for the first scheme is greater than that of the second scheme because the normal feeding time is 50% [(1080-720)/720] longer with consuming more basic energy of the machine tool [34]. Thus, the trade-off between the EC for feeding activities and the EC for spindle speed changes should be made when selecting the SRS. If the EC for spindle speed changes was neglected, the results would be skewed. This case suggests the requirement of developing an integrated EFTEC model and optimising the cutting parameters to achieve the minimum EFTEC. Especially, more feeding time reduction for \(A_{2}\) can benefit from the second scheme if the diameter of part (\(D_{0}\)) is increased. The feeding time affects the EFTEC. Therefore, the design parameter of diameter is included in the EFTEC model.

3 Modelling

3.1 Objective Function

The EFTEC model for single-pass end face turning is developed. As shown in Fig. 2, the EFTEC is comprised of the material-cutting EC, the EC for feeding activities, and the EC for spindle speed changes. Hence, the objective function of the model is expressed as:

$${\text{Minimise }}E_{\text{eft}} = E_{\text{mc}} + E_{\text{fa}} + E_{\text{sc}}$$
(1)

where \(E_{\text{eft}}\) is the total EFTEC for single-pass end face turning [J], \(E_{\text{mc}}\) is the material-cutting EC [J], \(E_{\text{fa}}\) is the EC for feeding activities [J], and \(E_{\text{sc}}\) is the EC for spindle speed changes [J]. They are modelled as follows.

  1. (1)

    Material-cutting EC (\(E_{\text{mc}}\))

The material-cutting power for end face turning varies with time, as shown in Fig. 2. Thus, \(E_{\text{mc}}\) is modelled as:

$$E_{{{\text{mc}}}} = \int\limits_{0}^{{t_{{{\text{mc}}}} }} {P_{{{\text{mc}}}} \left( t \right)} dt$$
(2)

where \(t_{\text{mc}}\) is total material-cutting duration for the end face turning [s], \(t\) is the time variable [s], and \(P_{\text{mc}} \left( t \right)\) is function of material-cutting power against time [W]. The material-cutting duration is divided into cutter entering duration and fully cutting duration, as shown in Fig. 1b. According to Jia et al. [21], \(t_{\text{mc}}\) is modelled as:

$$\left\{ {\begin{array}{*{20}l} {t_{\text{mc}} = t_{ce} + t_{fc} } \\ {t_{\text{ce}} = \frac{d \times \cot K}{n \times f/60}} \\ {t_{\text{fc}} = \frac{{D_{0} /2 - d \times \cot K}}{n \times f/60}} \\ \end{array} } \right.$$
(3)

where \(t_{\text{ce}}\) is cutter entering duration [s], \(t_{\text{fc}}\) is fully cutting duration [s], \(d\) is required cutting depth [mm], \(K\) is main angle of the cutter [°], \(n\) is SRS in material-cutting [rpm], \(f\) is feed rate [mm/r], and \(D_{0}\) is diameter of the part [mm].

According to Jia et al. [21], a multiple regression equation that has an accuracy of 95% is employed to model the \(P_{mc} \left( t \right)\), as follows:

$$P_{\text{mc}} \left( t \right) = C_{M} \cdot v\left( t \right)^{{w_{M} }} \cdot f^{{y_{M} }} \cdot d\left( t \right)^{{x_{M} }}$$
(4)
$$v\left( t \right) = \frac{\pi \times n}{1000}\left( {D_{0} - \frac{2 \times f \times n}{60} \times t} \right)$$
(5)
$$d\left( t \right) = \left\{ {\begin{array}{*{20}l} {d \times t/t_{\text{ce}} , } \hfill & {0 \le t < t_{\text{ce}} } \hfill \\ {d,} \hfill & { t_{\text{ce}} \le t < t_{\text{mc}} } \hfill \\ \end{array} } \right.$$
(6)

where \(C_{M}\) is coefficient in the model of material-cutting power, \(v\left( t \right)\) is function of cutting speed against time [m/min], \(d\left( t \right)\) is function of cutting depth against time [mm], and \(w_{M}\), \(y_{M}\), and \(x_{M}\) are exponents in the model of material-cutting power, respectively.

  1. (2)

    EC for feeding activities (\(E_{fa}\))

Single-pass end face turning has three feeding activities (\(A_{1}\), \(A_{2}\), and \(A_{3}\)) with the SRS of \(n\), as shown in Fig. 1a, thus \(E_{fa}\) is expressed as:

$$E_{fa} = \mathop \sum \limits_{j = 1}^{3} E_{fa}^{j}$$
(7)

where \(j\) denotes the index for a feeding activity and \(E_{fa}^{j}\) is EC for the \(j\)-th feeding activity [J]. Because there are two feeding modes (rapid and normal) in end face turning, two types of models for \(E_{fa}^{j}\) are provided. The feeding mode for \(A_{1}\) and \(A_{3}\) is rapid, and according to the model of Hu et al. [24], \(E_{fa}^{j}\) is expressed as:

$$E_{\text{fa}}^{j} = P_{\text{XR}} \times t_{\text{XR}}^{j} + P_{\text{ZR}} \times t_{\text{ZR}}^{j} + \left( {P_{0} + P_{\text{SR}} } \right) \times t_{j}$$
(8)
$$\left\{ {\begin{array}{*{20}l} {t_{j} = { \hbox{max} }\left\{ {t_{\text{XR}}^{j} ,t_{\text{ZR}}^{j} } \right\}} \\ {t_{\text{XR}}^{j} = \frac{{60 \times \Delta X_{j} }}{{1000 \times v_{\text{XR}} }}} \\ {t_{\text{ZR}}^{j} = \frac{{60 \times \Delta Z_{j} }}{{1000 \times v_{\text{ZR}} }}} \\ \end{array} } \right.$$
(9)
$$P_{\text{SR}} = B_{\text{SR}} \times n + C_{\text{SR}}$$
(10)

where the symbols in Expressions (8, 9, 10) are explained as below. \(P_{\text{XR}} ,P_{\text{ZR}}\): X-axial and Z-axial rapid feeding power, respectively [W]. \(t_{\text{XR}}^{j} ,t_{\text{ZR}}^{j}\): X-axial and Z-axial rapid feeding time for the \(j\)-th feeding activity, respectively [s]. \(P_{0}\): Basic power of the machine tool [W]. \(P_{\text{SR}}\): Spindle rotation power [W]. \(t_{j}\): Time consumption of the machine tool for the \(j\)-th feeding activity [s]. \(\Delta X_{j} ,\Delta Z_{j}\): Relative distances between the start and the end of the \(j\)-th feeding activity in X-axis and Z-axis, respectively [mm]. \(v_{\text{XR}} ,v_{\text{ZR}}\): X-axial and Z-axial rapid feeding speed, respectively [m/min]. \(B_{\text{SR}} ,C_{\text{SR}}\): Monomial coefficient and constant in the model of spindle rotation power.

The feeding mode for \(A_{2}\) is normal, and the feeding direction is X-axial. Based on the model of Hu et al. [24], \(E_{\text{fa}}^{2}\) is expressed as:

$$\left\{ {\begin{array}{*{20}l} {E_{\text{fa}}^{2} = \left( {P_{\text{XF}}^{2} + P_{\text{SR}} + P_{\text{CS}} + P_{0} } \right) \times \frac{{60 \times \Delta X_{2} }}{n \times f}} \\ {P_{\text{XF}}^{2} = A_{\text{XF}} \times \left( {n \times f} \right)^{2} + B_{\text{XF}} \times n \times f + C_{\text{XF}} } \\ \end{array} } \right.$$
(11)

where \(P_{XF}^{2}\) is X-axial feeding power for the second feeding activity [W], \(P_{CS}\) is coolant spray power [W], and \(A_{XF}\), \(B_{XF}\), and \(C_{XF}\) are quadratic coefficient, monomial coefficient, and constant, respectively, in the model of X-axial feeding power.

  1. (3)

    EC for spindle speed changes (\(E_{\text{sc}}\))

For single-pass end face turning, the SRS accelerates from 0 to n and then decelerates to 0. According to Lv et al. [33], \(E_{\text{sc}}\) is modelled as:

$$\left\{ {\begin{array}{*{20}l} {E_{\text{sc}} = E_{\text{sc}}^{1} + E_{\text{sc}}^{2} } \\ {E_{\text{sc}}^{1} = \mathop \smallint \limits_{0}^{{\frac{2\pi n}{{60\alpha_{A} }}}} \left[ {P_{0} + B_{\text{SR}} \times \left( {\frac{{30\alpha_{A} t}}{\pi }} \right) + C_{\text{SR}} + T_{s} \times \left( {\alpha_{A} t} \right)} \right]dt} \\ {E_{\text{sc}}^{2} = P_{0} \times \frac{{2\pi \left( {0 - n} \right)}}{{60\alpha_{D} }}} \\ \end{array} } \right.$$
(12)

where \(E_{\text{sc}}^{1}\) is EC for accelerating the spindle from 0 to \(n\) [J], \(E_{\text{sc}}^{2}\) is EC for decelerating the spindle from \(n\) to 0 [J], \(\alpha_{A}\) and \(\alpha_{D}\) are the spindle’s angular acceleration and deceleration, respectively [rad/s2], and \(T_{s}\) is the spindle’s acceleration torque [N m].

The objective function can output the value of EFTEC when inputting the values of \(n\) and \(f\).

3.2 Constraint Equations

The end face turning parameters should be selected with satisfying all constraint equations. By following Lu et al. [35], the constraint equations for the EFTEC model are developed considering the limits of cutting parameters, SRS, surface roughness, tool life [14], cutting force, and machine tool power. They are expressed as:

$$f_{L} \le f \le f_{U}$$
(13)
$$\left\{ {\begin{array}{*{20}c} { v_{L} \le v\left( {t_{0} } \right) \le v_{U} } \\ {v\left( {t_{0} } \right) = \frac{{\pi \times D_{0} \times n}}{1000}} \\ \end{array} } \right.\mathop \Rightarrow \limits^{\text{Conversion}} \frac{{1000v_{L} }}{{\pi D_{0} }} \le n \le \frac{{1000v_{U} }}{{\pi D_{0} }}$$
(14)
$$n \le n_{\text{max}}$$
(15)
$$\frac{{f^{2} }}{{8R_{N} }} \le R_{U}$$
(16)
$$T_{L} = \frac{{60 \cdot C_{L} }}{{v\left( {t_{\text{ce}} } \right)^{{w_{L} }} \cdot f^{{y_{L} }} \cdot d^{{x_{L} }} }} \ge N_{L} \times t_{\text{mc}} .$$
(17)

The cutting force and power at the moment when the fully cutting begins (\(t = t_{ce}\)) are selected to be restrained because the corresponding values are large [21]. The constraints are developed as:

$$F_{\text{cut}} = C_{Q} \cdot v\left( {t_{\text{ce}} } \right)^{{w_{Q} }} \cdot f^{{y_{Q} }} \cdot d^{{x_{Q} }} \le F_{U}$$
(18)
$$P_{\text{cut}} = P_{\text{mc}} \left( {t_{\text{ce}} } \right) + P_{\text{XF}}^{2} + P_{\text{SR}} + P_{\text{CS}} + P_{0} \le P_{U} .$$
(19)

The notations in Constraints (13–19) are described as follows.\(f_{L} ,f_{U}\): Lower and upper bounds of feed rates in material-cutting [mm/r]. \(v_{L} ,v_{U}\): Lower and upper bounds of cutting speeds in the beginning of material-cutting [m/min]. \(t_{0}\): Moment when the material-cutting begins. \(n_{ \hbox{max} }\): Maximum allowable SRS of the machine tool [rpm]. \(R_{N}\): Nose radius of the cutter [mm]. \(R_{U}\): Maximum allowable surface roughness [μm]. \(T_{L}\): Tool life model for end face turning [s]. \(C_{L}\): Coefficient in the tool life model. \(w_{L} ,y_{L} ,x_{L}\): Exponents of cutting speed, feed rate, and cutting depth, respectively, in the tool life model. \(N_{L}\): Minimum required passes of end face turning within one tool life. \(F_{cut}\): Material-cutting force when the fully cutting begins [N]. \(C_{Q}\): Coefficient in the model of material-cutting force. \(w_{Q} ,y_{Q} ,x_{Q}\): Exponents in the model of material-cutting force. \(F_{U}\): Maximum allowable cutting force [N]. \(P_{\text{cut}}\): Machine tool power when the fully cutting begins [W]. \(P_{U}\): Maximum available machine tool power [W].

4 Optimisation

After the EFTEC model is developed, SA is chosen as an optimisation method to get the optimal cutting parameters that lead to the minimum EFTEC. SA, proposed by Kirkpatrick et al. [36], is a probabilistic technique and meta-heuristic. It simulates the process of annealing in metallurgy to approximate the minimum internal energy (global optimum). To avoid premature convergence, the Metropolis criterion [37] about the acceptance of the inferior EFTEC value is incorporated. When the end temperature is reached, the optimal or near-optimal solutions are reported. Mahmoodpour and Masihi [38] presented the flowchart of SA.

According to the machine’s allowable accuracy, the EFTEC model based on the cutting parameters is discretised. For example, the SRS is divided to 600, 601, 602, …, 1098, 1099, and 1100 when the SRS constraint is \(600\,{\text{rpm}} \le n \le 1100\,{\text{rpm}}\). At the beginning, SA parameters are determined, including the initial and end temperatures, the temperature decrease function, and the length of Markov chain [39]. An initial solution including SRS and feed rate is random generated under the constraints (13–19). The candidate solution is generated through performing the random perturbations of SRS and feed rate on the last solution. This solution is checked to be accepted or rejected according to the EFTEC difference based on Expression (1) in each iteration. If the length of Markov chain is reached, the latest solution is returned with decreasing the temperature. Otherwise, the new candidate solution is generated. This iteration process is repeated unless meeting a stopping condition. The stop condition can be the specified end temperature that is reached.

Although SA consumes a short time to reach a solution, it cannot guarantee the global optimum in all trials due to the nature of meta-heuristics. Enumeration method, which is a deterministic algorithm, is used as the benchmark for verifying SA’s performance when minimising the EFTEC. Enumeration method lists all feasible schemes of SRS and feed rate under the Constraints (13)-(19) and then calculates the EFTEC according to Expressions (1–12) one by one [40]. By comparison, the scheme that leads to the minimum EFTEC is chosen as the optimum.

5 Case Study

The EFTEC optimisation for part A in Fig. 1 is demonstrated. Part A is made of 45#Steel. The required cutting depth \(d\) and diameter \(D_{0}\) are 1.7 mm and 43.6 mm, respectively. Part A is scheduled to be processed by a computer numerical control lathe (CK6153i). The cutter to use for end face turning is SNMG120408N-GU-AC725 whose nose radius and main angle are \(R_{N}\) = 0.8 mm and \(K\) = 45°, respectively. The experiment method and the experiment device for collecting force and power data were presented in Lv et al. [41]. According to Jia et al. [21] and Hu et al. [30], Table 1 lists the CK6153i’s parameters required for the EFTEC model. Because dry cutting is adopted, the coolant spray switch is OFF and corresponding power is \(P_{CS}\) = 0 W. Based on experiment measurements [41] and regression analysis [21], Table 2 provides the coefficients and exponents in the models of tool life and material-cutting force and power. The process parameters for part A are obtained from the process files and listed in Table 3. The maximum allowable cutting force \(F_{U}\) is determined by the stability and strength of the lathe (CK6153i) and the cutter (SNMG120408N-GU-AC725) [42]. The maximum allowable surface roughness \(R_{U}\) is determined by the technical criteria of part A [42].

Table 1 CK6153i’s parameters required for the EFTEC model
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Table 2 Coefficients and exponents in the material-cutting models
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Table 3 Process parameters for part A
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According to above data and Expressions (1)-(19), the EFTEC model for part A processed by CK6153i is developed in Table 4.

Table 4 The EFTEC model for part A processed by CK6153i
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To optimise the EFTEC, both SA and enumeration method were developed on Dev C ++ 5.11.0 software using the language C ++. The computing platform is the same as that of Hu et al. [30]. According to the allowable accuracy of CK6153i, the intervals of \(n\) and \(f\) are set to 0.1 and 0.001 in algorithms, respectively. The global minimum EFTEC of 20166.5 J for part A was obtained by enumeration method. The computation time of enumeration method is 6.179 s, and the optimal cutting parameters are: \(n = 594.6\,{\text{rpm}}\) and \(f = 0.306{\text{mm}}/{\text{r}}\). The values of SA parameters are: initial temperature = \(200\), end temperature = \(0.001\), temperature decrease function = \(200 \times 0.98^{k}\), and length of Markov chain = \(100\). SA was run 50 times, and the global minimum EFTEC of 20166.5 J is returned in all trials. An average computation time of SA is 0.0957 s, and the optimal cutting parameters also are: \(n = 594.6\,{\text{rpm}}\) and \(f = 0.306\,{\text{mm}}/{\text{r}}\). Figure 3 shows a search process of SA for the global minimum EFTEC. For the case, SA converges in 300 iterations. The performance of enumeration method and SA for part A is summarised in Table 5.

Fig. 3
figure 3

A search process of SA for minimising the EFTEC of part A

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Table 5 Results and performance of enumeration method and SA for case parts
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The methods are also tested on parts B, C, D, and E with the diameter \(D_{0}\) of 43.5 mm, 75.4 mm, 107.2 mm, and 107.3 mm, respectively, when the cutting depth \(d\) is 1.7 mm. Furthermore, they are tested on parts F, G, H, and I with the cutting depth \(d\) of 1.5 mm, 1.6 mm, 1.8 mm, and 1.9 mm, respectively, when the diameter \(D_{0}\) is 75.4 mm. By using SA and enumeration method, the results for optimising the EFTEC of these parts are listed in Table 5. It shows that the optimal SRS and feed rate are changed with the diameter and cutting depth, and the optimal SRS is between 534.7 rpm and 658.2 rpm. According to Table 5, SA has a 96% or more probability of getting the global optimal solutions.

By comparing the EFTEC in Table 5, it shows that the solution quality of SA is a little worse than that of enumeration method, but the computation time of SA is much less than that of enumeration method. For part B, the worst EFTEC obtained by SA is only 0.002% [(20122.9–20122.5)/20122.5] off from the global optimum, but the computation time of SA is 98.46% [(6.194–0.0953)/6.194] less than that of enumeration method. When the diameter increases from 43.5 mm to 107.3 mm, the computation time of enumeration method decreases from 6.194 to 2.560 s, but the computation time of SA slightly increases from 0.0953 to 0.1106 s. Among all cases, part E has the shortest computation time of enumeration method while the longest computation time of SA. For part E, the solution quality of SA is as good as that of enumeration method, but its computation time is 95.68% [(2.560–0.1106)/2.560] shorter than that of enumeration method. This verifies that SA can find good solutions in a short computation time. Thus, SA is recommended.

6 Discussion

In this section, the optimisation results are analysed. The relationship between the design parameters and the optimal SRS is described. Finally, the consequence of the EFTEC optimisation on the end face turning time is discussed.

6.1 Machining Energy Savings

The effect of our method in reducing the EFTEC is demonstrated. The traditional method of high SRS with medium feed rate serves as a benchmark to measure the energy savings [43]. The reason for selecting this method is that the high SRS can reduce the cutting time to improve the productivity while the medium feed rate can guarantee the surface quality of the workpiece [30]. The values of SRS and feed rate, generated by the traditional method, are \(n = 820\,{\text{rpm}}\) and \(f = 0.25\,{\text{mm}}/{\text{r}}\) for parts A, C and I, where the corresponding values of EFTEC are 23828.1 J, 44421.9 J, and 46867.3 J, respectively. According to Table 5, the MRR of the traditional method for parts A, C, and I is higher than that of our method. Kara and Li [44] suggested that the higher MRR led to the less unit process EC. However, considerable energy savings benefit from our method even though the MRR of the traditional method is higher than ours. Specifically, the EFTEC is reduced by 15.37% [(23828.1–20166.5)/23828.1], 14.03% [(44421.9–38189.4)/44421.9], and 7.30% [(46867.3–43447.3)/46867.3] for parts A, C, and I, respectively, by using our method. The reason for causing the conflict is analysed. The high MRR contributes to minimising the material-cutting EC. The traditional method increases the MRR by increasing the SRS, thereby increasing the EC for spindle speed changes and feeding activities. For the three cases, the increase of the EC for spindle speed changes and feeding activities exceeds the reduction of the material-cutting EC. Therefore, the total EFTEC is not minimised when adopting the high MRR with the high SRS.

6.2 Design Parameters Effects

In previous energy optimisation, only the material-cutting EC is concerned while the EC for spindle speed changes is neglected. As a result, the upper bound of SRS is regarded as the optimum, and the optimal SRS decreases with increasing the diameter of the part due to the SRS constraint. In actual machining, the EC for spindle speed changes is inevitable [25]. After adding the EC, the optimal SRS fluctuates with the diameter when the cutting depth is 1.7 mm, as shown in Fig. 4. At first the optimal SRS sharply increases from 585.4 rpm (lower bound of SRS of part B) to 594.6 rpm when the diameter only increases from 43.5 to 43.6 mm. Then, the optimal SRS declines until 54.2 mm, bottoming out at 577.4 rpm. For the diameter from 43.5 to 54.2 mm, the lower bound of the SRS constraint (14) and the cutting force constraint (18) jointly cause the fluctuation of the optimal SRS. After that, the optimal SRS rises until 86.0 mm, reaching the highest point at 639.0 rpm. The increase of the optimal SRS is caused by the increase of the proportion of material-cutting EC. From this point onwards, the optimal SRS falls to 578.4 rpm (upper bound of SRS of part E) from 86.0 mm to 107.3 mm. The decrease of the optimal SRS is caused by the upper bound of the SRS constraint (14). For any part shorter than 43.5 mm and longer than 107.3 mm, the optimal SRS is the lower and upper bounds of SRS, respectively. Besides, the relationship between the cutting depth and the optimal SRS when the diameter is 75.4 mm is shown in Fig. 5. Specifically, the cutting depth between 1.2 mm and 1.6 mm experiences a slight decrease in the optimal SRS from 541.3 rpm to the lowest point (534.7 rpm). After that, the optimal SRS goes up to 690.1 rpm from the cutting depth of 1.6–2.0 mm. The material-cutting EC becomes more dominant and causes the increase of the optimal SRS.

Fig. 4
figure 4

The relationship between the diameter and the optimal SRS

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Fig. 5
figure 5

The relationship between the cutting depth and the optimal SRS

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Overall, the relationship between the design parameters and the optimal SRS is not monotonously increasing or decreasing in the feasible solution space. For the next step, this relationship will be modelled. Thus, the optimal SRS can be directly calculated according to the values of diameter and cutting depth.

6.3 Machining Time Sacrifices

The effect of the end face turning energy optimisation on the end face turning time is discussed. The end face turning time based on the optimal cutting parameters of parts A (\(n = 594.6\,{\text{rpm}}\) and \(f = 0.306\,{\text{mm}}/{\text{r}}\)), C (\(n = 622.2\,{\text{rpm}}\) and \(f = 0.328\,{\text{mm}}/{\text{r}}\)), and I (\(n = 658.2\,{\text{rpm}}\) and \(f = 0.283\,{\text{mm}}/{\text{r}}\)) is 13.50 s, 17.53 s, and 18.98 s, respectively. By the aforementioned method of high SRS with medium feed rate, the cutting parameters of parts A, C, and I without considering the energy are generated. The end face turning time based on them (\(n = 820\,{\text{rpm}}\) and \(f = 0.25\,{\text{mm}}/{\text{r}}\)) is 13.63 s, 18.52 s, and 18.53 s for parts A, C, and I, respectively. Thus, 0.95% [(13.63-13.50)/13.63] and 5.35% [(18.52-17.53)/18.52] of the time reductions benefit from the energy minimisation of parts A and C, respectively. However, 2.43% [(18.98-18.53)/18.53] of the time increases suffer from the energy minimisation of part I, which verifies the conflicts between end face turning time and energy. If 2.43% of the time increases cannot cause the tardiness problem, the cutting parameters (\(n = 658.2\,{\text{rpm}}\) and \(f = 0.283\,{\text{mm}}/{\text{r}}\)) can be maintained for part I. Otherwise, the cutting parameters should be adjusted to optimise the trade-off between end face turning time and energy.

7 Conclusions

To realise sustainable manufacturing, reducing the machining EC is one of the challenges. It was approved that the cutting parameters optimisation is an effective approach for reducing the machining EC with constant MRR. However, this approach to reduce the machining EC with variable MRR did not receive attention. Besides, the EC for spindle speed changes was neglected in previous machining EC models. Thus, this paper develops the integrated EFTEC model including the sub-models of the material-cutting EC, the EC for feeding activities, and the EC for spindle speed changes. Especially, the design parameter of the diameter is included. In terms of optimisation, SA is employed and compared with enumeration method to verify its performance. In sum, it is a novel contribution to optimise the EFTEC with variable MRR considering the trade-off among the reductions of three portions of EC.

A case study was performed. Nine parts with changing diameters and cutting depths are machined by a lathe (CK6153i). The experiments show that SA has more than 96% probability of finding the global optima. For a case, the worst EFTEC obtained by SA is only 0.002% off from the global optimum, but the computation time of SA is 98.46% less than that of enumeration method. Hence, SA is effective for solving the EFTEC optimisation problem. By using the proposed method, 15.37%, 14.03%, and 7.30% EFTEC are decreased for parts A, C, and I, respectively. According to experiment results, the optimal SRS fluctuates with the cutting depth and diameter. For a case, 2.43% of end face turning time increases suffer from the EFTEC optimisation. Thus, the trade-off between the reductions of EFTEC and end face turning time should be considered.

The EFTEC model is sensitive to the coefficients derived from the initial experiments as well as the combination of the machining conditions including the workpiece material, the cutting tool, and the lathe. It is questionable whether the conclusions from the case study apply to other machining conditions or not. More experiments and validations will be performed to discover the trends. Besides, this paper has not considered some other machining operations with variable MRR including grooving, chamfering, and hobbing [45]. They are required in machining and have energy-saving potentials. The EC for these operations will be modelled. Furthermore, the EC for the combination of end face turning and other machining operations will be optimised [46]. In actual machining, the time, quality, and cost should be controlled when reducing the EFTEC. To achieve the optimal trade-off among these objectives, multi-objective optimisation will be conducted. In the future, a software module based the proposed method will be developed to automatically provide with the optimal end face turning parameters.