1 Introduction

High-performance metals, viz. super alloys, hardened steels, titanium, and its alloys, are known for their high creep and corrosion resistances, low thermal conductivity, hot hardness, chemical inertness, and resistance to thermal shocks. These properties have made their demand for the fabrication of parts in the manufacturing industries, such as automotive, aeronautical, and marine, very high. In the same breath, the same properties have made their machinability difficult. For instance, their hardness has been reported to accelerate the tool wear process [1] and introduce workpiece burnouts, which affect the quality of the final product [2]. Cold work hardened AISI D2 steel is an essential hard metal when it comes to making extrusion dies, slitting cutters, wire dies, burnishing rolls, gauges, and master tools. Its machinability has been greatly impeded by its mechanical properties, and as a result, its application in part production could have been much higher. Therefore, characterizing its machinability is an area that needs thorough research.

Efforts by manufacturing stakeholders have seen the development of several machining approaches for hardened AISI D2 steel. For a long time, grinding has been the primary method of fabricating parts from hard metals [3]. However, its low material removal rate, high-power consumption, long set-up times, and inability to produce complex geometries rendered it unreliable for hard machining [4]. This gave way to better methods such as hard turning [5]. Hard turning refers to the single-point cutting of metals with Rockwell hardness (HRC) of 45 and above, primarily in the range 58–68 HRC [6] [7], and is reliable for near-net shape and finish machining. The deployment of ultra-precision machine tools, tools, machining conditions, and production of parts with tight dimensional and geometrical tolerances characterizes ultra-precision hard turning (UHT) [8]. UHT is known for using small feeds and depths of cut, enabling it to achieve form accuracy of less than 1 µm and surface roughness of less than 0.1 µm. These accuracy levels are superior to the traditional hard turning methods [9].

Tool design and fabrication for the UHT process has been one of the critical research areas in hard machining. Efforts have been made to fabricate strong, tough tools with high thermal shock resistance and chemically inert at high temperatures. Cermet tools [10], ceramic tools, coated carbide inserts [11], and cubic boron nitride (CBN) tools [12] are some of the tools that have found recognition for use in UHT. However, due to their strength, toughness, and hardness (which is only second to diamond), CBN tools and their variants (PCBN) are the best for hard turning. In addition, their high melting temperature of 2730 °C and high stability of up to around 2000 °C have made them the best choice for hard turning processes [13]. Works by [14, 15] and [16] have shown the successful application of CBN tools in the hard turning of hardened steels.

The machinability of hardened AISI D2 can further be understood and improved through the prediction of machinability indices [17]. Force is a very significant machinability index when determining process stability. It has a strong correlation with almost all the other performance indices [18]. Force prediction, for a long period now, has been reported from analytical, numerical, and experimental (based on big data models) points of view. However, recently, industries have been rushing for intelligent machining to enhance productivity. Machine learning technologies, therefore, have been deployed to upheave production in line with the Industry 4.0 industrial revolution guidelines [19]. Machine learning (ML) models can be supervised, unsupervised, or reinforcement models. Supervised ML models such as Gaussian progressive relation (GPR), random forest (RF) regression, support vector machining (SVM), polynomial regression, gradient boosted trees (GBT), adaptive neuro-fuzzy inferencing system (ANFIS), decision trees (DT), and artificial neural network (ANN) have been the subject of interest in a bid to embrace intelligent machining. These ML models have been deployed largely for tool wear monitoring and machinability metrics forecasting both online and offline [20]. Process monitoring and prediction give prior knowledge of the process outcome, which can be informative when it comes to design and planning. It is through proper design and planning that the downtimes and production costs are reduced to improve production efficiency.

In addition to prediction, optimization of machinability indices is key in the planning and execution of sustainable and profitable production of AISI D2 steel products. Optimization can be objective as well as multi-objective [21]. Single objective optimization has been faulted for overlooking some of the process kinematics and the interdependence of machinability indices, hence not capturing the dynamic nature of processes. Multi-objective optimization finds a balance between selected machinability indices; hence, it can minimize vibration and acoustic emission while maximizing material removal rate, tool life, etc. Ant colony, teacher–learner-based optimization (TLBO), particle swarm optimization (PSO), grey relational method, multi-objective particle swarm optimization (MOPSO), firefly, reinforcement dung beetle algorithm, and genetic algorithm (GA) are some of the heuristics and metaheuristics optimization methods that have been reported this far [22,23,24,25].

Kumar et al. [26] studied the workability of heat-treated AISI D2 steel based on the cutting tool’s flank wear, product surface quality, and chip–tool interface temperature. They developed RSM and ANN models for forecasting the three machinability indices. Vahid et al. [27] developed an ACO (adaptive control optimization) system that relied on ANN and GP (geometrical programming) for prediction, and PSO for optimization when hard turning AISI D2 steel. According to Tabassum et al. [28], ANFIS prediction is superior to that of ANN and RSM when forecasting force from the hard turning of AISI 1045 (56 HRC) in both high-pressure coolant and dry cutting conditions. Non-dominated sorting genetic algorithm II (NSGA-II) was used by Meddour et al. [29] to find the optimal values of surface roughness and cutting forces simultaneously when hard turning AISI 4140 (60 HRC) using a mixed ceramics cutting tool. In addition, they developed RSM and ANN models for prediction. Similar work on ML modelling of AISI D2 processes was reported by Adizue et al. [30]. Further works on ML modelling have been reported by [31,32,33,34].

Parameter correlation studies by Patel and Gandhi [35], when hard turning AISI D2 steel using the CBN tool, showed that depth of cut was the most significant parameter in force generation. According to Rafighi et al. [17], variation of cutting edge radius had the most significant impact on cutting force when hard turning AISI D2 steel. They noted that the ceramic tool used led to a lower magnitude of forces than CBN inserts. According to [36], there is no correlation between the cutting velocity and cutting force when machining AISI D2 steel using the CBN tool.

Based on the literature review, there is a need for robust prognostic models and optimization methods to effectively characterize the UHT of AISI D2 steel using the CBN tool, based on the generated forces. Most of the reported works have been based on Taguchi’s design of experiments and, therefore, fail to investigate all the possible parameter combinations during the execution of the experiments, leading to incomprehensive and inaccurate conclusions. Thus, there is a need for more investigations based on full factorial experiments to exhaust all possible parameter combinations.

Due to the dynamic nature of the UHT process, no standard machine learning model or optimization method has been approved for force prediction when hard turning AISI D2 steel using the CBN tool. This is despite their prediction superiority over analytical and numerical methods [37]. The literature work on machine learning modelling and optimization of resultant force during UHT of AISI D2 steel is scarce. Hence, there is a need for either developing new, robust, and accurate models or evaluating and improving the existing ones for application during AISI D2 steel UHT using CBN tools.

Based on these gaps, this work analyses the kinematics of force components’ generation during UHT processes. In addition, it investigates the prediction of resultant forces using ANFIS during UHT of AISI D2 steel. The investigation is based on a full factorial design of experiments and an extra set of experiments for validation data for checking model robustness. Lastly, multi-objective particle swarm optimization (MOPSO) has been used to optimize the resultant force relative to material removal rate (MRR) and vibration to establish optimal parameters for sustainable and efficient production of parts made from cold work hardened AISI D2.

2 Materials and methods

2.1 Design of experiments

This work sought to characterize the UHT of cold work hardened AISI D2 steel through analysis, prediction, and optimization of force components based on full factorial experiments. Therefore, a theoretical analysis of the force components is conducted, and the resultant force is predicted through ANFIS modelling. The multi-objective optimization method is deployed to determine the optimal cutting parameters and resultant force.

UHT was conducted on a hollow, cold work hardened AISI D2 steel workpiece (62 HRC) of 12 mm internal and 60 mm external diameters and a length of 40 mm (Fig. 1). The internal hole provided an exit for the tool because as the diameter of the stock reduced, the machine tool’s rotational speed needed to be increased to maintain the constant cutting velocity. The hole, therefore, compensated for the limited rotational speed of the machine tool. AISI D2 steel was chosen because of its high-dimensional stability, good resistance to temper softening, and high compressive strength when hardened. These properties enhance its application in the development of extrusion dies and slitting cutters.

Fig. 1
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Workpiece of cold work hardened AISI D2 steel applied for the experiments

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A single tip, uncoated, full-face layer CBN insert with 50% CBN grade, 2 µm grain size, and bound by a ceramic binder was chosen due to its exceptional performance in high-speed hard machining processes. The CBN insert used was SECO DCGW11T308S-01020-L1-B CBN010 and is specifically meant for finish hard turning operation due to its excellent wear resistance, good thermal shock resistance, chemical inertness when machining steel, hot hardness and strength, good thermal conductivity, and high homologous temperature. It had two cutting edges, a clearance angle of 7°, a cutting edge effective length of 3.5 mm, an insert corner radius of 0.8 mm, a thickness of 3.75 mm, a weight of 0.0004 kg, and an included angle of 55°. Figure 2 shows the CBN insert.

Fig. 2
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CBN insert used for the experiment

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The machining process took place on a high-rigidity, ultra-precision CNC lathe (Hemburg-Mikroturn, 50 CNC) with a repetitive accuracy of ± 1 µm, maximum spindle speed of 6000 rpm, and a positional accuracy of 1 µm/150 mm. As shown in Fig. 3, the workpiece was held on the machine with an ultra-precision three-jaw pneumatic clamp, and its concentricity was set using a dial gauge indicator. A full factorial design of experiments was deployed to explore all the possible parameter combinations. As shown in Table 1, three cutting parameters (cutting velocity (m/min), feed (mm/rev), and depth of cut (mm)) with three levels were investigated. The cutting parameters’ levels were chosen relative to the manufacturer’s handbook, UHT lathe parameters, CBN tool geometry, and previous research work on hardened steel turning.

Fig. 3
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Experimental set-up

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Table 1 Levels of cutting parameters
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The three levels, therefore, meant that 27 runs of the experiment were conducted randomly. The 27 runs were carried out three times to ensure the accuracy of the final data, hence bringing the total number of experiments conducted to 81 runs. A further nine runs of experiments with unique parameter combinations were conducted to validate the models.

During the face hard turning process, three runs with different parameter combinations were carried on the surface of the workpiece cross-section. See Fig. 4. When transiting from one run to the other, the spindle was rotated five times without any cutting action. During these non-cutting phases, the machine tool’s controller adjusted the cutting parameters relative to the next run in the used G code. After machining a set of three runs, the machined face was flattened before machining the next set. This was repeated until all the 90 runs had been completed.

Fig. 4
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Schematic representation of the face hard turning strategy

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A three-axis Kistler dynamometer (Kistler, Type 9257A) was deployed to measure the signals of passive force (Fp), cutting force (Fc), and feed force (Ff) components. The directions of the force components are shown in Fig. 5. A transducer (Kistler, Type 9257) was applied for signal amplification. The force signals in Figs. 6, 7, and 8 show the individual signals for each run. These signals are separated by moments of change in parameter combination, as is shown in Fig. 8. During these moments of change in parameter combination, the workpiece was rotated five times without any tool feed.

Fig. 5
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Force components

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Fig. 6
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Cutting force (Fc) signal

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Fig. 7
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Feed force (Ff) signal

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Fig. 8
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Passive force (Fp) signal

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2.2 Adaptive neuro-fuzzy inferencing system (ANFIS)

This algorithm is a hybrid of two known machine learning algorithms, artificial neuro network (ANN) and fuzzy logic system (FLS). The hybrid utilizes the constituent algorithms’ strengths to establish a powerful and flexible model for complex nonlinear problems. Figure 9 shows the general ANFIS architecture. During modelling, the input layer receives the variables that are to be used for the prediction process. The membership functions (MF), for each of the variables, are then developed in the fuzzy layer. Equations (1) and (2) denote the nodal functions in the fuzzy layer.

$$C_{1,\, j} = \beta_{{a_{j} }} \left( x \right) j = 1,2, \ldots 6$$
(1)
$$C_{1, t} = \beta_{{b_{t} }} \left( x \right) t = 1,2, \ldots 6$$
(2)
Fig. 9
figure 9

ANFIS architecture

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The membership functions are denoted by aj and bt. C1, t represents the output from the tth node of the fuzzy layer. Nodes in the product layer generate the input parameters as per Eq. (3).

$$C_{2, t} = W_{t} = \beta_{{a_{t} }} \left( x \right) \times \beta_{{b_{t} }} \left( x \right) \ldots .t = 1, 2, \ldots 6$$
(3)

Normalization of the input parameters takes place in the third layer. The normalization function is denoted by Eq. (4), and it converts the input parameters to values between 0 and 1.

$$C_{3,t} = \overline{W} = \frac{{W_{t} }}{{W_{1} + W_{2} }} \ldots ..t = 1, 2, \ldots 6$$
(4)

Wt is determined by the firing strength of node t denoted by ω t. Equation (5) gives the output of the fuzzy system.

$$C_{4,t} = \overline{W}f_{i} = \overline{{W_{t} }} \left( {p_{t} y + q_{t} x + r_{t} } \right) \ldots . t = 1, 2, \ldots 6$$
(5)

Whereby pt, qt, and rt are the consequent parameters whereas x and y represent the prediction variables. The summation process at the defuzzification layer is summarized by Eq. (6).

$${\text{Output}} = \frac{{W_{1} }}{{W_{1} + W_{2} }}f_{1} + \frac{{W_{2} }}{{W_{1} + W_{2} }}f_{2} + \ldots + \frac{{W_{{\text{p}}} }}{{W_{{\text{p - 1}}} + W_{{\text{p}}} }}f_{{\text{p}}}$$
(6)

Training of the model involves the adjustment of the consequent parameters and MFs and is conducted by optimization algorithms such as backpropagation, least-square methods, and PSO [38] based on a cost function. The fuzzy output values, after solving Eq. (6), are clear, concise, numerical values that are suitable for data prediction and decision-making. ANFIS was chosen based on its excellent results when deployed by [39, 40] and [41] in their works.

2.3 Multi-objective particle swarm optimization (MOPSO)

MOPSO is an improved version of the heuristic particle swarm optimization (PSO) proposed by Coello and Lechunga [42] for multi-objective problem optimization. The MOPSO process begins with the initialization of a population of particles. Each particle is identified by its position (Pjk), velocity (Vjk), and fitness. Each particle’s position is a potential solution for the problem at hand. To optimize the hard turning process, the particles are considered a combination of predictors Pj = (vc j, fj, apj) whereas the velocities are Vj = (vj1, vj2, vj3). The new position of the jth particle in k + 1 iteration, Pj k+1, is updated by the new velocity, Vjk+1, as shown by Eq. (7). Equation (8) shows the updating of the previous velocity Vjk. The new position is incorporated in the objective function to determine the fitness value. This fitness value is compared to the previous values, and if it is superior, it is considered the personal best; otherwise, the previous fitness is considered the best. The best individual position in the population is considered the global best position.

$$P_{j}^{k + 1} = P_{j}^{k} + V_{j}^{k + 1}$$
(7)
$$tV_{j}^{{k + 1}} = \omega \times V_{j}^{k} + r_{1} c_{1} \left( {P_{{{\text{best}}}} ^{k} - P_{j} ^{k} } \right) + r_{2} c_{2} \left( {P_{{{\text{global}}}} ^{k} - P_{j}^{k} } \right),$$
(8)

Whereby Pbestk is the personal best position, Pglobalk is the best overall position (solution), \(\omega\) is the inertia weight, c1 represents the personal learning coefficient, and c2 refers to the global learning coefficient, whereas r1 and r2 are random values in the range [0, 1].

As an extension to PSO, MOPSO deploys the concept of Pareto optimality. It involves the creation of external repositories for each particle at each iteration to store its non-dominated solutions. Another repository is created for the entire population for each iteration and one global repository for the entire population for the entire optimization process. A selection criterion is used to choose the best non-dominated solution, from the first two repositories. A leader or global solution is later chosen from the global repository. Figure 10 shows the process of executing MOPSO.

Fig. 10
figure 10

Flowchart of MOPSO algorithm

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MOPSO was chosen for this work because of its fast convergence, great compatibility, high efficiency, and feasibility [43].

3 Results and discussion

The signal extraction process was performed in MATLAB R2022b software through coding for all 81 runs and an additional 9 for validation data. Numerical signal values were obtained from arbitrary ranges within the relevant run signal, as shown in Fig. 7. The non-cutting ranges shown in Figs. 6 and 7 contain signals originating from machine excitation (due to tool acceleration towards the workpiece) and vibrations due to chuck rotation. These non-cutting signals were not considered during numerical extractions of signal values. The force components’ and vibrations’ values in Table 2 are the averages extracted from the signals of the three experimental replicates. Fr and MRR values were determined using Eqs. (9) and (10), respectively.

$$F_{r} = \sqrt {F_{f}^{2} + F_{c}^{2} + F_{p}^{2} }$$
(9)
$${\text{MRR}} = v_{c} \times f \times a_{{\text{p}}}$$
(10)
Table 2 Experimental results
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3.1 Force analysis

Generally, as shown in Table 2, passive force had the largest magnitude. This is a distinctive characteristic of hard turning processes, which can be attributed to the exceptional circumstances of chip formation at hard turning (e.g. spring-back effect) [35]. The spring-back effect results from the elastic and plastic deformation phenomenon that the workpiece experiences during machining [1]. According to Rath et al. [44], chip formation in steel hard turning results from crack initiation on the surface of the workpiece. The crack is initiated ahead of the tool and propagates as the compressive stresses induced by the tool increase due to the feed force to form saw-tooth chips. In the wake of chip formation, the workpiece experiences some element of elastic recovery. The elastic recovery (spring-back) raises the magnitude of the passive force. According to Bartaya et al. [45], the high passive force can also be associated with the lower depth of cut relative to the CBN insert’s cutting edge radius and the negative rake angle of the tool.

Compared to conventional turning, the cutting forces magnitude in hard turning is smaller. This is probably due to residual thermal stresses induced by the high temperatures generated and the low feed rates (relative to cutting speed) associated with this process. The high temperatures soften the workpiece during a given machining pass. As a result, the subsequent passes (either close to the previous pass or covering some part of the previous pass due to small feeds and relatively high cutting speeds) tend to occur on softer surfaces. The low cutting forces can also result from the low depths of cut and feed deployed in hard turning processes [46].

Analysis of variance in Table 3 indicates that all three cutting parameters were significant in resultant force generation, as their p-values were below 0.05 confidence level. Feed has the highest contribution (69.74%), whereas cutting velocity (3.79%) is the least, and the interaction of feed and depth of cut has the strongest influence on resultant force generation. From Figs. 11 and 12, an increment in the depth of cut leads to an increment in the resultant force. This can be attributed to an increase in cutting edge angle, which leads to an increment in the coefficient of friction, and subsequent increment in compressive stresses by the tool. Similarly, the feed directly correlates with the resultant force, as shown in Figs. 12 and 13. This is probably due to the increase in the size of the theoretical chip area with an increase in feed. The changes in the chip area have a direct correlation with the coefficient of friction hence leading to the significant changes in the magnitude of generated force components. Cutting velocity and resultant force have an inverse correlation (Figs. 11 and 13). This can be attributed to increase in temperature with increasing cutting velocity at the primary shear zone. The increasing temperature softened the workpiece ahead of the tool, leading to low compressive stresses asserted by the tool, hence the generation of low resultant force [47].

Table 3 Analysis of variance for resultant force
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Fig. 11
figure 11

Variation of resultant force with depth of cut and cutting velocity at a constant feed of 0.125 mm/rev

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Fig. 12
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Variation of resultant force with depth of cut and feed at a constant cutting speed of 125 m/min

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Fig. 13
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Variation of resultant force with cutting velocity and feed at a constant depth of cut of 0.06 mm

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As shown in Table 4, all the cutting parameters and their two-factor interactions were significant in determining material removal rates. All of them had p-values of 0.0. However, feed was the most significant parameter with a contribution of 60.10%. Depth of cut had the least influence at 8.45%, whereas velocity had a contribution of 21.63%. The interaction of velocity and feed had the strongest influence on MRR amongst the parameters’ interactions. It can be inferred from Figs. 14, 15, and 16 that all the cutting parameters had a positive correlation with MRR. Increasing cutting velocity could have led to temperature increment at the primary cutting zone, hence softening the workpiece, making it easier and quicker to machine. Likewise, the increment of feed increased the chip area and subsequently volume of the material to be machined per unit time. Similarly, increasing the depth of cut implied increased engagement between the workpiece and the cutting tool that translated to large volume of material to be machined.

Table 4 Analysis of variance for MRR
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Fig. 14
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Variation of MRR with cutting velocity and feed at a constant depth of cut of 0.06 mm

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Fig. 15
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Variation of MRR with feed and depth of cut at a constant cutting velocity of 125 m/min

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Fig. 16
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Variation of MRR with cutting velocity and depth of cut at a constant feed of 0.125 mm/rev

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3.2 Tool vibration analysis

The correlation between vibration and the cutting parameters is quite stochastic, and no clear trend is observed in Figs. 17, 18, and 19. This can be attributed to the complexity of the vibration signal. Apart from tool vibration, the accelerometer often records signals from other unwanted sources within the machining environment. These unwanted signals (noise) may be challenging to get rid of during signal processing. The ANOVA (Table 5), however, shows that feed and its interaction with cutting velocity are significant in determining tool vibration. Their contributions are 62.42% and 31.42%, respectively. At a confidence level of 0.05, feed has a p-value of 0.001 whereas its interaction with cutting velocity has a p-value of 0.02. The high influence of feed can be attributed to its direct correlation with the cutting force and the subsequent direct correlation between the cutting force and vibration. As the theoretical chip area increases with the increase in feed, the force generated becomes more due to the high compressive stresses asserted by the advancing tool. The significant impact of the interaction of feed and velocity on vibration can be attributed to possible tool wear. The variation of feed and cutting velocity determines the coefficients of friction and cutting temperature at the shear zone, and this influences tool wear progression. The tool vibration, therefore, will vary with the tool wear status. Figure 20 shows the vibration signals.

Fig. 17
figure 17

Variation of vibration with cutting velocity and feed at a constant depth of cut of 0.06 mm

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Fig. 18
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Variation of vibration with cutting velocity and depth of cut at a constant feed of 0.125 mm/rev

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Fig. 19
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Variation of vibration with depth of cut and feed at a constant velocity of 125 m/min

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Table 5 Analysis of variance for tool vibration
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Fig. 20
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Vibration signal

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3.3 ANFIS model development and evaluation

Development of the ANFIS model for resultant force prediction was conducted in MATLAB R2022b using data in Table 2. The training and testing processes were conducted using data from 19 and 8 experimental runs, respectively. Data from nine experimental runs were later used for the validation of the model. The training process involved variation of the number of membership functions (MF) allocated to the inputs and output parameters, variation of MF type, variation of the type of training algorithms (backpropagation or hybrid (backpropagation and least-squares method)), as well as variation of the fuzzy inferencing system (FIS) (Sugeno or Mamdani). These variations gave different models whose suitability was gauged using the model performance parameters, mean absolute percentage error (MAPE), and coefficient of correlation, R. The adopted model comprised four Gaussian MFs for one input and two each for the other two inputs. Its output parameter was allocated a constant MF and was based on the Sugeno inferencing system. The hybrid algorithm trained the model using the weighted average of rules. Figure 21 shows the model’s structure.

Fig. 21
figure 21

ANFIS model structure

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The results of the model performance analysis in Table 6 showed that the model developed was highly reliable for prediction. According to [48], the accuracy of a model is considered low if its MAPE is above 50%, satisfactory if it is between 20 and 50%, good if it is between 10 and 20%, and high if it is below 10%. Similarly, a coefficient of determination of 1 shows a strong positive correlation between the measured and predicted values whereas an R2 value of − 1 indicates a strong negative correlation. On the other hand, 0 indicates a weak correlation. The MAPE of the model on the trained and test data was 1.47% and 4.81%, respectively, whereas the R2 values were 0.9 (trained data) and 0.9 (test data). The high R2 of 0.8 on validation data indicated a strong positive correlation between the measured and predicted resultant forces. This strong correlation was further confirmed by a MAPE of 10.66% on the validation data. The high MAPE and R2 values (close to 1) on validation data indicated that the model was robust enough to predict resultant force during UHT of cold work hardened AISI D2 using the CBN tool. Figure 22 demonstrates the prediction ability of the developed model. Similar results are observed with the validation data as is shown in Fig. 23.

Table 6 Average MAPE and R
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Fig. 22
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Comparison of measured against ANFIS model predicted values

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Fig. 23
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Comparison between validation and ANFIS-predicted data

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3.4 MOPSO results

A relatively high MRR in UHT, relative to the optimal values of other process parameters, increases the production rate by lowering the lead-time on a single product. Therefore, maximization of MRR relative to other machinability metrics can upheave effective and efficient production [49]. Tool vibration is also a very important parameter in the determination of production quality and cost. Tool vibration correlates with the cutting force, surface roughness, tool wear, and MRR amongst other process parameters. Sahu et al. [50] demonstrated that surface roughness increased with an increase in tool vibration. Similarly, Guleria et al. [51] used a tool vibration signal processing technique to classify surface roughness. Tool vibrations introduce chatter marks on the product surface, hence lowering the surface’s quality. In addition to these, tool vibration is also a very critical cause of tool failures and breakages. Therefore, keeping tool vibration relatively minimal is reliable for the stability of the machining process and the acquisition of quality products. This work, therefore, opted to optimize UHT resultant force relative to MRR and vibration with the aim of maximizing MRR and minimizing both the resultant force and vibration.

MOPSO sought to find the best cutting parameter values that would give an optimal magnitude of the resultant force, MRR, and vibration. Coding of the MOPSO algorithm was conducted in MATLAB R2022b using Eqs. 11, 12, and 13 as the objective functions. These equations were developed by the response surface methodology approach in Minitab 21. The algorithm was best tuned according to the parameters in Table 7.

$$F_{r} = 67.3 - 0.2203v_{c} + 348f + 862a_{{\text{p}}} + 7432fa_{{\text{p}}}$$
(11)
$$\begin{aligned} {\text{MRR}} = & 750 - 6v_{C} - 1000f - 9375a_{{\text{P}}} \\ & + 80v_{c} f + 75v_{c} a_{{\text{p}}} + 125000fa_{{\text{p}}} \\ \end{aligned}$$
(12)
$$Vb = 0.2972 + 0.1329f + 0.1152 v_{c} f$$
(13)
Table 7 Tuned parameters’ values
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Figure 24 shows the distribution of particles in the search space during the iteration processes of MOPSO. A series of red and black circular marks in the search space represent a series of Pareto fronts. The most optimal Pareto solution (leader) was selected from the many Pareto fronts. Table 8 shows the optimal values for the cutting parameters (cutting velocity, feed, and depth of cut) and response parameters.

Fig. 24
figure 24

Distribution of particles (possible solutions) in the search space

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Table 8 Optimal parameters
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The large MRR value is desirable for enhancing the production rate of hardened AISI D2 steel components in an industrial setting. Since there is a direct correlation between vibration and the cutting forces [12], the low vibration value attained in this work shows that the acquired resultant force is the most optimal to warrant machine stability during UHT of AISI D2 steel within the investigated parameter range.

4 Conclusion

A comprehensive literature review on UHT of hardened AISI D2 steel is presented in this work. An in-depth analysis of the variation of force components during UHT of AISI D2 steel has been reported. ANFIS machine learning model for force prediction was developed and evaluated. The resultant force was later optimized relative to MRR and vibration using MOPSO. Consequently, the following conclusions were made:

  • The proposed ANFIS model was developed, and its performance evaluation was conducted using mean absolute percentage error (MAPE) and the coefficient of determination (R2). According to MAPE values of 10.66% and R2-values of 0.8 on validation data, the model’s prediction was highly satisfactory (Table 6). The model, therefore, can be incorporated into digital twin models for monitoring machining stability during the UHT process of AISI D2 metals.

  • From the optimization process, a feed of 0.125 mm/rev, a cutting velocity of 158.8 m/min, and a depth of cut of 0.074 mm will result in a stable UHT of hardened AISI D2 steel when using the CBN tool. The corresponding optimal resultant force, MRR, and vibration related to these cutting parameters were 224.8 N, 2603.6 mm3/min, and 0.03 m/s2 (Table 8). These values are bound to improve the machining rate as well as lower the machining costs on a shop floor.

  • All the cutting parameters are significant in resultant force generation during UHT of AISI D2 steel, with feed being the most significant parameter with a contribution of 69.74%. Cutting velocity is a minor contributor (Table 3).

  • Passive force is the largest force component during UHT of hardened AISI D2 steel using the CBN tool (possibly due to the spring-back effect experienced during UHT). In contrast, the feed and cutting forces have low magnitudes (probably due to the high temperature generated at the primary shearing zone), see Table 2. The knowledge of the magnitude of force components can be of help during tool design and tool choice for UHT processes.

  • Feed and cutting velocity are the most significant factors in the determination of MRR and vibration during UHT. Feed’s influence dominates that of cutting velocity.

The reported findings, therefore, offer invaluable insight into kinematics, forecasting, and optimization of UHT processes. The results can contribute immensely to the development of precision engineering in the machining of difficult-to-cut metals and other related materials.

4.1 Future directions

The authors recommend future studies to be carried out on composite materials with an interest in the chip removal mechanisms relative to the composite material’s structures. A digital twin model for monitoring machine stability, amongst other machinability indices, can be developed using the developed AI model.