1 Introduction and background

CNC milling is amongst the best metal-cutting processes in the modern manufacturing world. Its fastness, accuracy, automation and flexibility have made it stand above most conventional and non-conventional metal-cutting methods. A successful milling process is affordable and characterized by high product quality. The cost of production is a function of power consumption, and the latter is a function of cutting force. According to Kienzle-Victor [1, 2], cutting force in milling is a result of machining parameters, cutting conditions (presence or absence of coolant), tool and workpiece materials, tool path, chip thickness as well as tool–workpiece interaction amongst many other factors as shown in Eq. (1);

$$F = f\left( {w_{p} \ldots w_{q} , tm_{r} \ldots tm_{s} , tg_{t} \ldots tg_{u} , r_{k} \ldots r_{l} ,{\text{LR}}} \right)$$
(1)

whereby F is the force. \(w_{p} , \ldots ,w_{q}\) are the cutting parameters related to the workpiece’s material, \(tm_{r} , \ldots ,tm_{s}\) are the parameters that are related to the cutting tool’s material, LR stands for the lubrication conditions, \(tg_{t} , \ldots ,tg_{u}\) are the parameters that are related to the geometry of the tool. Likewise, \(r_{k} , \ldots ,r_{l}\) are the process parameters and other related machining conditions. This relationship is complex, and all these factors cannot be studied simultaneously. Because of this, research on the appropriate cutting force models has been endless. This far, the research work reported in the literature has chosen to keep some parameters constant as the desired parameters are under investigation. For instance, [3] investigated the impact of three cutting parameters and cutting fluids on surface integrity during the milling process of aluminium 7075.

Cutting force modelling for CNC milling processes can be either experimental-oriented or analytically oriented. Analytical cutting force modelling focuses on tool geometry, tool–workpiece interaction and other process physics. Some of the analytical models include work carried out by [4, 5] and many others. Despite their good levels of predictability, the rate of adaptation of these models by machinists is low due to a large number of assumptions made and rigorous calculations involved in their development and application. Experimental-oriented models are data based. The data collected from CNC milling processes are numerical signals in the time domain. They are preprocessed and processed to acquire the response parameters’ numerical values for process characterization.

In CNC milling, the cutter tooth–workpiece interaction is intermittent due to pitch angles between the cutters’ teeth as well as entry (\({\varnothing }_{{\text{entry}}}\)) and exit angles (\({\varnothing }_{{\text{exit}}}\)) during the cutting process [6].

As shown in Fig. 1, both the entry and exit angles are in the milling zone. These angles can be used to express, mathematically, the force components, Fx, Fy and Fz as shown in Eq. (2);

$$\left[ {\begin{array}{*{20}c} {F_{x} \left( {\emptyset_{x} } \right)} \\ {F_{y} \left( {\emptyset_{x} } \right)} \\ {F_{z} \left( {\emptyset_{x} } \right)} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}l} {\cos \left( {\emptyset_{x} } \right)} \hfill & {\cos \left( {\emptyset_{x} } \right)} \hfill & 0 \hfill \\ {\sin \left( {\emptyset_{x} } \right)} \hfill & { - \cos \left( {\emptyset_{x} } \right)} \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 1 \hfill \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {F_{{\text{t}}} } \\ {F_{{\text{r}}} } \\ {F_{{\text{a}}} } \\ \end{array} } \right]$$
(2)
Fig. 1
figure 1

End mill entry and exit angles

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The three forces, Ft (tangential), Fr (radial) and Fa (axial), have varying impacts on the cutters’ lifespan, the process stability and the surface integrity of the final product. The tangential cutting force, Ft, induces torsional deformation on the cutting edge. Radial cutting force Fr induces bending deformation characterized by cutter run-outs and deflections (if it turns out to be high magnitude). These two forces (tangential and radial) are periodical due to the pitch angles of the teeth of the cutters. Fa, on the other hand, induces compressive deformation. It is a continuous force that ensures that the cross section of the cutter is in constant contact with the workpiece within the cutting zone during the cutting [7].

The resultant cutting force (FR) is determined as shown in Eq. (3);

$$F_{{\text{R}}} = \sqrt {F_{x}^{2} + F_{y}^{2} + F_{z}^{2} }$$
(3)

\({F}_{x}\), \({F}_{y}\) and \({F}_{z}\) are the instantaneous cutting forces in the cutting zone.

1.1 Feed rate optimization

Feed optimization in CNC milling processes is the best way of minimizing machining time and power consumption and increasing the potential for efficient machining [8, 9]. Impact of increasing or decreasing feed is directly proportional to feed rates. Feed rate optimization can be based on a response parameter constraint or on a feed rate threshold value. In the former, the feed rate monitoring is relative to the predetermined value of a response parameter like cutting force. The second approach aims to optimize the feed rate relative to a predefined optimal feed rate value [10]. Feed rate optimization can be online or offline. Online optimization of feed rate is optimization on the fly. The feed rate is optimized in this case as the milling process is ongoing. It, therefore, relies on sensors and feedback loops. Research work on online feed rate optimization has been reported by [11, 12] and [13], amongst others. Despite handling the dynamic nature of the milling process, online feed rate optimization is expensive because of sensors and developed controllers used. Installation of the sensors in the cutting zone is also challenging, considering the messy nature of the milling process in the milling zone [14].

Research work on force modelling and feed rate optimization during milling of aluminium 7075 is scanty in literature records. Robson et al. [15] modelled cutting force based on feed rate and cutting velocity. Their work also reported the effect of noisy variables on the measured signals. Similarly, Sethupathy et al. [9] modelled cutting force based on tool geometry and process parameters. Their work investigated the correlation between cutting force and feed rate, cutting speed and axial depth of cut. In their work, [16] modelled cutting force and surface roughness as functions of cutting speed, axial depth of cut, feed rate and radial depth of cut for milling Al.2024-T351. Their work reported that feed rate was one of the most influential parameters in determining the cutting force magnitude and surface quality of the final product. Similar results were reported by [17] and [18]. Most of the work recorded in literature has cutting force modelled with cutting speed, feed rate and depth of cut, as the predictors.

Our research opted to include radial depth of cut in our model because it influences the chip removal possibility during the milling process. It influences chip thickness and cannot be ignored in force prediction. We also adopted the TLBO algorithm for optimization because it is robust and does not rely on algorithm-specific parameters. Most of the time the algorithm-specific parameters lead to local convergence during optimization.

The organization of this paper is as follows; Sect. 1 introduces the research topic and reviews the relevant previous publications. Section 2 describes the adopted design of experiment, the machine tool used, the machining conditions, the measurement instruments and the workpiece and tool materials. This section also explains the concept, structure and the application of TLBO algorithm. Section 3 presents the experimental results. In Sect. 4, the acquired data are analysed and discussed with emphasis on force modelling and validation, parameter correlations and feed optimization. Section 5 summarizes the findings and outlines their possible application.

2 Experimentation and methodology

2.1 Design of experiment and set-up

The milling process experiment took place on a CNC milling machine tool (Z-540B, KC201706149). The main process parameters were: cutting velocity (v), radial depth of cut (ar), axial depth of cut (ap) and feed (f). Each process parameter had four levels, as shown in Table 1

Table 1 Levels of the process parameters
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The levels were based on the machine tool’s maximum feed rate and spindle speed. The radial and axial depths of cut were based on the cutting tool’s and workpiece’s materials as well as the geometry of the cutting tool. The design of the experiment was conducted in a Minitab 15 environment and L16 Taguchi array was adopted. The experiment had 20 runs. The machine tools’ acceleration and jerk levels were negligible during the milling process.

A three-axis stationary piezoelectric Kistler dynamometer (Kistler Instrumente AG, Winterthur Switzerland, Type 9139 AA and serial Number (SN: 5,241,854)) together with a charge amplifier (Kistler LabAmp Type 5167A) recorded the force components’ signals. A personal PC displayed time domain signals collected by the dynamometer. We used DynoWare Type 2845D-03 for data acquisition and processing. The material used for this experiment was an aluminium 7075 billet of 130 mm by 100 mm by 80 mm. Its mechanical properties included a fatigue strength of 159 MPa, ultimate tensile strength of 500 MPa, shear strength of 331 MPa, modulus of elasticity of 1.7 GPa and a hardness of 175 HB. It is also an excellent thermal conductor with a high specific density and machinability. The cutter was a solid, carbide flat-end mill cutter (Type GB/T6117.1-1996; 10 × 10 × 72). The cutter was of diameter 16 and was a four-tooth centre cutting type. The set-up of the experiment is shown in Fig. 2.

Fig. 2
figure 2

Experimental set-up

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2.2 Teaching learning-based optimization algorithm (TLBO)

Many optimization algorithms are parameter-specific. Parameters like mutation rate and selection criteria for genetic algorithm (GA), inertia weight, cognitive and social parameters for particle swarm optimization (PSO) and number of bees for artificial bee colony (ABC) make the application of these algorithm complex due to the rigour of tuning them. TLBO relies on the standard optimization algorithm parameters, that is, population size and the number of iterations [19]; hence, its application is easier and it can converge easily to the global solution. TLBO imitates the learning phenomenon in a class setting. Therefore, it is characterized by a teacher who tries to improve the mean of their learners, in a given subject (s), by imparting knowledge to them. In this algorithm, all the possible solutions (population) to the objective function are analogous to the total number of learners in the class. The learner with the best marks (optimal solution) is considered the teacher and he/she teaches their classmates based on his/her ability. The number of subjects taken are analogous to the number of variables being investigated. Apart from the teaching imparted by the teacher to the learners, the learners can also learn from each other. Thus, a brighter student will share his knowledge with a classmate who knows little [20, 21]. TLBO, therefore, is executed in two phases, viz. (i) Teacher phase and (ii) Learners phase. TLBO execution is as follows [22];

2.2.1 Teacher phase

At jth iteration, it can be assumed that there are ‘p’ learners (size of the population, t = 1,2, … p) taking ‘s’ subjects (design variables), and Ar, j is the average results of the learners in a given subject ‘r’ (r = 1, 2 … s). The algorithm considers the best learner (tbest), with the best results (Ytotal − tbest, j), the class teacher. The algorithm determines the difference between the average of each subject and the corresponding marks of the tutor in each of the subjects as illustrated in Eq. (4).

$${\text{Difference}}\_{\text{mean}}_{r, t, j} = r_{j} \left( {Y_{{r, {\text{tbest}},j}} - T_{{\text{F}}} A_{r, j} } \right)$$
(4)

whereby the results of the teacher are \({Y}_{r, {\text{tbest}}, j}\), \({r}_{j}\) is a random number between 0 and 1 and \({T}_{{\text{F}}}\) is the tutoring factor. \({T}_{{\text{F}}}\) determines the value of the class average value that needs to be changed and it is not an input parameter of TLBO. \({T}_{{\text{F}}}\) value ranges between 1 and 2 and is randomly decided by Eq. (5);

$$T_{{\text{F}}} = {\text{round}}\left[ {1 + {\text{rand}}\left( {0,1} \right)\left\{ {2 - 1} \right\}} \right]$$
(5)

Preliminary experiments by Rao et al. [23] showed that the algorithm worked well when the \({T}_{{\text{F}}}\) range was between 1 and 2, but it worked the best when \({T}_{{\text{F}}}\) was either 1 or 2. Subsequently, to simplify the execution of the algorithm, Rao et al. proposed \({T}_{{\text{F}}}\) to be either 1 or 2, depending on the rounding off of the results of Eq. (5). The existing solutions (results of learners, (Y)) in this phase of TLBO are updated by adding \({{\text{Difference}}\_{\text{mean}}}_{{\text{r}},\mathrm{ t},\mathrm{ j}}\) to give better solutions as shown in Eq. (6).

$$Y^{\prime }_{r, t, j} = Y_{r, t, j} + r_{j} \left( {Y_{{r, {\text{tbest}},j}} - T_{{\text{F}}} M_{r, j} } \right)$$
(6)

whereby \(Y^{\prime }_{r, t, j}\) give the updated solution. If \(Y^{\prime }_{r, t, j}\) gives a better solution, \(Y_{r, t, j}\), is dropped, and \(Y^{\prime }_{r, t, j}\) is adopted for the learning phase. The initial solution (s), \(Y_{r, t, j}\), are obtained randomly based on the variable and problem constraints.

2.2.2 Learners phase

After several iterations in the teachers phase, the best learners in each iteration are grouped into a pool of best learners in the learners phase. In the learners phase, classmates learn from each other through random interaction. A learner will only learn from their classmate if their classmate has more knowledge than them. If there are two students, E and F, such that \(Y^{\prime }_{{{\text{total}}, E, j}} \ne Y^{\prime }_{{{\text{total}}, F, j}}\), then the governing equations in the learner phase are Eqs. (7) and (8);

$$Y^{\prime \prime }_{r, E, j} = Y^{\prime }_{r, E, j} + r_{r,j} \left( {Y^{\prime }_{r, E, j} - Y^{\prime }_{r, F, j} } \right)\;{\text{if}}\;Y^{\prime }_{r, E, j} < Y^{\prime }_{r, F, j}$$
(7)
$$Y^{\prime \prime }_{r, E, j} = Y^{\prime }_{r, E, j} + r_{r,j} \left( {Y^{\prime }_{r, F, j} - Y^{\prime }_{r, E, j} } \right)\;{\text{if}}\;Y^{\prime }_{r, F, j} < Y^{\prime }_{r, E, j}$$
(8)

These two equations are for the minimization problems, and \(Y^{\prime }_{r, E, j}\), is adopted only if its value is better than \(Y^{\prime }_{r, E, j}\). Otherwise, if the problem is a maximization problem, the applicable equations are (9) and (10);

$$Y^{\prime \prime }_{r, E, j} = Y^{\prime }_{r, E, j} + r_{r,j} \left( {Y^{\prime }_{r, E, j} - Y^{\prime }_{r, F, j} } \right)\;{\text{if}}\;Y^{\prime }_{r, F, j} < Y^{\prime }_{r, E, j}$$
(9)
$$Y^{\prime \prime }_{r, E, j} = Y^{\prime }_{r, E, j} + r_{r,j} \left( {Y^{\prime }_{r, F, j} - Y^{\prime }_{r, E, j} } \right)\;{\text{if}}\;Y^{\prime }_{r, E, j} < Y^{\prime }_{r, F, j}$$
(10)

The learners’ phase marks the end of the TLBO algorithm, with the best result from the best learner being the best solution for the objective function. In this paper, the work reported was a minimization problem; hence, Eqs. (7) and (8) were adopted in MATLAB R2013a for TLBO optimization coding. Figure 3 shows the flow chart of the minimization TLBO implemented in this paper.

Fig. 3
figure 3

Flow chart of the minimization TLBO

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3 Results

Table 2 presents the experimental results obtained using DynoWare Type 2845D-03 software. The software was used to preprocess the acquired signals from the dynamometer by detrending and denoising. The numerical values of the force components Fx, Fy and Fz were then extracted from regions within the entire signal that contained the actual cutting signals. The range from which the extraction was conducted was chosen arbitrarily, see Fig. 4. The non-cutting signals represented electrical, milling table linear motion and spindle rotation signals, and were not considered for force components acquisition. Resultant force, Fr, was computed using Eq. (3).

Table 2 Table of results
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Fig. 4
figure 4

Cutting force signal in the time domain

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4 Data analysis and discussion

4.1 Force modelling

The force models in this work were based on Eq. (11).

$$F = cv_{c}^{x} f^{y} a_{p}^{z} a_{r}^{p}$$
(11)

whereby vc, f, ap and ar are the process parameters and c, x, y, z and p, are model exponents that can be determined by any suitable statistical technique. F is force.

Results obtained from the first 16 runs in Table 2 were used to establish instantaneous and resultant force components’ models. The results from the other four runs were used for model verification. The following force components’ models were determined using a multivariable regression technique in MATLAB R2013a.

$$F_{x} = 3413.181v_{c}^{ - 0.17573} f^{0.70714} a_{p}^{0.45701} a_{r}^{0.1705}$$
(12)
$$F_{y} = 3078.507v_{c}^{ - 0.54747} f^{0.9728} a_{p}^{0.59651} a_{r}^{1.6341}$$
(13)
$$F_{z} = 1216.46v_{c}^{ - 0.38129} f^{0.54675} a_{p}^{0.10034} a_{r}^{0.22174}$$
(14)
$$F_{R} = 4370.357v_{c}^{ - 0.23628} f^{0.7502} a_{p}^{0.50304} a_{r}^{0.40094}$$
(15)

4.2 Significance of the force components’ models based on ANOVA

Analysis of variance (ANOVA) determined adequacy of the models in prediction. F value test determined the significance of the models. According to this test, a regression equation is significant if only \(F>{{{F}}}_{{m }({{n}}-{{m}}-1)}\) at a significance level of 0.05. On the other hand, if the significance level was at 0.01 and \(F>{F}_{m, (n-m-1)}\), then the regression equation was considered very significant. In this case, m referred to the number of process parameters under investigation, whereas n was the total number of experimental runs. In this experiment, the value of m was four, whereas n was 16. From the distribution table of F values, \({F}_{m, (n-m-1)}={F}_{4, (16-4-1)}={F}_{4, 11}=3.36\) at 0.05 significance level and \({F}_{4, 11}=5.67\) at 0.01 significance level. The results of these significance tests on Eqs. (12), (13) and (14) were tabulated (see Tables 3 and 4). The table shows that the force components in the three directions were very significant based on comparing their F values and those obtained from the F distribution table. The three models were, therefore, reliable for prediction. Feed had the highest impact on force components. The radial depth of cut, axial depth of cut and cutting speed followed in that order. Analysis of variance for resultant force, as shown in Table 3, showed similar results.

Table 3 ANOVA for the resultant force
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Table 4 Force components’ ANOVA
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The Pearson correlation practical significance test substantiated the ANOVA results on the significance of the process parameters on force generation, see Table 5.

Table 5 Pearson correlation significance results
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4.3 Effects of process parameters on force components

In order to study the effects of each process parameter on the force components (using the models above), the rest of the parameters were held at constant values. The values were:

vc = 80 m/min, f = 0.02 mm/rev, ap = 2 mm and ar = 3 mm.

4.3.1 Main effect of cutting velocity on force components

The correlation of cutting velocity and force components was studied with the values of axial depth of cut, feed and radial depth of cut held at 2 mm, 0.02 mm/rev and 3 mm, respectively.

Figure 5 clearly shows that force components decreased whenever there was an increment in cutting velocity. The increment in cutting velocity led to increased temperature levels in the cutting zone. The increment in temperature lowered the plastic deformation resistance coefficient, making the metal soft and easy to cut. The cutter met little resistance as it cut through the aluminium billet, hence the low magnitudes of force components. This decrement in force magnitude due to an increment in cutting velocity was more pronounced in Fx and Fy than in Fz.

Fig. 5
figure 5

Force component (N) versus cutting velocity (m/min)

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4.3.2 Main effect of feed on force components

Figure 6 illustrates the effects of feed. An increase in feed led to an increment in force components’ magnitude. An increment in feed led to an increment in chip thickness per tooth. This increment in chip thickness meant a subsequent increment in plastic deformation resistance coefficients and friction coefficients. As a result, force components increased. From the figure, it can be seen that the increment in force components is more pronounced in Fx and Fy. The cumulative effect is, however, gentle in Fz.

Fig. 6
figure 6

Force component (N) versus feed (mm/rev)

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4.3.3 Main effect of axial depth of cut on force components

It could be inferred from Fig. 7 that increment in the axial depth of cut led to an increment in magnitude of the components of force in the three directions. With the increase in the axial depth of cut, the axial cutting action along the cutter also increased due to the increased cutter–workpiece interaction (an increase of chips along the flutes). This led to an increment in deformation resistance and friction resistance. Cutter’s torsional deformation also increased due to the increased torque needed to facilitate the cutting process. The increment was witnessed in the three components of the resultant cutting force but was quite gentle on the axial component, Fz.

Fig. 7
figure 7

Force component (N) versus axial depth of cut (mm)

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4.3.4 Main effects of radial depth of cut on force components

An increment in radial depth of cut led to an increment in Fx, Fy and Fz, as shown in Fig. 8. Increasing the radial depth of cut led to an increment in the material feed per tooth. It also increased uncut chip thickness, subsequently leading to an increment in both friction and plastic deformation resistance coefficients.

Fig. 8
figure 8

Force component (N) versus radial depth of cut (mm)

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4.4 Two factors’ interaction effects on force components

This section focused on the simultaneous impact of any of the two factors on force components. The effects were studied in the environment of MATLAB R2013a using the models established, and the findings were presented on 3D surface plots.

4.4.1 Interaction effects of cutting velocity and feed on force components

Figure 9 illustrates the interaction effects of cutting velocity and feed on force components. The axial and radial depth of the cut were kept at 2 mm and 3 mm, respectively. In this study, force components would decrease with increased cutting velocity. It, however, had a direct proportionality with feed. As mentioned in the previous effect analysis, an increment in cutting velocity led to temperature rise and subsequent softening of the aluminium billet.

Fig. 9
figure 9

Effects of cutting velocity (mm/min) and feed (mm/rev) on force components

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On the other hand, an increment in feed rate meant an increment in the tool’s feed impulse and chip thickness, hence raising the component forces. Therefore, according to Fig. 9, an optimal force component could be achieved by raising the cutting velocity and lowering the feed to reasonable values, with the material removal rate kept constant.

4.4.2 Interaction effects of cutting velocity and axial depth of cut on force components

Figure 10 illustrates the interaction impact of cutting velocity and axial depth of cut on force components. Here the constant values of feed and radial depth of cut were kept at 0.02 mm/rev and 3 mm, respectively. As mentioned during the previous analysis, cutting velocity and force components have inverse proportionality in their correlation. On the contrary, the axial depth of cut and force components have direct proportionality. Increasing the axial depth of the cut meant an increment in cutter–workpiece contact, increasing both the plastic deformation resistance coefficient and friction coefficient hence increment in force components. In order to keep the cutting force low when it comes to the industrial application (to avoid high chatter levels, high rates of tool wear, high power consumption and poor surface finish, amongst others), the axial depth of cut could be reasonably low when the cutting velocity is reasonably high at a constant material removal rate.

Fig. 10
figure 10

Effects of cutting velocity (m/min) and axial depth of cut (mm) on force components

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4.4.3 Interaction effects of cutting velocity and radial depth of cut on force components

This analysis kept the feed and depth of cut values at 0.02 mm/rev and 2 mm, respectively. As mentioned in the previous effect analysis, an increment in the radial depth of cut implied that there was more material to cut hence an increment in force components, as shown in Fig. 11. This figure also confirmed the inverse proportionality between cutting velocity and cutting force. It could be inferred from Fig. 11 that when the material removal rate was kept constant, an increment in cutting velocity and a decrement in the radial depth of cut would lead to a decrement in force components.

Fig. 11
figure 11

Effects of cutting velocity (m/min) and radial depth of cut (mm) on force components

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4.4.4 Interaction effects of feed and axial depth of cut on force components

The cutting velocity and radial depth of cut were kept at 80 m/min and 3 mm, respectively. Increasing the feed and axial depth of cut led to force components’ magnitudes increment (due to increment in material to cut in the direction of feed and axially, respectively), as shown in Fig. 12. The repercussion of this was an increment in coefficients of friction and plastic deformation resistance hence the increment in force components’ magnitude. For a low force magnitude that would ensure low power consumption and long tool life, these two parameters must be reasonably low during any milling process of aluminium 7075 metal.

Fig. 12
figure 12

Effects of feed (mm/rev) and axial depth of cut (mm) on force components

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4.4.5 Interaction effects of feed and radial depth of cut on force components

In this analysis, cutting velocity and axial depth of cut were kept at 80 mm/min and 2 mm, respectively. The inferences made in Fig. 13 regarding feed and radial depth of cut were almost identical to those made in Fig. 12. An increment in radial depth of cut led to an increment in the number of teeth engaged in the material removal process. The consequence was an increment in force components’ magnitude due to an increment in cutter–workpiece interaction. Figure 13 shows that feed and force components have a direct proportionality. Thus, with a predefined material removal rate, an increment in these two parameters would have resulted in an increment in the force components’ magnitude.

Fig. 13
figure 13

Interaction effect of feed (mm/rev) and radial depth of cut (mm) on force components

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4.4.6 Interaction effect of axial depth of cut and radial depth of cut on force components

Cutting velocity and feed were kept at 80 m/min and 0.02 mm/rev, respectively. Figure 14 shows that a decrement in force components’ magnitudes was due to a decrement in both radial and axial depth of cut. The reduction of these two factors meant a decrease in cutter–workpiece interaction, leading to decreased friction and deformation resistance. It, therefore, meant that when the material removal rate was kept constant, these two factors would have led to an increment in force components. This observation was similar to the one made by [24].

Fig. 14
figure 14

Interaction effect of axial depth of cut (mm) and radial depth of cut (mm) on force components

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4.5 Verification of resultant force model

Verification of Eq. (15)’s accuracy in force prediction was carried out. Resultant force was considered because it gives the comprehensive resistance of workpiece to cutter intrusion. Four more runs (17, 18, 19 and 20), shown in Table 2, were used for the verification process. The parameter values in the four runs did not resemble any combinations in the 16 runs used to acquire the models. Figure 15 shows the results of the resultant force verification analysis.

Fig. 15
figure 15

Verification plot for resultant force’s model

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As shown in Fig. 15, the predicted and actual values obtained from the experiment were not the same. Their differences were, however, minimal, as can be seen in the percentage error plot for the resultant cutting force. The calculation of percentage error was as shown in Eq. (16);

$$\% {\text{residual}} = \left( {\frac{{{\text{Predicted}}\;{\text{Force}} - {\text{Actual}}\;{\text{Force}}}}{{{\text{Actual}}\;{\text{Force}}}}} \right) \times 100$$
(16)

Fr (resultant force model) had only one residual error above 8%. The rest of its residuals were below 5%, as shown in Fig. 15. These minimal errors verified the model’s reliability in predicting cutting force for feed optimization. The residuals could have been minimum if the inherent experimental errors were minimized or eliminated.

4.6 TLBO optimization results

TLBO was deployed to get the optimal cutting parameters. In the execution of TLBO in MATLAB R2013a, a population of 100 learners and 1000 iterations were considered. Equation (15) was considered the objective function, and the algorithm was implemented in MATLAB R2013a. The resultant optimum combination of process parameters were as follows: feed was 0.02 mm/rev, axial depth of cut was 1.5 mm, radial depth of cut was 2 mm and the cutting velocity was 125 m/min. The optimal resultant force turned out to be 120.1619 N. Therefore, by milling Al 7075 (under the same machining environment and parameter range) with these optimal process and response parameters, there will be less tool wear, less power consumption and probably a quality surface finish.

4.7 Optimization of feed

Based on the optimal resultant force (obtained by TLBO), the initial feed and actual force, feed can be optimized using Eq. (17). Figure 16 compares the initial feed in Table 2 with the optimized feed (calculated by Eq. (17)) and the optimal feed (obtained through TLBO optimization).

$$f_{O} = f_{i} \times \frac{{F_{{r\left( {{\text{optimal}}} \right)}} }}{{F_{{{\text{actual}}}} }} = f_{i} \times \frac{120.1619}{{F_{{r\left( {{\text{actual}}} \right)}} }} = 120.1619\frac{{f_{i} }}{{F_{{\text{r}}} }}$$
(17)
Fig. 16
figure 16

Feed optimization

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4.8 Validating the optimal effect of optimized feed on force

This section depicts the effects of optimized feed on force, as an affirmative to the correlations shown by the force versus feed plots in the previous sections. Based on initial feeds in validation experiments 17, 18, 19 and 20, the optimized feeds were calculated using Eq. (17). The optimized feeds were then incorporated into Eq. (15) to get the predicted optimized force values. Table 6 and Fig. 17 compare the experimental force, predicted optimized force and predicted force (based on the initial feed). From the table and the figure, it is clear that the optimized feeds resulted to lower magnitudes of forces.

Table 6 Comparative analysis of resultant force values
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Fig. 17
figure 17

Comparative analysis of resultant force values

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5 Conclusion

This work conducted a comprehensive review of the forces generated during milling, with emphasis on the force components and the resultant forces. A comprehensive review of feed optimization and the application of TLBO in determining the optimum cutting parameters was also conducted. The following conclusions were made:

  • Through ANOVA and Pearson correlation analysis, it was found that feed had the highest significance in force generation and its impact on the magnitude of force components was directly proportional, as shown in Figs. 6, 9, 12 and 13. Therefore, optimization of feed (within the investigated parameter range) in the milling process of aluminium 7075 will optimize force components (as shown in Fig. 17 and Table 6), consequently, reducing: tool and workpiece vibration, power consumption, tool wear and jerks. Lower magnitudes of force are likely to reduce residual stresses on the resultant Al 7075 products, hence upholding their high fatigue strength for successful applications in structural components.

  • The developed force components models, Eq. (12), (13) and (14) as well as the resultant force model (Eq. (15)), can be used for further prediction of forces when milling Al 7075 under similar milling conditions.

  • The optimum cutting parameters, as determined by TLBO, were: 0.02 mm/rev, 1.5 mm, 2 mm and 125 m/min for feed, depth of cut, radial depth of cut and cutting velocity, respectively. The optimal resultant force was 120.1619 N and it was used as a constraint parameter for feed optimization, as demonstrated by Eq. (17); hence, under similar milling conditions and within the same parametric range, Eq. (17) can be used to optimize the feed when milling Al 7075.

The findings in this work can be used to give technical support on any shop floor during CNC milling of Al 7075. The findings can also be used as a basis for further research on CNC milling processes.