1 Introduction

A titanium alloy matrix is the titanium metal matrix composite (TiMMC) foundation. It is reinforced with fibers, particles, or whiskers to give them extraordinary qualities, including high specific strength, high specific modulus, high-temperature resistance, and potential weight reduction. TiMMCs have replaced Ni-based alloys in aerospace applications for 30 years [1, 2]. Waterjet machining is one of the machining techniques that can process hard materials and delicate components with great precision. This makes it worthwhile in automobiles, renewable vitality, aviation, therapeutic gadgets, etc. [3,4,5,6].

The development of exceptional, cutting-edge technology for abrasive water jet machining (AWJM) is receiving much attention. As with any manufacturing process, there are downsides to the creation of abrasive jets. These include secondary erosive wear, abrasive fouling, taper profile, jet diffusion, inappropriate kerf geometry, and so on [7, 8]. Water fly mass, water jet angle of attack, angle of contact, feed rate, abrasive particle density, machining time, abrasive particle mass flow rate, and distance are the main determinants of abrasive jet machining [9, 10].

2 Related works

The rate of material rejection (MRR), roughness of the cut surface (Ra), kerf width, and kerf point are frequently used to measure the performance of machining operations. The performance of waterjet penetration in AWJM in aluminium–silicon carbide composites is primarily influenced by the speed of navigation and waterjet pressure [11, 12]. As a result of being machined, TiMMCs likely emit particles and severely wear down the tools. During the machining of TiMMCs in various lubricated modes, a recent study examined dust formation at the micro- and nanoscale. Although cutting speed has little effect on the specific surface concentration and mass concentration of ultrafine particles, it significantly impacts the generation of fine particles (2.5 M aerodynamic particle diameter or smaller) (size range of 0.1 M). Turning with lubricated tools reduces particle emission, as predicted, and coated inserts with high flow rates (300 ml/min) emit less ultrafine particles than their uncoated counterparts. [13, 14]. The roughness of the ground can also be affected by the concentration of reinforcing particles in the composite. The roughness of MMCs’ surface is influenced by the number of reinforcing particles used [15,16,17]. The roughness does, however, lessen with TiMMCs. The matrix and particle composition likely contribute to the observed asymmetry in the roughness trend. Titanium carbide (TiC) particles are more durable than silicon carbide (SiC) particles, while titanium alloy (Ti6 Al4V) is more malleable than an aluminum alloy (Al alloy). Grooves and clusters can’t form on TiC particles’ surfaces. The rugged character of TiC particles encourages detachment rather than breaking, which helps to reduce roughness [18,19,20,21,22,23]—an analysis of how ANN is used in traditional machining to forecast the roughness of the cut surface.

According to the literature that is currently accessible, ANN’s use for atypical machining processes is underdeveloped, and its application to AWJM is relatively limited. ANN provides a better method for estimating the process parameters based on the selection of the most crucial and advantageous values of the parameters, as shown by reports on the prediction of roughness of the cut surface and jet velocity for a given pressure, rate of abrasive mass stream, and thickness of the target material. With the help of numerical models, Yang’s reports can be used to assess the AWJM technique’s cutting capacity for a range of engineering materials [24,25,26,27,28]. The report on granite machining includes a study of the outcomes from empirical modeling and parameter optimization utilizing a hybrid strategy. The effectiveness of abrasive waterjet machining was measured using ANN. Neural networks provide an overview of unconventional machining techniques for abrasive waterjet cutting. The application of micro-channel property prediction at extremely high navigation speeds is covered in the current work. The neural network systems mentioned are versatile and can be used in different AWJM applications to enhance machining performance and efficiency [29, 30]. The utilization of high-pressure waterjet help has substantially enhanced the longevity of tools [31]. In this paper, experiments are carried out with AWJM to improve the tool’s life.

3 Proposed methodology

The AWJM employed in the research projects is capable of producing a maximum working pressure of 400 MPa and a maximum pump capacity of 20 hp. The water jet travels at a high velocity of 900 m/s, which is a 3 Mach number. Figure 1a shows the experimental setup, and Fig. 1b shows the machined components.

Fig. 1
figure 1

a Experimental setup of abrasive water jet machining. b Machined components

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Sigma Aldrich purchases titanium, whose properties are shown in Table 1. The material being utilized for the actual product is a composite of TiMn metal hydrides. The measured dimensions of an 8-mm-thick TiMMC are 250 mm. The AWJM method produced a clean 30-mm-long through-cut.

Table 1 Properties of titanium
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Figure 2 shows four AWJM process control parameters, including water fly mass, distance between water and object, steam rate, and navigation speed, after a meticulous selection technique.

Fig. 2
figure 2

ANN model

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The impact angle and focusing length have been held steady at 90 degrees and 0.76 mm throughout the machining process. The abrasive particle is a TiMMC with a mesh size of 80 (or 165 mm in length). The cutting head has been moved with a precision of 0.025 mm, and the nozzle has a diameter of 0.70 mm. MARR, SUR, KEW, and KEA are the four performance indicators chosen to assess the AWJM process’s machining efficacy on TiMMC composites. Using Eq. (1), we can determine that MRR equals the material removed from the workpiece in one machining cycle. Using the roughness of the cut surface tester, we determined the SR value by averaging the results of three separate runs on the top, middle, and bottom of the machined surface of TiMMC composites. Top kerf width (TKEW) and bottom kerf width (BKEW) were measured using a 100X optical microscope at three positions along the cut length on both the top and bottom surfaces. After determining the sum of the two, we have the KEW. The Kerf taper, often known as KEA (ϴ), is an essential measure of the precision of machined parts, which has been calculated using Eq. (2).

$$ MARR = KEW \times TUS \times TH $$
(1)
$$ \theta = \tan^{ - 1} \left( {\frac{{TH_{KEW} - B_{KEW} }}{2TH}} \right) $$
(2)

3.1 An artificial neural fuzzy logic algorithm (ANFLA)

Developing a matrix of choices containing n criteria (responses) and alternative answers is the first stage in the ANFLA procedure. The methodology starts with experimental trials of the ANFLA. Figure 3 provides a comprehensive breakdown of the ANFLA application procedure, detailing each stage.

Fig. 3
figure 3

A proposed ANFLA application process framework

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Step 1: Establishing a baseline for the matrix of choices.

Eliminating variability and making the dataset dimensionless requires normalizing the elements of the matrix of choices. Specifically, we need to put each component into a range between 0 and 1 that is consistent with the remainder of the range.

The following equations can be used, with care given to the sort of quality feature being analyzed: For those of you who believe the bigger, the better!

If the bigger-is-better kind.

$$ c_{ij}^{*} = \left( {c_{ij} - \min c_{ij} } \right)/ (\max c_{ij} - \min c_{ij} ) $$
(3)

where i = 1,2, 3….m; j = 1,2, 3…n.

To get the best results, go for the smallest size possible.

$$ c_{ij}^{*} = \left( {maxc_{ij} - c_{ij} } \right)/ (\max c_{ij} - \min c_{ij} ) $$
(4)

\({{c}_{ij}, c}_{ij}^{*}\) are the measured and normalized values for the ith alternative for the jth criterion.

Step 2: Calculation of relevant FRC (Fuzzy Relational Coefficient) values.

We can derive the FRC values from the normalized data for all responses by Eq. (5). They represent the relationship between the optimal (target) and actual normalized results.

$$ \rho_{ij} = \left( {\gamma_{min} + \beta \gamma_{max} } \right)/\left( {\gamma_{ij}^{0} + \beta \gamma_{max} } \right) $$
(5)

where \({\gamma }_{ij}^{0}\) is the variation between \({c}_{ij}^{0}\) (ideal sequence) and \({c}_{ij}^{*}\). Contrarily, β is the differentiating coefficient, and it can take on values between 0 and 1, with 0.5 being the most optimal. To a large extent, it determines whether the range for FRC values widens or narrows. In addition, \({\gamma }_{min}\) and \({\gamma }_{max}\) denote the global minimum and maximum values in a given data set. A larger FRC number indicates a closer potential solution.

Step 3: Determination of FRG values via calculation.

Now, we compute the FRG values by averaging the FRCs for each criterion concerning each alternative, and the results are shown in Fig. 4.

$$ F_{i} = \frac{1}{n} \mathop \sum \limits_{j = 1}^{n} \rho_{ij} $$
(6)
Fig. 4
figure 4

Normalized values of the experimental results

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For each given issue, the solution with the highest FRG value stands out as superior to all others. The experimental information needed to determine the FRC and FRG is shown in Fig. 5. The experimental findings are normalised to a range of 0–1 using either Eqs. (3) or (4), depending on the quality feature being addressed. Figure 5 displays the results of calculating the FRC and FRG for each experimental trial using the normalised data and the corresponding Eqs. (5) and (6). The third experiment with the highest FRG value was chosen as the optimal one. However, a fuzzy logic method boosts the solution’s superiority and lessens the experimental findings’ uncertainty and fuzziness.

Fig. 5
figure 5

Computation of FRC and FRG value

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3.2 Modelling with fuzzy-based rules

The Fuzzy Set Theory was developed primarily to resolve disagreements brought on by erroneous data when trying to reach a consensus on a course of action. In this paper, we combine ANFLA with fuzzy logic to remove the doubts that arise while weighing the relative merits of higher and smaller values for quality indicators. Fuzzy membership functions translate qualitative phrases like “low,” “mid,” “high,” etc., into quantitative values for application in fuzzy set theory. A fuzzy set can be considered a collection of membership functions, where each membership function maps an element x into a set of objects, X, and the resulting mapping is a natural integer R between 0 and 1. Using fuzzy logic, the ambiguity of neuro theory can be eliminated. In other words, a fuzzy multi-performance instrument can be made using this technique, also known as neuro-fuzzy logic. The fuzziness near the center method is typically employed to convert the multi-response fuzzy value to the crisp value of OFRG.

$$ OFRG = \frac{{\sum F\mu_{{F_{0} }} \left( F \right)}}{{\sum \mu_{{F_{0} }} \left( F \right)}} $$
(7)

After, all uncertainties and ambiguities in the empirically observed data have been removed, the OFRG values can be rated from best to worst, with the best choice being selected. Ultrasonic-assisted electrical release machining, warm penetrating, and many others can all benefit from the ANFLA method combined with fuzzy logic, as it is a straightforward and effective strategy for addressing complex multi-criteria problems involving the identification of optimal parametric combinations. To find the perfect spot for all the variables in the AWJM process, this work uses a multi-response optimization method, namely an artificial neural-fuzzy approach.

3.3 Artificial neural fuzzy approach

This article describes how an artificial neural-fuzzy system was used to determine the optimal combination of processing parameters for AWJM on TiMMCs. An analysis is carried out to isolate the effect of each process parameter on the OFRG results. Table 2 shows the calculated OFRG value of the response.

Table 2 Response table for the calculated OFRG value
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In addition, the surface plots are created based on a regression equation that links the input parameters to the estimated OFRG values.

The impact of the AWJM process parameters on the results is also shown through interaction plots, as shown in Fig. 6.

Fig. 6
figure 6

Interaction plots of the calculated OFRG value

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Table 3 displays the results of variance analyses performed on the calculated OFRG values. A factor’s p-value must be less than or equal to 0.05 (p 0.05) to be considered statistically significant. With 60.8% of the variance in OFRG values attributable to the traversal speed, it is evident that this is the most crucial control parameter.

Table 3 Analysis results of OFRG values
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3.4 Validation test

To verify whether the Neurofuzzy method-based approach is successful in determining the optimal parameters for the AWJM process under discussion, we calculate the predicted OFRG (OFRGp) value for this combination using Eq. (8).

$$ OFRG_{p} = OFRG_{m} + \mathop \sum \limits_{i = 1}^{n} (\overline{{OFRG_{i} }} - OFRG_{m} ) $$
(8)

OFRGi denotes the ideal level of the examined process parameters. The mean OFRG value is OFRGm, where n is the total number of experimental trial runs, and OFRGi is the mean OFRG value.

OFRG is higher than in Fig. 7, established for the first machining condition. This step involves running a test to verify the derived optimal parametric blend.

Fig. 7
figure 7

Optimization results using OFRG

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Table 4 shows how the measured responses differ between the default and best parameter values. Using the data in the table above, we can see that increasing the parameters of the water fly mass to 165 MPa, the distance from water and object to 3 mm, the steam rate to 200 g/min, and the speed of navigation to 50 mm/min results in increases of 67.8% in MARR, 3.61% in SUR, 30.24% in KEW, and 86.86% in KEA. The proposed OFRG technique has improved 66.91% compared to the first machining.

Table 4 Comparison of response values
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4 Conclusion

This study analyses an AWJM method for machining OFRG composites using water fly mass, distance from water and object, steam rate, and navigation speed as inputs and MARR, SUR, KEW, and KEA as outputs. The following conclusions are drawn. Navigation speed is the most influential process parameter, contributing 60.98% of OFRG values, followed by water fly mass.

  • There is a maximum improvement of 86.86% in MARR, 3.61 times improvement in SUR, 30 times improvement in KEW, and 3.61 times improvement in KEA when using the ideal parametric mix.

  • Fuzzy systems demand less information to analyze an unknown process’ behavior while delivering an unbiased estimate. Because fuzzy neural model generation requires minimal data, it may need to describe system dynamics fully. Fuzzy logic is used to improve the fuzzy system’s capabilities. Other evolutionary algorithms often determine optimal parametric mixtures that aren’t fixable in AWJM. Tuning parameters (Speed) affect the optimization performance of these methods.

  • AWJM can machine thin, non-corrosive, difficult-to-cut materials. Therefore, it’s used in manufacturing, coal mining, automotive, and aerospace industries. Thus, applying an artificial neuro-fuzzy method-based approach with a solid mathematical basis can enable process engineers to derive the ideal parametric mix for the AWJM process while exploring its full cutting potential.