Research Progress on Optimization Method of Magnetic Grinding Process for Inner Surface of Aircraft Engine Bend Pipe

Abstract

The level of magnetic grinding technology determines the accuracy and efficiency of magnetic grinding on the inner surface of aircraft engine bend pipes. This article analyzes the optimization methods of magnetic grinding process parameters for the inner surface of aircraft engine bent pipes, such as the multiple regression prediction method, the response surface method, and the grey relational analysis method. It is pointed out that the current optimization methods for magnetic grinding technology on the inner surface of aircraft engine bent pipes do not consider the nonlinear characteristics between various grinding process parameters, resulting in defects such as low precision and efficiency of magnetic particle grinding technology. An optimization approach was proposed to accurately predict the optimal magnetic grinding process parameters for the inner surface of aircraft engine bent pipes, establish a nonlinear mapping relationship that reflects the roughness of the inner surface of the bent pipe and the main process parameters, optimize the BP neural network model based on the genetic algorithm, design magnetic grinding experiments on the inner surface of aircraft engine bend pipes, and explore the magnetic grinding process that is beneficial for improving the accuracy and efficiency of magnetic grinding on the inner surface of aircraft engine bend pipes. It can achieve efficient and accurate prediction of magnetic grinding of the inner surface of aircraft engine bend pipes. It provides a basis for the manufacturing and maintenance of high-precision aircraft engine bend pipes.

1. Introduction

There are hundreds of conduits outside the aircraft engine, which are designed as various curved pipes with different radii of curvature due to the limited space. In addition, the external conduits of the aircraft engine are the carriers of fuel, lubrication, oil, and gas mixtures, connecting various accessories of the aircraft engine. They have high reliability requirements for operation. Due to the surface defects and uneven surface quality caused by the existing cold bending process, gradient changes in the flow field, velocity field, and pressure field are generated, causing unstable operation of aircraft engines and resulting in vibration and noise. Under actual working conditions, the uneven distribution of pressure load at the bend of the conduit exacerbates the stress concentration of the bend. The complex working environment can also lead to stress corrosion cracking, which is one of the main causes of failure and even major accidents, leading to fatigue fracture of the conduit (as shown in Figure 1) [1]. To ensure the reliability of high-performance aircraft engines and improve the quality of the inner surface of aircraft engine bends, it is necessary to grind the inner surface of the bends.

Common surface finishing techniques for aircraft engine bend pipes include traditional manual mechanical polishing, electrochemical polishing, and shot peening. Aircraft engine bend pipes are mostly space irregular pipes, and the existing polishing process is outdated, greatly limiting the development of high-performance aircraft engine manufacturing and the repair industry. Magnetic grinding is the process of placing a certain amount of magnetic abrasive particles with magnetic permeability and processing capability between the magnetic pole head and the workpiece with a reasonable processing gap. Through a magnetic field-generating device, the magnetic abrasive particles exert a certain amount of force on the workpiece, and complex relative motion is generated between the workpiece and the abrasive particles through a certain form of motion. Under the action of force and relative motion, magnetic abrasive particles have a comprehensive effect on the surface of the workpiece, including microgrinding, squeezing, abrasive wear, and electrochemical wear [2,3]. By changing the geometric features of the workpiece surface and improving the physical and mechanical properties of the surface, the surface quality of the workpiece can be enhanced, thus achieving the goal of improving the performance and lifespan of the workpiece. Magnetic particle grinding technology can effectively solve the problem of surface finishing of bent pipes and also ensure the cleanliness of the inner surface of bent pipes [4]. It has advantages such as no need for specialized processing equipment, high precision, good flexibility and imitation, and easy automation. The research on the application of magnetic particle grinding technology has just begun. Efficient and high-precision magnetic particle grinding technology has become a bottleneck in the manufacturing and repair of high-performance aircraft engines [5]. In order to improve the service life of aviation engine bend pipes, reduce the occurrence of accidents, and complete the manufacturing and repair tasks of high-performance aviation engines faster and better, there is an urgent need for in-depth research on the precision magnetic particle grinding process for the curved inner surface of aircraft engines to develop a comprehensive high-precision magnetic particle grinding process for the inner surface of curved pipes in high-performance aircraft engines [6,7].

Since the 1970s and 1980s, people have been developing experimental equipment to study the principle of magnetic powder grinding on the inner surface of bent pipes. Extensive research has been conducted on the impact of major influencing factors such as magnetic field strength, magnet distribution, and relative motion between the workpiece and the abrasive on grinding quality [8,9]. Under the action of a strong external magnetic field, the magnetic abrasive inside the pipe is arranged along the direction of the magnetic field lines and adsorbed on the inner surface of the bent pipe to form an abrasive brush. It applies a certain pressure to the surface of the workpiece [10,11]. The rotating magnetic field drives the magnetic abrasive particles inside the bent pipe to rotate and roll, generating relative motion and friction with the stationary inner surface of the bent pipe. To further improve the processing efficiency of magnetic particle grinding, auxiliary magnetic poles are added inside the bent pipe to drive the magnetic abrasive material inside the pipe to grind the inner surface [12,13]. Introducing the finite element method into the theoretical calculation of magnetic pole mechanisms can optimize the magnetic particle grinding mechanism and process [14,15]. In recent years, many scholars have applied the theory of magnetic grinding to study the influence of process parameters on the accuracy and efficiency of magnetic grinding of the inner surface of aircraft engine bent pipes [16,17].

Advanced methods such as multiple regression, the response surface methodology, and grey relational analysis were used for process parameter optimization research [18,19]. For example, Chaudhari Ashokkumar R and Judah Kesarabhai B conducted magnetic grinding experiments using an L25 orthogonal table design. The effects of process parameters such as the work gap, rotational speed, mixing ratio, and mesh number on surface roughness (SR) and material removal (MR) were studied [20,21]. Yu Zhenghao et al. studied the effects of process parameters on reducing surface roughness (Ra) and improving microsurface morphology [21,22]. HENG L and GAO Y conducted similar studies [23,24]. Han Bing et al. used grey correlation theory and the response surface methodology to analyze the factors affecting the processing effect of magnetic grinding technology. They conducted preliminary research on process parameters under multiple performance indicators. Optimized process parameters were obtained using multiple and individual performance indicators [25,26,27].

There are many process factors that affect the surface quality of magnetic grinding tubes, such as magnetic pole speed, grinding time, machining gaps, magnetic pole movement speed, etc. Therefore, it is necessary to analyze the influence of different nonlinear process parameters on magnetic grinding accuracy and study the optimization methods of process parameters to improve the quality of magnetic grinding of bent pipes [28,29]. At present, the optimization of magnetic grinding technology includes methods such as multiple regression prediction, the response surface methodology, and grey relational analysis. They do not take into account the nonlinear characteristics of the mutual influence between various process parameters. It is impossible to accurately analyze their impact on the accuracy and efficiency of magnetic powder grinding, nor can they predict the optimal combination of process parameters and surface roughness [30,31]. A BP neural network not only has strong fault tolerance but also extraordinary nonlinear mapping ability. By establishing a BP neural network prediction model based on genetic algorithm optimization, the optimal magnetic grinding process parameters for the inner surface of aircraft engine bend pipes can be accurately predicted.

Based on industrial robots and industrial cameras, we optimize the calculation of pose and trajectory for magnetic particle grinding of curved pipes in aircraft engines, design magnetic grinding experiments on the inner surface of aircraft engine bend pipes, establish a BP neural network prediction model based on genetic algorithm optimization through deep learning of experimental data, and use the predicted optimal grinding process parameters to grind the inner surface of the aircraft engine bend pipe in order to verify the accuracy of the prediction model. It can solve the problem of inaccurate quality control in magnetic grinding, improve grinding efficiency, and enrich the research content on the precision of the magnetic grinding mechanism, making it beneficial for improving the accurate and efficient prediction ability of BP neural network prediction models optimized based on genetic algorithms. Efficient and accurate prediction of magnetic grinding of the inner surface of aircraft engine bends can be achieved [32,33,34,35].

2. Materials and Methods

Based on industrial robots and industrial cameras (Anshan, China), we optimize the calculation of the pose and trajectory for magnetic particle grinding of curved pipes in aircraft engines. The device structure (Anshan, China) for grinding the inner surface of the bent pipe of the magnetic particle grinding engine used in the experiment is shown in Figure 2. The main body consists of a multi-degree-of-freedom robotic arm and a magnetic particle grinding device. The point set collected for calibration is performed using the Matlab 7.0 software (Changsha, China) to process cubic spline interpolation and linear interpolation operations. The aviation engine bend pipe is manufactured and provided by ChengFei Group. When processing the bent pipe, we fix it on the workbench and place a magnetic abrasive and an appropriate amount of oily abrasive solution inside the pipe. The magnetic abrasive sizes used in the experiment are 150 (100 mesh), 178 (80 mesh), and 250 μm (60 mesh). The robotic arm drives the grinding device to move back and forth along the axis of the pipeline. The reciprocating speed adjustment accuracy of the robotic arm is 10 mm/min. The test piece is a TC₄ pipe with dimensions of 20 mm (outside diameter) × 18 mm (inside diameter) × 400 mm.

Table 1. Chemical composition of the TC₄ titanium alloy tube (wt.%) [9].

Ti	Fe	C	N	H	O	Al	V
margin	≤0.30	0.10	0.05	0.015	0.20	5.5~6.8	3.5~4.5

and

Table 2. Physical and mechanical properties of the TC₄ titanium alloy tube.

Density g/cm3	Tensile Strength σb/MPa	Residual Stress σr/MPa	Elongation δs (%)	Reduction in Area ψ (%)
4.5	≥895	≥825	≥10	≥25

show the main chemical composition and physical and mechanical properties of the TC₄ titanium alloy material for slender tubes. We use NdFeB (N35) permanent magnets with dimensions of 15 mm × 15 mm × 10 mm. The magnetic gathering device is made of carbon steel with dimensions of 15 mm × 15 mm × 10 mm and a taper of 25°. We use the Ansoft Maxwell 11 software (Anshan, China) to analyze the magnetic field lines and magnetic induction intensity of magnetic poles. We use the oil-based honing oil of Laolian SR-991 as the grinding fluid [9,36,37].

We use the JB-8E surface roughness measuring instrument to measure the axis direction along the TC4 pipe fittings. We measure the inner surface of the pipe fittings before and after the experiment. The resolution of the roughness profile is 0.001 µm. The surface microstructure was observed using a VHX-500F depth-of-field microscope. We use the HV-1000 microhardness tester to test the Vickers hardness of the workpiece surface [9,38,39].

We use the JB-8E stylus roughness measuring instrument to measure the surface roughness value of the sample. The sampling length of the sample is set to 10 mm. We measure the length at different positions on the surface of the sample 7 times, and then take the average of the results [10,40,41,42].

The ultra-deep 3D microscope can observe the changes in surface texture and geometric features of the sample before and after processing. It can more intuitively and accurately evaluate the surface quality of the processed sample. The maximum magnetic induction intensity is generated when the taper of the magnetic pole head is 60 degrees [10,43,44,45].

2.1. The Magnetic Grinding Process Based on a Multiple Regression Prediction Model

2.1.1. The Design of Experiments

The material of the slender tube is the TC4 titanium alloy. The experiment uses 3 factor and a 4-level L16(43) orthogonal table [14,15,16]. The orthogonal experimental design method arranges multi-factor experiments through orthogonal tables. Each level of each factor can be balanced among all levels of other factors. It can effectively analyze the impact of multiple factors on experimental results. Its advantages include a small number of experiments, easy data analysis, and high experimental effectiveness. The magnetic grinding experiment on the inner surface of the tube is shown in

Table 3. Experimental design for magnetic grinding of the inner surface of the tube with 3 factors and 4 levels [9].

Level	A	B	C
	Test Piece Speed (rpm)	Grinding Particle Diameter (μm)	Processing Time (min)
1	1000	150	10
2	1500	200	20
3	2000	250	30
4	2500	375	40

[10].

Table 3 is the experimental design table, which was designed using 3 factors and 4 levels. Four values were used for the process factors of magnetic pole rotation speed, namely 1000 r/min, 1500 r/min, 2000 r/min, and 2500 r/min. Four values were used for the process factors of grinding particles, namely 150 μm, 200 μm, 250 μm, and 375 μm. Four values were used for grinding time and process factors, namely 10 min, 20 min, 30 min, and 40 min. Three factors and four levels can be designed as 16 different combinations of the orthogonal experiments [10,15,16].

2.1.2. Establish a Prediction Model for the Roughness of Inner Surface Magnetic Grinding

We find the common logarithm of the surface roughness values of 16 sets of slender tubes after magnetic grinding. In $\widehat{\beta }={\left(X^{/}X\right)}^{-1}{X}^{/}Y$, it can be obtained as

$$\widehat{\beta }=\left[\begin{array}{c}{\widehat{\beta }}_0\\ {\widehat{\beta }}_1\\ {\widehat{\beta }}_2\\ {\widehat{\beta }}_3\end{array}\right]=\left[\begin{array}{c}2.9039\\ -0.5950\\ -0.1815\\ -0.6268\end{array}\right]$$

(1)

$\widehat{\beta }$ is a 4 × 1 matrix composed of regression coefficients in a linear regression model.

The regression equation can be determined from the above equation (Equation (1)):

$$y=2.9039-0.595x_1-0.1815x_2-0.6268x_3$$

(2)

Among them, $y=\mathrm{l}g{R}_a$, $x_1=\mathrm{l}gn,x_2=\mathrm{l}gd,\mathrm{a}\mathrm{n}\mathrm{d}{x}_3=\mathrm{l}gd$.

The final surface roughness prediction model obtained is

$$R_a={10}^{2.9039}{n}^{-0.595}{d}^{-0.1815}{t}^{-0.6268}$$

(3)

We use the regression function in the analysis software Matlab 7.0 to perform multiple linear regression analyses on the model. The call instructions for the regress function in Matlab are [b, bint, r, rint, stats] = regress (Y, X). The STATS in the regression function is a vector containing four components. The return value includes confidence $\left(R^2\right)$, statistic (F), significance (p-value), and the estimated variance of the regression model error (${\sigma }^2)$. The range of the R value is from 0 to 1. The closer it is to 1, the more obvious the linear relationship [10,25,30].

$$R^2=0.9567,F=88.3355,p=1.9019\times {10}^{-8},{\sigma }^2=0.0017$$

According to the calculated values, the $R^2$value is approximately 1, with p < 0.05. The multiple regression model for predicting surface roughness meets the requirements.

The influence of four process parameters on surface roughness in the predictive model is determined [46,47,48,49,50,51]. The order from strong to weak is as follows: grinding time > magnetic pole speed > the particle size of grinding particles. We analyze the final predictive model. Magnetic grinding process parameters have a “negative impact” on the surface roughness value of the inner surface of the tube [52,53,54,55]. This prediction model can effectively predict the surface roughness of the inner surface of the pipe. It has high prediction accuracy and positive significance in guiding the optimization of magnetic grinding process parameters [10,53].

2.2. The Magnetic Grinding Process Based on Grey Relational Theory

2.2.1. Grey Correlation Analysis

Grey relational analysis (GRA) studies the magnitude of data correlation. It measures the degree of correlation between data through correlation. It is a method that assists in decision-making. The basic idea is to determine the geometric similarity between the reference data column and several comparison data columns. It can determine whether their connections are tight. It reflects the degree of correlation between curves. We design magnetic grinding experiments on the inner surface of aircraft engine bend pipes using the Taguchi orthogonal method and analyze the effects of magnetic pole shape, magnetic pole disc speed, magnetic pole spacing, and grinding time on roughness (Ra), surface hardness (Hv), and grinding removal amount (M). The analysis method combining the orthogonal method and grey correlation theory can optimize the process parameters of multiple performance indicators. We take the shape of the magnetic pole, the speed of the magnetic pole disc, the distance between the magnetic poles, and the processing time as the four factors for the experiment. Each factor has three levels. The specific levels of each factor are shown in Table 4 [11,13]. The comprehensive evaluation indicators include the inner surface roughness value, the microhardness value, and the grinding removal quality. We observe the surface microstructure using an ultra-deep 3D electron microscope (VHX-500F), measure the surface roughness value of the workpiece using the JB-8E stylus surface roughness measuring instrument, measure microhardness values using a microhardness tester, and measure the material removal amount using an electronic analytical balance [11,18]. The flowchart of GRA is shown below.

Table 4. Magnetic grinding test conditions [11].

Serial Number	Factor	Level 1	Level 2	Level 3
A	Magnetic pole shape	taper 30°	taper 35°	Auxiliary magnetic pole slotting
B	Magnetic pole disc speed (r/min)	1200	2000	2800
C	Magnetic pole gap, d/mm	10	20	30
D	Grinding time (t/min)	10	20	30

Table 4. Magnetic grinding test conditions [11].

Serial Number	Factor	Level 1	Level 2	Level 3
A	Magnetic pole shape	taper 30°	taper 35°	Auxiliary magnetic pole slotting
B	Magnetic pole disc speed (r/min)	1200	2000	2800
C	Magnetic pole gap, d/mm	10	20	30
D	Grinding time (t/min)	10	20	30

Grey correlation degree is used to characterize the degree of correlation between factors that are not unified between two units or different systems over time or objects. It is the degree of synchronous change between the two. The grey correlation method requires data preprocessing of the various factors involved in grey correlation, and their range is between [0, 1]. The smaller the experimental data value of surface roughness, the better the effect of magnetic grinding. We preprocess the raw data as shown in Equation (4).

$$x_i\left(k\right)={\displaystyle \frac{max{y}_i\left(k\right)-y_i\left(k\right)}{max{y}_i\left(k\right)-min{y}_i\left(k\right)}}$$

(4)

The larger the amount of grinding removal and the surface hardness value, the better the effect of magnetic grinding. We preprocess the raw data as shown in Equation (5).

$$x_i\left(k\right)={\displaystyle \frac{y_i\left(k\right)-min{y}_i\left(k\right)}{max{y}_i\left(k\right)-min{y}_i\left(k\right)}}$$

(5)

In the formula, $x_i$ is the preprocessed data, $k$ is the sequence number, and $y_i$ is the test result.

The grey correlation coefficient ${\epsilon }_i$ represents the degree of correlation between the reference sequence and the comparison sequence. The reference sequence can be seen as a certain result, while the comparison sequence can be understood as the sequence that causes this result. Its value range is 0–1. The correlation coefficient ${\epsilon }_i$is defined as the degree of causal relationship between the two. The reference sequence includes the surface roughness value, surface hardness, and the grinding removal amount. The comparison sequence includes the magnetic pole shape, the magnetic pole disc speed, the magnetic pole gap, and grinding time.${\epsilon }_i$is calculated by Equation (6).

$${\epsilon }_i(k)={\displaystyle \frac{{∆}_{min}+\rho {∆}_{max}}{{∆}_{oi}\left(k\right)+\rho {∆}_{max}}}$$

(6)

In the formula, ${∆}_{oi}\left(k\right)$ is the difference between the reference value and the comparison value of the pre-processing number. It can be understood as the correlation between the reference sequence and the comparison sequence of sequence K. Its calculation is shown in Formula (7), with,

$${∆}_{oi}\left(k\right)=\left|x_o\left(k\right)-x_i\left(k\right)\right|$$

(7)

where ${∆}_{min}$ is the minimum difference,${∆}_{max}$ is the maximum difference, and ρ is the resolution coefficient that reflects the resolution ability. The grey correlation number ${\epsilon }_i\left(k\right)$ is the difference between the reference value and the comparison value at sequence K, which has multiple values. We take the average of multiple ε_{i (k)} values at each sequence K according to Equation (8) to obtain the grey correlation degree ${\gamma }_i$.

$${\gamma }_i={\displaystyle \frac{1}{n}}\sum _{i=1}^n{\epsilon }_i\left(k\right)$$

(8)

Zhou Chuanqiang, Han Bing, and others designed these experiments. The goal is to conduct theoretical analysis and experimental verification of grayscale correlation. By analyzing the results of the Taguchi orthogonal table experiment using grayscale correlation theory, it can not only comprehensively consider the influence of multiple factors on the experimental results but also determine the optimal level of a single influencing factor. The factors affecting the magnetic grinding process, in descending order, are magnetic pole taper, magnetic pole spacing, magnetic pole speed, and grinding time [11,17,18].

2.2.2. Experimental Verification

Based on the grey relational theory, the optimization of the magnetic grinding process parameters was determined, and the optimal combination of the magnetic grinding process parameters for magnetic grinding of the inner surface of the pipe is a magnetic pole taper of 35 degrees. For slotting, the magnetic pole rotation speed is 2000 r/min, while the magnetic pole-to-pipe outer surface distance is 3 mm, with a grinding time of 30 min. The selected optimal process parameter combination is tested and verified.

Using the HV-1000 microhardness tester, each group of data is measured seven times, and the average value is taken. The surface roughness value and microhardness are measured in seven different areas, and then the average is taken. Vickers hardness testing was conducted, with a test force of 9.81 N and a holding time of 10 s. The results were compared with the hardness of the magnetic grinding bend pipe after optimizing the process parameters. After optimization, the hardness of the workpiece increased by 29.4%.

As shown in Figure 3 and Figure 4, the optimized experimental results are superior to those in the orthogonal experimental table in terms of surface roughness, surface hardness, and material removal. The original roughness value of the workpiece is about 0.95 μ m, and the Vickers microhardness value is about 161 MPa. Under optimized process parameters, the surface roughness value of the workpiece processed is about $R_a$—0.58 μm. The surface hardness value is about $H_v-$208 MPa. The material removal amount of the workpiece is about M-5.5 mg. The optimized experimental results showed that all performance indicators were superior to the experimental results in the orthogonal table. The optimization of magnetic grinding process parameters has been achieved in a previous study [26].

2.3. The Magnetic Grinding Process for a Space Bending Pipe Based on Response Surface Analysis

Using the Box–Behnken response surface design method, we analyze the process factors and influencing trends. They affected the quality of magnetic grinding on the inner surface of bent pipes. The experimental design of magnetic particle grinding is shown in

Table 5. Experimental design of magnetic grinding on the inner surface of bent pipes based on response surface analysis [26].

Parameter	Lower-Level	High-Level
Magnet revolution N/(r·min−1)	450	1050
Magnet size D/μm	150	250
Axial feed rate V (mm·s−1)	0.50	1.50

[21,27,34].

The specific corresponding functions of surface roughness and various factors and their interactions can be obtained through regression analysis of experimental data, as shown in Equation (9) of the Princeton grinding and polishing equation.

Y = 0.13 + 10⁻³ (12X₁ + 4.27X₂ + 12X₃ − 5.9X₁X₂ + 3.8X₁X₂ − 0.55X₂X₃ + 56X₁² + 56X₁² + 3.083X₂² + 18X₃²)

(9)

In the formula, Y represents surface roughness, X₁ is the magnetic pole speed, X₂ is the abrasive particle diameter, and X₃ is the axial feed rate.

The influence of interaction under the corresponding surface on the roughness of the inner surface of the bent pipe is shown in Figure 4. We select X₃ as 1 mm/s and increase X₁ and X₂, causing Y to first decrease and then increase, as shown in Figure 4a. The reasons were analyzed. According to the Princeton equation of grinding and polishing in Formula (9), it can be seen that as the relative rotational speed between the magnetic abrasive particles and the workpiece increases, the amount of material removed also increases. Thus, the roughness of the pipe surface was reduced. But as the rotational speed continues to increase, the centrifugal force increases, while the grinding pressure decreases, and the roughness of the pipe surface will increase again. When the diameter of the abrasive particles is too small, the grinding depth decreases. When the particle size is too large, it scratches the inner surface. Only by selecting the appropriate particle size can the best magnetic grinding effect be achieved. We choose a grinding particle diameter of 200 μm and increase X₃ and X₁, causing Y to first decrease and then increase, as shown in Figure 4b. X₃ and X₁ values that are too low or too high are not conducive to uniform grinding and reduce grinding efficiency [18,19,21].

When X₁ selects 550 r/min and increases X₂, Y first decreases and then increases, as shown in Figure 4c. Due to the increase in X₃, grinding is more thorough. However, after exceeding the limit value, uneven grinding may occur, resulting in grooves and a decrease in grinding efficiency [21,27].

When analyzing the conditional factors of magnetic pole speed, abrasive particle size, and axial feed rate, the interaction of various variables were taken into account. It can be concluded that when the magnetic pole speed is 550 r/min, the abrasive particle size is 200 μm, and the axial feed rate is 1 mm/s, so the machining effect is the best. The machining efficiency can be improved while achieving the desired effect.

There are many surface defects such as cracks and pits on the surface of unprocessed bent pipes, with significant height differences and prominent surface points. This is because space bending pipes need to be bent according to their usage during the production process. Due to production process reasons, there will inevitably be defects such as cracks and pits on the inner surface. After processing for 30 min, the pits, scratches, and surface defects on the inner surface of the workpiece were completely removed. The surface texture was more uniform. The surface height difference was small. The surface quality was improved. It achieved the same effect as the previous 40 min processing. This is because when using magnetic grinding technology to polish the original surface of the titanium alloy bent pipes, due to the sharp point effect of the process itself, the surface protrusions will be removed first. As a result, the peaks generated by the residual height will be smoothed out, making them consistent with the height of the valleys. The surface processing texture will be significantly improved. At the same time, due to the optimal selection of processing factors, the processing process is in the best state. It resulted in uniform processing effects and a rapid improvement in surface quality. And it increased processing efficiency. The surface roughness value of the bent pipe decreased from 10 μm to 0 after magnetic grinding. After magnetic grinding, the surface of the bent pipe changes from uneven to shiny [19,21,27].

2.4. Process Optimization of BP Neural Network Prediction Modelling Based on Genetic Algorithm Optimization

The BP neural network not only has strong fault tolerance but also extraordinary nonlinear mapping ability. The genetic algorithm is an optimization algorithm generated by simulating biological genetics and the theory of “survival of the fittest” biological evolution. By evaluating the fitness function, the selection operation, crossover operation, and mutation operation are iteratively updated to select individuals with high fitness from the population, while individuals with low fitness will be eliminated. The use of the genetic algorithm for global optimization can improve the disadvantage of the BP neural network, which is prone to falling into local minima. The fitness function of the genetic algorithm is not constrained by conditions such as continuity and differentiability. It has a wide range of applications. By determining the topology structure of the BP neural network through input and output parameter sequences, the number of weights and thresholds to be optimized by the genetic algorithm is determined. It is the population size of the genetic algorithm [4,29]. By optimizing the initial weights and thresholds of the BP neural network through the genetic algorithm, an accurate prediction of surface roughness of magnetic particle grinding can be achieved. Comparing the prediction error (the difference between the predicted value and the expected value) of the BP neural network before and after evolution, the closer the prediction error is to 0, the more accurate the model prediction is. The algorithm flow is shown in Figure 5 [4,55].

This model has a three-layer network structure, as shown in Figure 6. Using magnetic pole speed, machining gap, abrasive particle size, and feed rate as inputs to the network, the difference in surface roughness ΔRa before and after grinding is used as the output. The number of nodes in the input layer is u = 4, and the number of nodes in the output layer is v = 1. The number of iterations for the network is set to 200; the learning rate is set to 0.2, and the training error is set to 0.0001. At present, the number of hidden-layer nodes is mostly determined using empirical Formula (10) [4,30,31,32].

$$m=a+\sqrt{u+v}$$

(10)

In the formula, m represents the number of neuron nodes in the hidden layer.

If there is a decimal, it needs to be rounded up to one digit. A generally selects a constant of 1–10, so the range of nodes in the hidden layer is 3–13 [33,55].

3. Discussion

(1) The BP neural network optimized based on the genetic algorithm has high accuracy and good predictability.

The average percentage of error between the predicted value and the true value is used as the basis for determining whether the number of hidden-layer nodes is reasonable, as shown in Equation (11).

$$s={\displaystyle \frac{1}{n}}{\sum }_{i=1}^n\left({\displaystyle \frac{Z_i-Y_i}{Z}}\right)$$

(11)

In the formula, S represents the average percentage of error in network prediction, N represents the number of samples in the test sample, Zi represents the i-th predicted value in the sample, and Yi represents the i-th true value in the sample. The closer the S value is to zero, the higher the prediction accuracy of the network. Therefore, S can be used to represent the prediction accuracy of neural networks. When the number of hidden-layer nodes is different, the value of S is shown in Figure 7 [33].

From Figure 7, it can be concluded that when the number of hidden-layer neurons is five and the value of S is closest to zero, the prediction error of the neural network is minimized. It can be inferred that the prediction accuracy is best when the number of hidden-layer nodes is five. The network structure with five hidden-layer neuron nodes is shown in Figure 8 [4].

We randomly select 15 sets of data to train a BP neural network, which can be used to express nonlinear function outputs after training. The remaining five sets of data were used as tests for the fitting performance of the neural network, and the fitting results are shown in Figure 9 [29]. It can be seen that except for the large error of 9% in the fourth group of data, the predicted values of the other groups are not significantly different from the actual values. The prediction performance is relatively ideal, reflecting that the trained neural network has a certain degree of accuracy and good predictive ability.

(2) The BP neural network based on the genetic algorithm can predict experimental data well and obtain the optimal surface quality of the bent pipe of the aircraft engine.

We select four process parameters for research, including magnetic pole speed, machining gap, abrasive particle size, and feed rate. Based on the empirical values of various process parameters, we select the range of values for each factor. The experimental plan is shown in

Table 6. Experimental programme [33].

Test Number	Spinning Speed/(r·min−1)	Machining Gap/mm	Abrasive Particle Size/μm	Axial Feed Rate/(mm·min−1)	Change in Surface Roughness/μm
1	450	1.0	150	30	0.96
2	450	1.5	178	60	1.02
3	450	2.0	250	90	0.99
4	450	1.0	178	30	1.04
5	450	1.5	150	60	0.13
6	750	2.0	150	90	1.14
7	750	1.0	178	30	1.08
8	750	1.5	250	60	1.15
9	750	2.0	178	90	1.11
10	750	1.0	150	30	1.04
11	1050	1.5	150	60	0.96
12	1050	2.0	178	90	0.94
13	1050	1.0	250	30	0.97
14	1050	1.5	178	60	1.04
15	1050	2.0	150	90	0.95
16	450	1.5	150	30	0.98
17	750	1.5	178	60	1.00
18	1050	2.0	250	90	0.96
19	450	1.0	178	90	1.10
20	750	1.5	150	60	1.14

[33]. Using wire cutting, we cut the pipe fittings used in the experiment along the axial direction and measure multiple parts of the original surface of the workpiece using a roughness meter. Due to the lack of significant differences in the surface roughness of the inner circumferential surface of the workpiece, a multi-point measurement method was used to take the average value to determine the final surface roughness value. The average surface roughness value is 1.34 μm. We fix the cut workpiece together, perform magnetic particle grinding on the inner surface of the pipe fitting, and then disassemble the fixed pipe fitting and re-measure it using a roughness meter. We determine the surface roughness after processing by taking the average of multiple measurements. The difference in surface roughness before and after processing can intuitively reflect the effect of magnetic particle grinding on pipe processing.

Using the obtained grinding process parameters to grind the inner surface of the TC₄ bend pipe, we verify the accuracy of the BP neural network genetic algorithm. We measure the inner surface of the TC₄ bent pipe after processing using the JB-8E stylus roughness measuring instrument and observe its surface microstructure using a super-depth microscope. During the manufacturing process of pipe fittings, some pits or protrusions may occur due to processing methods. From Figure 10a, it can be seen that the original surface is rough and uneven with deep grooves. Its surface roughness is 1.341 μm, as shown in Figure 10b. We use the obtained optimal process parameters for grinding, as shown in Figure 10c [4]. After grinding, its surface is very smooth, as its original grooves are removed, presenting a dense and uniform state. After measurement, its surface roughness is 0.113 μm, as shown in Figure 10d. The similarity with the predicted value indicates that the BP neural network genetic algorithm can predict the experimental data well and obtain the optimal value.

(3) The BP neural network optimized based on the genetic algorithm can predict and optimize the optimal process parameters for magnetic grinding of the inner surface of aircraft engine bend pipes.

The range of the process parameters is shown in

Table 7. Process parameter value range [33].

Process Parameters	Spinning Speed/(r·min−1)	Machining Gap/mm	Abrasive Particle Size/μm	Axial Feed Rate/(mm·min−1)
range	450~1050	1.0~2.0	150~250	30~90

. Two hundred individuals are randomly generated with values from the range of the process parameters, which are used as the initial population [33].

By utilizing the global optimization function of the genetic algorithm, after 200 iterations of genetics and evolution, the optimal process parameters were finally obtained, with Y_max = 1.2147. The optimal surface roughness is 0.1253 μm. The optimal parameter combination for obtaining magnetic particle grinding TC₄ bent pipes is X = [576.798, 1.7718, 172.9354, 77.364]. The final optimal process parameters obtained are a magnetic pole speed of 570 r/min, a machining gap of 2.0 mm, an abrasive particle size of 178 μm (80 mesh), and a feed rate of 80 mm/min.

(4) The BP neural network optimized based on the genetic algorithm is also applicable to magnetic grinding of bent pipe materials other than the TC4 titanium alloy. The method of optimizing the BP neural network using the genetic algorithm is not only applicable to the optimization of magnetic grinding process parameters for TC4 titanium alloy bent pipes but can also be applied to the optimization of magnetic grinding process parameters for bent pipes of other materials. This method establishes a nonlinear mapping model and uses the genetic algorithm to globally optimize the optimal process parameters, yielding high accuracy and applicability.

4. Conclusions

(1)

The magnetic grinding process based on a multiple regression prediction model: In the actual process of magnetic grinding of bent pipes, there are many factors that affect the grinding effect and efficiency. There are also intersecting effects between each factor. A large number of research experiments on magnetic grinding of bent pipes should be conducted. There are significant differences in the selection of process parameters for different lengths and diameters of aircraft engine bent pipes. The establishment of a surface roughness prediction model based on multiple regression is still in its infancy. Further research is needed to use other prediction methods to establish the prediction model so as to better predict the surface roughness values after magnetic polishing finishing and to optimize the magnetic polishing process parameters.

(2)

The magnetic grinding process theory based on grey relational theory: It can improve the accuracy of magnetic grinding of aircraft engine bend pipes to a certain extent. It cannot achieve accurate modelling and prediction of the process parameters of magnetic grinding and the surface roughness of bend pipes. So it cannot improve the efficiency of magnetic grinding. Further research is needed to establish predictive models using other prediction methods so as to accurately predict the surface roughness values after magnetic abrasive finishing and to quickly obtain the optimal magnetic abrasive process parameters.

(3)

The magnetic grinding process for spatial bent pipes based on response surface analysis: It can effectively improve the processing efficiency of the grinding and polishing device on the inner surface of spatial bent pipes and save processing time. However, the nonlinear characteristics between various grinding process parameters were not considered. An effective model to predict the optimal surface roughness of the inner surface of the bent pipe after magnetic grinding was not established. It cannot quickly obtain the optimal magnetic grinding process parameters for the inner surface of the bent pipe. Therefore, it is not possible to achieve precise and efficient magnetic grinding of the inner surface of the bent pipe.

(4)

BP neural network prediction modelling based on genetic algorithm optimization: By designing different processing parameter ratios through orthogonal experiments, experiments were conducted on the inner surface of magnetic particle grinding aircraft engine bend pipes. The BP neural network prediction model optimized by the genetic algorithm can achieve accurate prediction of the surface roughness of the inner surface of aircraft engine bend pipes. By predicting the surface roughness of magnetic particle grinding on the inner surface of aircraft engine bent pipes, the surface processing effect can be improved, the scrap rate can be reduced, and the product’s processing efficiency can be enhanced.

By comparing the current advanced magnetic grinding process prediction with the BP neural network prediction model optimized by the genetic algorithm, the average relative error of the current advanced magnetic grinding process prediction is 11.3%. While the BP neural network optimized by the genetic algorithm has higher prediction accuracy, with an average relative error of 4.2%, it can greatly avoid the disadvantage of large errors in the current advanced magnetic grinding process prediction.

The BP neural network optimized by the genetic algorithm can predict surface roughness more accurately in the field of magnetic particle grinding compared to traditional magnetic particle grinding technology. It has been confirmed in the grinding experiment of the inner surface of nickel-based alloy aircraft engine bent pipes. It can be extended to magnetic particle grinding processing of different materials and processing technologies.

5. Future Directions

Extensive research has been conducted on the optimization of magnetic grinding technology for aircraft engine bend pipes. Future research focus on optimizing the magnetic particle grinding process for aircraft engine bend pipes should be carried out from the following four aspects:

(1)

Mechanism aspect

Analysis and verification have been conducted on magnetic pole speed, abrasive particle diameter, axial feed rate, etc., but the depth is not sufficient. For example, the optimization of the magnetic particle grinding process did not fully consider the non-correlated influence of magnetic particle grinding process factors on the accuracy and efficiency of magnetic particle grinding. Further research is needed on the mechanism and experimental verification of modelling and optimizing the factors of the magnetic grinding process to improve the accuracy and efficiency of magnetic particle grinding.

(2)

In terms of predictive modelling

The research on magnetic particle grinding technology for the inner surface of aircraft engine bend pipes often uses the finite element method, the orthogonal method, and the response surface method for analysis, but the accuracy and efficiency of magnetic particle grinding are not high. Due to the nonlinear relationship between surface roughness and process parameters, the research on the influence of magnetic particle grinding process parameter combinations on the quality of magnetic particle grinding is more complex. The impact of prediction models on the accuracy and efficiency of magnetic particle grinding is worth further research.

(3)

Monitoring and evaluation of magnetic particle grinding process progress and effectiveness

Improving the accuracy and efficiency of detection, thereby achieving precise and reliable integrated qualitative and quantitative discrimination, is a major trend in the optimization of magnetic particle grinding technology. In addition, the integration of artificial intelligence technology in the monitoring of the magnetic particle grinding process on the inner surface of aircraft engine bend pipes can break and improve the limitations currently existing in detection. Realizing intelligent real-time monitoring of the magnetic particle grinding process on the inner surface of aircraft engine bend pipes is important.

(4)

Application of magnetic grinding technology

The urgent problems that need to be solved in the development of magnetic particle grinding equipment for the inner surface of aircraft engine bent pipes are high precision, high efficiency, and stable magnetic field output. With the increasing complexity of bend shapes for different types of aircraft engines and the expansion of the scope of magnetic particle grinding, the automation requirements for magnetic particle grinding process equipment are also becoming higher and higher. The automation of precise magnetic particle grinding and the development of intelligent robotic magnetic grinding systems have become the direction for later engineering applications. In the application of magnetic particle grinding technology, it is necessary to adjust the process parameters in real time according to the diverse requirements of magnetic grinding on the inner surface of bent pipes for different types of aircraft engines.