A Cutting Parameters Selection Method in Milling Aero-Engine Parts Based on Process Condition Matching

Abstract

An optimal selection method of process parameters based on process condition matching is proposed, for the difficulty of the process parameters selection in the milling of complex structure and difficult-to-cut material parts. The factors of process parameters selection are analyzed, process condition vector and process parameter vector are defined, and their quantitative expressions are proposed. The mapping of existing process condition vectors to the process parameter vectors is established, based on the process data accumulated in practical production. Then, process condition matching degree is defined. In the calculation of the matching degree, Analytic Hierarchy Process (AHP) is adopted to determine the affecting weights of process condition factors, and leveling matrix is adopted to eliminate the differences of dimensions and numerical scales between process condition factors. The optimal process parameters are achieved through matching the actual process condition to the existing process condition. A group of typical aero-engine part milling processes is taken as instance, and the feasibility and effectiveness of this method are verified. A typical aero-engine part CNC machining process database system has been designed and developed based on this method.

1. Introduction

CNC machining process system includes three parts mainly, machine tool, workpiece, and cutting tool. CNC machining process is the removing of material from workpiece by the cutting tool controlled by NC program, which is accompanied by a series of changes in geometric shape and physical parameters, such as cutting force, cutting temperature, tool wear, material hardening, and residual stress. Therefore, the machining quality of the part not only depends on the NC program, but also is restricted by various variables and parameters in machining process. They are classified into four categories, control variables, noncontrol variables, procedure variables, and output variables, according to the characteristics of various variables in cutting process [1]. Noncontrol variables (workpiece material, workpiece shape, and workpiece state) reflect the physical and geometrical characteristics of workpiece and are selected and designed according to the function requirement of the part. In order to ensure that the output variables (precision, surface quality, tool life, and chip) meet the requirement, the control variables (machine tool, cutting tool, cooling, process parameters, and fixtures) need to be selected legitimately, to achieve the purpose of controlling the procedure variables (vibration, cutting force, cutting temperature, and tool wear). Therefore, the effects of the control variables on CNC machining must be analyzed, to achieve a reasonable selection of the process parameters.

Aero-engine parts have the characteristics of a wide variety, complex structure, high requirement of machining accuracy and surface quality, and great impact on engine performance. For the part composed of the complex freeform surfaces, due to its high geometric accuracy and difficult machining, the method of subregional milling is usually adopted in its machining process the process parameters in different regions have great differences. The common aero-engine materials include titanium alloys, nickel-based alloys, and carbon-based, ceramics-based, and metal-based composites. These materials have a poor machinability and are sensitive to the process parameters in machining. Correct and reasonable selection of process parameters is very important to improve productivity and reduce manufacturing costs and has a great significance for ensuring product quality and service reliability.

In recent years, the studies on the CNC milling process parameters optimization and selection focus mainly on the optimization model, the solving of optimization model, and the creation of machining process database. Wherein, Tolouei-Rad and Bidhendi took the production costs and efficiency as the goals, took the machine tool power, surface quality, and cutting force as the restrictions, and then proposed a cutting parameters optimization model [2]. Yan and Li took into account three evaluation indexes: the material removal rate, cutting energy, and surface roughness, then selected the spindle speed, feed rate, cutting depth, and cutting width [3]. Chen and Zhang took the profit rate per unit time as the goal and proposed an optimization model [4]. These studies all take the production costs or production efficiency as the optimization goal; there is no investigation of the loss of cutting tool and machine tool. For the issues of tool life, Iqbal et al. investigated the effect of workpiece material hardness, tool helix angle, milling direction, and coolant on the tool life in hard milling and then proposed a milling parameters optimization expert system based on experimental data [5]. In his another study, the effect of the hardened steel microstructure, cutting tool tilt angle, cutting speed, and radial cutting depth on the tool life in milling process was studied, but the specific method of process parameters selection was not presented [6]. Choudhury and Appa Rao established a tool life equation based on experimental data and adhesive wear model for turning process and took the tool wear as the goal to optimize and select the cutting speed and feed rate [7]. However, the part quality was ignored in the selection of process parameters. Vidal et al. took the maximum material removal rate as the goal and designed a milling process parameters optimization system based on the optimal machining costs, in the basis of the considering of workpiece material, surface roughness, machine tool, and cutting tool [8]. But this study did not take into account the type characteristics of the specific part, and it lacked specificity. On the basis of local cutting feature Li proposed a feed rate optimization method based on local condition, in her master degree thesis [9]. Rai et al. simulated the material removal process through the finite element method, took the material removal rate as the goal, took the machine tool power, cutting tool deflection, and feed rate in curve as the restrictions, and used the genetic algorithms to optimize the cutting parameters [10]. The effects of workpiece material and cutting tool parameters on machining process were ignored in the parameter optimization. Zhang et al. took the surface roughness, cutting force, and material removal rate as the goal and proposed a cutting parameters optimization method based on the fuzzy analysis [11]. Zhuang et al. proposed the cutting force model of plunge milling and evaluated the cutting stability of plunge milling using the frequency domain method. Then a strategy of cutting parameters optimization was proposed based on the prediction of cutting force and cutting stability [12]. This method was only for the plunge milling process, and the application range was narrow. Kuram et al. studied the effect of the cutting fluid type on machining quality through experiments. The process model was presented on the basis of considering the cutting fluid type, and a cutting parameters optimization method taking the surface roughness and tool life as the goal was proposed [13]. Vivancos et al. established a surface roughness model for the high speed milling of injection mold. The surface roughness was taken as the goal and the cutting parameters were optimized through designing experiments [14]. The cutting force and workpiece deflection were not analyzed in the parameter optimization, so this method could not be applied to the milling process of thin-walled and weakly rigid parts. After establishing the appropriate optimization model, most researchers used artificial intelligence methods, such as fuzzy logic [15, 16], neural network [17], PSO [18], and simulated annealing genetic algorithm [19], to solve the optimization model. Other researchers analyzed the accumulated cutting process parameters and then established the cutting process parameter database. Ghahramani et al. proposed a web-based parameter selection system, and the system contained the client, JSP pages, and the database [20]. Users could search milling process data through the network. Wu and Liao studied the process parameter database for high speed machining [21]. Xu et al. developed cutting parameters optimal selection database based on genetic algorithm for the milling efficiency of the key parts in radar [22].

However, there are three shortcomings in current numerous studies. (1) The selection of process parameters, which was for general machining, could not satisfy the CNC milling requirements of complex structure and difficult-to-cut material aero-engine part. (2) Process parameters were optimized and selected for a single goal, and there was no comprehensive analysis of the various factors in actual machining. This led to the fact that the resulting process parameters were not globally optimal parameters, and the availability of resulting process parameters was low in the actual production. (3) The artificial intelligence method was adopted to calculate and select the process parameters, and there was no mining and utilization of the large amounts of data accumulated naturally in practical production.

In response to these problems, this study takes the milling process of typical aero-engine part as the object and proposes an optimal selection method of process parameters based on actual machining process condition matching, on the basis of previous studies. In this method, the affecting factors of machining process and process parameters are analyzed comprehensively and then the milling process condition vector and process parameter vector are defined. The process condition vector is used to describe the various factors, and the process parameter vector is used to describe the various process parameters in machining process. The process data accumulated in practical production is arranged, the mapping of the existing process conditions to the process parameters is established, and then the large amounts of process data verified by practical production are stored in the process parameter database. The process condition matching degree is defined and its calculation is presented. In its calculations, AHP is adopted to determine the affecting weights of process condition factors, and leveling matrix is adopted to eliminate the differences of dimensions and numerical scales between process condition factors. Finally, the optimal process parameters are selected through the matching of actual process conditions to the existing process conditions.

2. Milling Process Condition Division and Expression

CNC milling of complex freeform surface parts involves many factors; these factors will have some direct or indirect impacts on machining process and machining result. In this paper, the set of these factors in CNC milling is defined as process condition and it is expressed as condition vector. The factors are divided into six subsets according to the needs of the milling process, characteristics of the machined part, and milling process method. These subsets are machine tool, cutting tool, material of machined part, cooling, characteristics of the machined part, machining procedure, and machining method.

2.1. Machine Tool

In the milling process, all the metal cutting is carried out on the machine tool. The machine tool, tool and workpiece compose a process system with multi-degree of freedom together. The capability of the machine tool determines the process quality and efficiency. The factors of the machine tool are considered from power, spindle torque, maximum speed of revolution, maximum feed rate, and position precision. These factors are written in the form of condition subvector, as shown in

$\mathbf{M}\mathbf{t}=\left\{P,T,n_{\max },f_{\max },\text{Pr}\right\},$

(1)

wherein the various factors are power, spindle torque, maximum speed of revolution, maximum feed rate, and position precision.

2.2. Cutting Tool

In the milling process, this tool removes the allowance and forms the machined surface, so the profile of tool is critical to machine the workpiece accurately. To describe the geometry of the tool, it must be modeled firstly; the profile model of milling cutter can be determined through seven parameters {r, R _c , R _r , R _z , a, b, h}, as shown in Figure 1 (a). These parameters can express the profile information of the milling cutter accurately. Various types of milling cutters are defined by giving different values of these parameters. For instance, flat mill (Figure 1 (b)) can be defined as $\{R_c=R_z=\mathrm{0},R_r=r,a=b=\mathrm{0},h\}$ ; ball end mill (Figure 1 (d)) can be defined as $\{R_c=R_z=r,R_r=\mathrm{0},a=b=\mathrm{0},h\}$ .

Figure 1:

The profile of arbitrary mill. (a) Definition of tool geometry. (b) Flat mill. (c) Toroidal mill. (d) Ball end mill. (e) Cone mill.

The overhang length of tool, flute number, and edge geometry are also extremely important, in addition to the tool profile information. The angles of tool include rake angle, flank angle, and helix angle. These parameters will affect the machining quality and efficiency. These factors are written in the form of condition subvector, as shown in

$\mathbf{T}=\left\{r,R_c,R_r,R_z,a,b,h,z,{\gamma }_{\mathrm{0}},{\alpha }_{\mathrm{0}},\beta \right\},$

(2)

wherein the first six items represent the geometric parameters of tool profile, which is defined in Figure 1. The other five items are overhang length of tool, flute number, rake angle, flank angle, and helix angle.

2.3. Workpiece Material

Part material determines the performance of part and also determines the machinability of part. The parts in aviation and aerospace industry are mostly titanium alloy, superalloy, and other difficult-to-cut materials, which leads to difficulty to machine, and more rigor of selection of process parameters. The workpiece material factors are written in the form of condition subvector, as shown in

$\mathbf{M}=\left\{\text{HRC},E,G,\mu ,\rho ,{\sigma }_s,{\sigma }_b,k\right\}.$

(3)

Various factors in the formula are Rockwell hardness, elastic modulus, shear modulus, Poisson ratio, density, yield strength, tensile strength, and thermal conductivity.

2.4. Cooling

In machining process, the functions of cooling are to lubricate and cool the milling region, it affects the cutting force and cutting temperature directly, and improves the machining quality of results and tool life. Therefore, the effect of cooling cannot be ignored, the factors of cooling are written in the form of condition subvector, as shown in

$\mathbf{C}=\left\{S,p,L,t\right\}.$

(4)

Various factors in the formula are cooling mode, pressure, flow rate, and temperature (the temperature of the coolant when in the coolant mode, the temperature of the air when in the air cooling mode, and the temperature of the environment when in the dry cutting mode).

2.5. Part Feature

Traditional data of milling parameters are for simple structures, rigid workpiece, and straight cutting motion. These data are obtained through experiments or statistics; they are not suitable for machining of weakly rigid part and complex freeform surface structure. The geometrical characteristics of workpiece affect the milling process significantly. If such information did not persist in the cutting parameter records, it cannot match the specific process condition with the conditions recorded in the database, and it cannot judge whether the process data in the database can be utilized. Thus, the structure feature information of workpiece needs to be described in addition to the workpiece material, when describing the process condition. The features of machined part include part category (open blisk, closed blisk, blade, etc.) and local feature (flow passage, blade body, wheel hub, etc.), and its form of condition subvector is shown in

$\mathbf{F}=\left\{C,\text{Lf}\right\}.$

(5)

The two factors in the formula are part category and local feature.

2.6. Machining Procedure and Method

In milling process, it is generally divided into roughing milling, semi-finishing milling, and finishing milling. In different procedures, the main purpose of milling is also different. The main purpose of roughing milling is fast and efficient removal of allowance. The task in semi-finishing milling is to machine further on the results of roughing milling and to make the allowance of finishing milling well-distributed to prepare for finishing milling. The complete surface of part is formatted in finishing milling. Therefore, the process parameters used in different machining procedure or different machining method have a great difference. Its form of condition subvector is shown in

$\mathbf{P}=\left\{\text{wp},m\right\}.$

(6)

The two factors in the formula are machining procedure and machining method.

2.7. Condition Vector

The factors analyzed above are mutually independent, so we take these factors as the subvectors of condition, and they compose condition vector of milling process. Its expression is shown in

$\mathbf{C}\mathbf{c}=\left\{\mathbf{M}\mathbf{t},\mathbf{T},\mathbf{M},\mathbf{C},\mathbf{F},\mathbf{P}\right\}.$

(7)

The six subvectors in the formula represent subvector of machine tool, subvector of cutting tool, subvector of workpiece material, subvector of cooling, subvector of part feature, and subvector of machining procedure and method, respectively.

Then, a condition vector space consisting of six dimensions is established, and the six dimensions are composed of its condition subvector, respectively.

3. Milling Process Parameters

The selection of cutting parameters has a direct effect on machining quality, machining efficiency, and tool life. The cutting will produce different results using different cutting parameters in the same process condition. The process parameters selected in milling process include spindle speed of revolution, cutting depth, cutting width, and feed rate, as shown in Figure 2.

Figure 2:

Cutting parameters in milling.

3.1. Process Parameter Vector

Spindle Speed of Revolution. Spindle speed of revolution determines the velocity of cutting edge relative to the workpiece, namely, cutting speed. Since cutting speed has the great effect on tool life, the selection of cutting speed relates to the durability of tool closely. Too low or too high cutting speed will cause the tool life to decline dramatically. Meanwhile, in the milling of thin-walled workpiece, spindle speed of revolution has a significant effect on the stability of cutting. Therefore, the spindle speed of revolution should be selected discreetly in milling process.

Cutting Depth and Cutting Width. Cutting depth and cutting width are restricted by spindle power, transmission power of machine tool, material type, tool parameters, coolant, machining procedure, and the stiffness of machine tool-tool-workpiece system. And, they have a great effect on tool life. Therefore, they should be selected reasonably according to machining quality, machining efficiency, and machining procedure. Generally, machining efficiency is the first goal in roughing machining, so a larger cutting depth and cutting width should be selected. Quality of workpiece surface is the main goal in finishing machining, so a less cutting depth and cutting width should be selected.

Feed Rate. Feed rate is the velocity of feed move of the cutting tool relative to workpiece in milling process. Generally, linear feed rate is adopted in practical production and it is defined as feed per minute. The feed rate of milling will affect the machining accuracy, surface quality, deformation of the workpiece, and tool life directly. And it is also restricted by tool parameters, workpiece material, tool path, stiffness of machine tool, and performance of feed system. In machining process, the feed rate of milling is selected according to part material, geometry features, quality requirements, and the capability of machine tool.

The process parameters mentioned above can be written in the form of vector, as shown in

$\mathbf{P}\mathbf{p}=\left\{n,a_p,w_e,f\right\},$

(8)

wherein these parameters represent spindle speed of revolution, cutting depth, cutting width, and feed rate.

Then, a process parameter vector space consisting of four dimensions is established.

3.2. The Mapping of Process Condition to Process Parameter

Process parameters are the concrete instance of process knowledge in the machining of practical workpiece and process knowledge is implicit in the process data. Process data accumulated in practical production is used, amended, and validated repeatedly through long time, and then the natural evolutional process knowledge is achieved. Therefore, the process data accumulated in practical production implies a large amount of field-proven process knowledge. These data should be mined deeply and the process knowledge should be used in the selection of process parameters.

The experiential process data of practical production is analyzed and arranged according to the proposed condition vector and process parameter vector, and then these data are accumulated into process database. The process parameter vectors are matched with the condition vectors, and the mapping of condition vector space to process parameter vector space is constituted. The process of natural evolution of process parameters and accumulation of data is shown in Figure 3.

Figure 3:

Natural evolution of process parameters and accumulation of data.

4. Method for Optimal Selection of Process Parameters

4.1. Matching Selection of Process Parameters

The typical parts of aero-engine are mostly of complex structure and freeform surface, there are too many uncertain factors in the machining of these parts, and these uncertain factors will have a direct or indirect effect on machining process and machining result. So in a complex milling process, the process parameters must be selected reasonably, combining these uncertain factors and considering the requirements of various factors to process parameters, to enable the machining quality and machining efficiency to reach optimum.

The traditional method of cutting parameters selection aims at the general machining condition. This method applies to the machined part with relatively simple structure, less requirement of machining quality, and test cutting. However, the experiential process data accumulated in practical production implies a large amount of process knowledge, with a high availability. Therefore, the actual process condition is analyzed and matched to process conditions in the database, to find the process condition in the database which has the highest matching degree of the actual process condition. When the highest matching degree exceeds the threshold, the relevant process parameter vector could be used as the optimal process parameters in this actual process condition. The principle of the matching and optimal selection of process parameters is shown in Figure 4.

Figure 4:

Matching and optimal selection of process parameters.

4.2. Calculation of Process Condition Matching Degree

4.2.1. Process Condition Matching Degree

The matching degree of the actual process condition Cc and a specific process condition Cc₀ is defined as

$\text{Mdr}={\left(\mathbf{D}\times \mathbf{W}\times {\mathbf{D}}^T\right)}^{-\mathrm{1}/\mathrm{2}},$

(9)

wherein D is the difference vector of Cc and Cc₀, W is the weight matrix of the differences vector, and their expression is

$\mathbf{D}=\left\{D_{\text{Mt}},D_T,D_M,D_C,D_F,D_P\right\}.$

(10)

The components of D are the various differences of condition subvector, respectively,

$\mathbf{W}=\left[\begin{array}{cccc}{w}_{\mathrm{1}}& \mathrm{0}& ·& \mathrm{0}\\ \mathrm{0}& w_{\mathrm{2}}& ·& \mathrm{0}\\ ⋮& ⋮& \ddots & ⋮\\ \mathrm{0}& \mathrm{0}& ·& w_{\mathrm{6}}\end{array}\right],$

(11)

wherein w₁, w₂, …, w₆ are the various weights of the difference of condition subvector, respectively, and their sum is 1.

4.2.2. Difference Vector of Process Condition

The component D _q (q = Mt, T, M, C, F, P) of difference vector of process condition D denotes the difference of machine tool subvector, cutting tool subvector, workpiece material subvector, cooling subvector, part feature subvector, and machining procedure and method subvector, respectively.

Since the dimensions and scales of various factors in subvector are different, there is no comparability between the various factors. Therefore, the leveling matrix of difference scale is introduced, to level the scale of various factors and remove the dimensions.

The difference of various subvectors is

$D_q={\left(\left(\left(\mathbf{q}-{\mathbf{q}}_{\mathrm{0}}\right)\times \mathbf{S}{\mathbf{L}}_q\right)\times {\mathbf{W}}_q\times {\left(\left(\mathbf{q}-{\mathbf{q}}_{\mathrm{0}}\right)\times \mathbf{S}{\mathbf{L}}_q\right)}^T\right)}^{\mathrm{1}/\mathrm{2}},$

(12)

wherein q = Mt, T, M, C, F, P represents the various subvectors in process condition vector, q and q₀ are the various components of the actual process condition and a specific process condition, respectively. SL _q is the scale leveling matrix of various condition subvectors, and W _q is the weight matrix of various condition subvectors, and their expression is

$\mathbf{S}{\mathbf{L}}_q=\left[\begin{array}{cccc}\text{s}{\text{l}}_{q\mathrm{1}}& \mathrm{0}& ·& \mathrm{0}\\ \mathrm{0}& \text{s}{\text{l}}_{q\mathrm{2}}& ·& \mathrm{0}\\ ⋮& ⋮& \ddots & ⋮\\ \mathrm{0}& \mathrm{0}& ·& \text{s}{\text{l}}_{qn}\end{array}\right],$

(13)

wherein sl_q1, sl_q2, …, sl _qn (q = Mt, T, M, C, F, P) denote the scale leveling coefficients of the various factor differences in condition subvectors. n is the number of the factors in various condition subvectors, when q = Mt, T, M, C, F, P, the corresponding values of n are n = 5, 11, 8, 4, 2, 2, respectively.

Consider

${\mathbf{W}}_q=\left[\begin{array}{cccc}{w}_{q\mathrm{1}}& \mathrm{0}& ·& \mathrm{0}\\ \mathrm{0}& w_{q\mathrm{2}}& ·& \mathrm{0}\\ ⋮& ⋮& \ddots & ⋮\\ \mathrm{0}& \mathrm{0}& ·& w_{qn}\end{array}\right],$

(14)

wherein w_q1, w_q2, …, w _qn (q = Mt, T, M, C, F, P) denote the weights of the various factor differences in condition subvectors and their sum is 1. n is the number of the factors in various condition subvectors, when q = Mt, T, M, C, F, P, and the corresponding values of n are n = 5,11,8, 4, 2, 2, respectively.

It should be noted that the cooling mode in subvector of cooling, the part category and local feature in subvector of part feature, and the machining procedure and machining method in subvector of machining procedure and method cannot be quantified to describe. Therefore, different discrete values are used to represent these factors and the values and meanings of the factors which cannot be quantified are listed in Table 1.

Table 1:

Values and meanings of the factors which cannot be quantified.

Condition factor	1	2	3	4	5	6
Cooling mode	Coolant	Air cooling	Dry cutting
Part category	Open blisk	Closed blisk	Blade	Impeller
Local feature	Flow passage	Blade body	Wheel hub	Blade edge	Damper platform	Tenon
Machining procedure	Roughing	Semi-finishing	Finishing
Machining method	Slotting plunging	Slotting	Cylindrical milling	End milling	Point milling	Clean-up

For the factors of condition subvector represented by discrete values, if the discrete values are the same, their difference is 0, else, their difference is 1, when calculating the difference of condition subvectors.

4.2.3. Calculation Process

The determined weight matrixes of the difference and leveling matrixes are brought into (9), and the matching degree of the actual process condition and a specific process condition in database is calculated and obtained; the calculation process is shown in Figure 5.

Figure 5:

Calculation of condition matching degree.

4.3. Determination of the Weight Matrix of Difference and Scale Leveling Matrix

4.3.1. Weight Matrix of Difference

Using the method to determine the weights of various factors in AHP [23], the correlation between various subvectors in condition vector and process parameters is analyzed and the effect of various subvectors in condition vector on the difference of process condition is determined, respectively. Through analyzing the various subvectors in condition vector, using the nine-point scale in AHP [23] to compare the factors of various subvectors pairwise, the comparison matrix A is obtained. Each row of the comparison matrix indicates the importance of one factor relative to the other factors.

Consider

$\mathbf{A}=\left[\begin{array}{cccccc}\mathrm{1}& \frac{\mathrm{1}}{\mathrm{6}}& \frac{\mathrm{1}}{\mathrm{7}}& \frac{\mathrm{1}}{\mathrm{5}}& \frac{\mathrm{1}}{\mathrm{5}}& \frac{\mathrm{1}}{\mathrm{9}}\\ \mathrm{6}& \mathrm{1}& \frac{\mathrm{1}}{\mathrm{4}}& \mathrm{1}& \mathrm{1}& \frac{\mathrm{1}}{\mathrm{5}}\\ \mathrm{7}& \mathrm{4}& \mathrm{1}& \mathrm{3}& \mathrm{2}& \frac{\mathrm{1}}{\mathrm{3}}\\ \mathrm{5}& \mathrm{1}& \frac{\mathrm{1}}{\mathrm{3}}& \mathrm{1}& \frac{\mathrm{1}}{\mathrm{2}}& \frac{\mathrm{1}}{\mathrm{5}}\\ \mathrm{5}& \mathrm{1}& \frac{\mathrm{1}}{\mathrm{2}}& \mathrm{2}& \mathrm{1}& \frac{\mathrm{1}}{\mathrm{4}}\\ \mathrm{9}& \mathrm{5}& \mathrm{3}& \mathrm{5}& \mathrm{4}& \mathrm{1}\end{array}\right].$

(15)

The maximum eigenvalue of the comparison matrix A is calculated, it is λ_max = 6.2654, and the corresponding normalized eigenvector is p = (0.0264, 0.0968, 0.2308, 0.0861, 0.1190, 0.4408) ^T . Consistency index is CI = (6.2654 – 6)/(6 – 1) = 0.0531, the random consistency index obtained from the table in [23] is RI = 1.24, the consistency ratio is CR = CI/RI = 0.0531/1.24 = 0.0428 < 0.1, and this eigenvector satisfies the requirement of consistency as the weight vector corresponding to the weight matrix of difference [23]:

$\mathbf{w}={\left(w_{\mathrm{1}},w_{\mathrm{2}},\dots ,w_{\mathrm{6}}\right)}^T\hspace{1em}=\mathbf{p}={\left(\mathrm{0.0264,0.0968,0.2308,0.0861,0.1190,0.4408}\right)}^T.$

(16)

The same applies to machine tool, cutting tool, workpiece material, cooling, part feature, machining procedure, and method subvector; the comparison matrixes, maximum eigenvalues, normalized eigenvector (weight vectors), and consistency ratios are obtained and listed in Table 2, wherein the comparison matrixes of various condition subvectors are

$\begin{array}{c}{\mathbf{A}}_{\text{Mt}}=\left[\begin{array}{ccccc}\mathrm{1}& \mathrm{2}& \mathrm{1}& \mathrm{1}& \mathrm{7}\\ \frac{\mathrm{1}}{\mathrm{2}}& \mathrm{1}& \frac{\mathrm{1}}{\mathrm{2}}& \mathrm{1}& \mathrm{5}\\ \mathrm{1}& \mathrm{2}& \mathrm{1}& \mathrm{1}& \mathrm{5}\\ \mathrm{1}& \mathrm{1}& \mathrm{1}& \mathrm{1}& \mathrm{5}\\ \frac{\mathrm{1}}{\mathrm{7}}& \frac{\mathrm{1}}{\mathrm{5}}& \frac{\mathrm{1}}{\mathrm{5}}& \frac{\mathrm{1}}{\mathrm{5}}& \mathrm{1}\end{array}\right],\\ {\mathbf{A}}_T=\left[\begin{array}{ccccccccccc}\mathrm{1}& \mathrm{2}& \mathrm{2}& \mathrm{2}& \mathrm{2}& \mathrm{2}& \mathrm{3}& \mathrm{1}& \mathrm{2}& \mathrm{2}& \mathrm{2}\\ \frac{\mathrm{1}}{\mathrm{2}}& \mathrm{1}& \mathrm{1}& \mathrm{1}& \frac{\mathrm{1}}{\mathrm{2}}& \frac{\mathrm{1}}{\mathrm{2}}& \mathrm{2}& \frac{\mathrm{1}}{\mathrm{2}}& \mathrm{1}& \mathrm{1}& \mathrm{1}\\ \frac{\mathrm{1}}{\mathrm{2}}& \mathrm{1}& \mathrm{1}& \mathrm{1}& \frac{\mathrm{1}}{\mathrm{2}}& \frac{\mathrm{1}}{\mathrm{2}}& \mathrm{2}& \frac{\mathrm{1}}{\mathrm{2}}& \mathrm{1}& \mathrm{1}& \mathrm{1}\\ \frac{\mathrm{1}}{\mathrm{2}}& \mathrm{1}& \mathrm{1}& \mathrm{1}& \frac{\mathrm{1}}{\mathrm{2}}& \frac{\mathrm{1}}{\mathrm{2}}& \mathrm{2}& \frac{\mathrm{1}}{\mathrm{2}}& \mathrm{1}& \mathrm{1}& \mathrm{1}\\ \frac{\mathrm{1}}{\mathrm{2}}& \mathrm{2}& \mathrm{2}& \mathrm{2}& \mathrm{1}& \mathrm{1}& \mathrm{3}& \frac{\mathrm{1}}{\mathrm{2}}& \mathrm{2}& \mathrm{2}& \mathrm{2}\\ \frac{\mathrm{1}}{\mathrm{2}}& \mathrm{2}& \mathrm{2}& \mathrm{2}& \mathrm{1}& \mathrm{1}& \mathrm{3}& \frac{\mathrm{1}}{\mathrm{2}}& \mathrm{2}& \mathrm{2}& \mathrm{2}\\ \frac{\mathrm{1}}{\mathrm{3}}& \frac{\mathrm{1}}{\mathrm{2}}& \frac{\mathrm{1}}{\mathrm{2}}& \frac{\mathrm{1}}{\mathrm{2}}& \frac{\mathrm{1}}{\mathrm{3}}& \frac{\mathrm{1}}{\mathrm{3}}& \mathrm{1}& \frac{\mathrm{1}}{\mathrm{3}}& \frac{\mathrm{1}}{\mathrm{2}}& \frac{\mathrm{1}}{\mathrm{2}}& \frac{\mathrm{1}}{\mathrm{3}}\\ \mathrm{1}& \mathrm{2}& \mathrm{2}& \mathrm{2}& \mathrm{2}& \mathrm{2}& \mathrm{3}& \mathrm{1}& \mathrm{3}& \mathrm{3}& \mathrm{2}\\ \frac{\mathrm{1}}{\mathrm{2}}& \mathrm{1}& \mathrm{1}& \mathrm{1}& \frac{\mathrm{1}}{\mathrm{2}}& \frac{\mathrm{1}}{\mathrm{2}}& \mathrm{2}& \frac{\mathrm{1}}{\mathrm{3}}& \mathrm{1}& \mathrm{1}& \frac{\mathrm{1}}{\mathrm{2}}\\ \frac{\mathrm{1}}{\mathrm{2}}& \mathrm{1}& \mathrm{1}& \mathrm{1}& \frac{\mathrm{1}}{\mathrm{2}}& \frac{\mathrm{1}}{\mathrm{2}}& \mathrm{2}& \frac{\mathrm{1}}{\mathrm{3}}& \mathrm{1}& \mathrm{1}& \frac{\mathrm{1}}{\mathrm{2}}\\ \frac{\mathrm{1}}{\mathrm{2}}& \mathrm{1}& \mathrm{1}& \mathrm{1}& \frac{\mathrm{1}}{\mathrm{2}}& \frac{\mathrm{1}}{\mathrm{2}}& \mathrm{3}& \frac{\mathrm{1}}{\mathrm{2}}& \mathrm{2}& \mathrm{2}& \mathrm{1}\end{array}\right],\\ {\mathbf{A}}_M=\left[\begin{array}{cccccccc}\mathrm{1}& \mathrm{8}& \mathrm{7}& \mathrm{6}& \mathrm{8}& \mathrm{3}& \mathrm{3}& \mathrm{4}\\ \frac{\mathrm{1}}{\mathrm{8}}& \mathrm{1}& \frac{\mathrm{1}}{\mathrm{2}}& \frac{\mathrm{1}}{\mathrm{2}}& \mathrm{3}& \frac{\mathrm{1}}{\mathrm{4}}& \frac{\mathrm{1}}{\mathrm{4}}& \frac{\mathrm{1}}{\mathrm{3}}\\ \frac{\mathrm{1}}{\mathrm{7}}& \mathrm{2}& \mathrm{1}& \frac{\mathrm{1}}{\mathrm{2}}& \mathrm{3}& \frac{\mathrm{1}}{\mathrm{6}}& \frac{\mathrm{1}}{\mathrm{6}}& \frac{\mathrm{1}}{\mathrm{3}}\\ \frac{\mathrm{1}}{\mathrm{6}}& \mathrm{2}& \mathrm{2}& \mathrm{1}& \mathrm{3}& \frac{\mathrm{1}}{\mathrm{5}}& \frac{\mathrm{1}}{\mathrm{5}}& \frac{\mathrm{1}}{\mathrm{2}}\\ \frac{\mathrm{1}}{\mathrm{8}}& \frac{\mathrm{1}}{\mathrm{3}}& \frac{\mathrm{1}}{\mathrm{3}}& \frac{\mathrm{1}}{\mathrm{3}}& \mathrm{1}& \frac{\mathrm{1}}{\mathrm{6}}& \frac{\mathrm{1}}{\mathrm{6}}& \frac{\mathrm{1}}{\mathrm{3}}\\ \frac{\mathrm{1}}{\mathrm{3}}& \mathrm{4}& \mathrm{6}& \mathrm{5}& \mathrm{6}& \mathrm{1}& \mathrm{1}& \mathrm{2}\\ \frac{\mathrm{1}}{\mathrm{3}}& \mathrm{4}& \mathrm{6}& \mathrm{5}& \mathrm{6}& \mathrm{1}& \mathrm{1}& \mathrm{2}\\ \frac{\mathrm{1}}{\mathrm{4}}& \mathrm{3}& \mathrm{3}& \mathrm{2}& \mathrm{3}& \frac{\mathrm{1}}{\mathrm{2}}& \frac{\mathrm{1}}{\mathrm{2}}& \mathrm{1}\end{array}\right],\\ {\mathbf{A}}_C=\left[\begin{array}{cccc}\mathrm{1}& \mathrm{9}& \mathrm{9}& \mathrm{9}\\ \frac{\mathrm{1}}{\mathrm{9}}& \mathrm{1}& \mathrm{4}& \mathrm{1}\\ \frac{\mathrm{1}}{\mathrm{9}}& \frac{\mathrm{1}}{\mathrm{4}}& \mathrm{1}& \frac{\mathrm{1}}{\mathrm{3}}\\ \frac{\mathrm{1}}{\mathrm{9}}& \mathrm{1}& \mathrm{3}& \mathrm{1}\end{array}\right],\\ {\mathbf{A}}_F=\left[\begin{array}{cc}\mathrm{1}& \frac{\mathrm{1}}{\mathrm{3}}\\ \mathrm{3}& \mathrm{1}\end{array}\right],\hspace{1em}\hspace{1em}{\mathbf{A}}_P=\left[\begin{array}{cc}\mathrm{1}& \frac{\mathrm{1}}{\mathrm{3}}\\ \mathrm{3}& \mathrm{1}\end{array}\right].\end{array}$

(17)

Table 2:

Weight matrixes of the differences of condition sub-vectors.

Condition sub-vector	Comparison matrix	Maximum eigenvalue	Normalized eigenvector	Consistency index	Consistency ratio	As weight vector
Machine tool	AMt	5.0723	p Mt	0.0181	0.0162	Yes
Cutting tool	A T	11.2024	p T	0.0202	0.0134	Yes
Workpiece material	A M	8.4055	p M	0.0579	0.0411	Yes
Cooling	A C	4.2072	p C	0.0691	0.0768	Yes
Part feature	A F	2	p F	0	0	Yes
Machining procedureand method	A P	2	p P	0	0	Yes

The normalized eigenvectors corresponding to maximum eigenvalue of various comparison matrixes are

$\begin{array}{c}{\mathbf{p}}_{\text{Mt}}={\left(\mathrm{0.2829,0.1770,0.2660,0.2311,0.0430}\right)}^T,\\ {\mathbf{p}}_T=(\mathrm{0.1534,0.0678,0.0678,0.0678,0.1182,0.1182},\hspace{1em}{\mathrm{0.0368,0.1645,0.0617,0.0617,0.0821})}^T,\\ {\mathbf{p}}_M=(\mathrm{0.3631,0.0402,0.0448,0.0580},\hspace{1em}{\mathrm{0.0253,0.1854,0.1854,0.0977})}^T,\\ {\mathbf{p}}_C={\left(\mathrm{0.7373,0.1141,0.0452,0.1034}\right)}^T,\\ {\mathbf{p}}_F={\left(\mathrm{0.25,0.75}\right)}^T,\\ {\mathbf{p}}_P={\left(\mathrm{0.25,0.75}\right)}^T.\end{array}$

(18)

Therefore, the weight vectors corresponding to the weight matrix of difference of various condition subvectors are

$\begin{array}{c}{\mathbf{w}}_{\text{Mt}}={\left(w_{\text{Mt}\mathrm{1}},w_{\text{Mt}\mathrm{2}},\dots ,w_{\text{Mt}\mathrm{5}}\right)}^T={\mathbf{p}}_{\text{Mt}},\\ {\mathbf{w}}_T={\left(w_{T\mathrm{1}},w_{T\mathrm{2}},\dots ,w_{T\mathrm{11}}\right)}^T={\mathbf{p}}_T,\\ {\mathbf{w}}_M={\left(w_{M\mathrm{1}},w_{M\mathrm{2}},\dots ,w_{M\mathrm{8}}\right)}^T={\mathbf{p}}_M,\\ {\mathbf{w}}_C={\left(w_{C\mathrm{1}},\hspace{0.17em}{w}_{C\mathrm{2}},w_{C\mathrm{3}},w_{C\mathrm{4}}\right)}^T={\mathbf{p}}_C,\\ {\mathbf{w}}_F={\left(w_{F\mathrm{1}},w_{F\mathrm{2}}\right)}^T={\mathbf{p}}_F,\\ {\mathbf{w}}_P={\left(w_{P\mathrm{1}},w_{P\mathrm{2}}\right)}^T={\mathbf{p}}_P.\end{array}$

(19)

4.3.2. Leveling Matrix

The differences of various factors are transformed to values without dimension in the range of [anjan48] according to their actual range. The leveling vector is used to indicate the main diagonal elements of the leveling matrix.

The leveling vectors of various condition subvectors are

$\begin{array}{c}\mathbf{s}{\mathbf{l}}_{\text{Mt}}={\left(\text{s}{\text{l}}_{\text{Mt}\mathrm{1}},\text{s}{\text{l}}_{\text{Mt}\mathrm{2}},\dots ,\text{s}{\text{l}}_{\text{Mt}\mathrm{5}}\right)}^T={\left(\frac{\mathrm{1}}{\mathrm{2}},\frac{\mathrm{1}}{\mathrm{25}},\frac{\mathrm{1}}{\mathrm{1000}},\frac{\mathrm{1}}{\mathrm{2500}},\mathrm{500}\right)}^T,\\ \mathbf{s}{\mathbf{l}}_T={\left(\text{s}{\text{l}}_{T\mathrm{1}},\text{s}{\text{l}}_{T\mathrm{2}},\dots ,\text{s}{\text{l}}_{T\mathrm{11}}\right)}^T={\left(\frac{\mathrm{1}}{\mathrm{2}},\mathrm{1},\frac{\mathrm{1}}{\mathrm{2}},\mathrm{1,1},\frac{\mathrm{1}}{\mathrm{3}},\frac{\mathrm{1}}{\mathrm{20}},\mathrm{2},\frac{\mathrm{1}}{\mathrm{3}},\frac{\mathrm{1}}{\mathrm{2}},\frac{\mathrm{1}}{\mathrm{4}}\right)}^T,\\ \mathbf{s}{\mathbf{l}}_M={\left(\text{s}{\text{l}}_{M\mathrm{1}},\text{s}{\text{l}}_{M\mathrm{2}},\dots ,\text{s}{\text{l}}_{M\mathrm{8}}\right)}^T={\left(\frac{\mathrm{1}}{\mathrm{2}},\frac{\mathrm{1}}{\mathrm{10}},\frac{\mathrm{1}}{\mathrm{4}},\mathrm{40,2},\frac{\mathrm{1}}{\mathrm{50}},\frac{\mathrm{1}}{\mathrm{50}},\frac{\mathrm{1}}{\mathrm{10}}\right)}^T,\\ \mathbf{s}{\mathbf{l}}_C={\left(\text{s}{\text{l}}_{C\mathrm{1}},\text{s}{\text{l}}_{C\mathrm{2}},\text{s}{\text{l}}_{C\mathrm{3}},\text{s}{\text{l}}_{C\mathrm{4}}\right)}^T={\left(\mathrm{10,10},\frac{\mathrm{1}}{\mathrm{10}},\frac{\mathrm{1}}{\mathrm{10}}\right)}^T,\\ \mathbf{s}{\mathbf{l}}_F={\left(\text{s}{\text{l}}_{F\mathrm{1}},\text{s}{\text{l}}_{F\mathrm{2}}\right)}^T={\left(\mathrm{10,10}\right)}^T,\\ \mathbf{s}{\mathbf{l}}_P={\left(\text{s}{\text{l}}_{P\mathrm{1}},\text{s}{\text{l}}_{P\mathrm{2}}\right)}^T={\left(\mathrm{10,10}\right)}^T,\end{array}$

(20)

wherein sl_q1, sl_q2, …, sl _qn (q = Mt, T, M, C, F, P) represent the leveling coefficients of the differences of various factors in condition subvector.

If the absolute values of the differences of various factors, which are transformed through leveling transform, are still greater than 10, then 10 is taken as the absolute value, as the maximum difference. The factors of various leveling vectors correspond to the main diagonal elements of corresponding leveling matrixes; thus the leveling matrixes of various condition subvectors are obtained when the leveling vectors are determined.

5. Example and Application

Some process condition vectors and process parameters are taken out, for instance, to validate this method. Parts of condition subvectors in the process parameter database are listed in Tables 3, 4, 5, 6, 7, and 8.

Table 3:

Sub-vector of machine tool.

ID	Power (KW)	Spindle torque (Nm)	Maximum revolution (rpm)	Maximum feed rate (mm/min)	Position precision (mm)
3	24	162	15000	15000	0.010
5	12	70	10000	10000	0.012

Table 4:

Sub-vector of cutting tool.

ID	Radius (mm)	R c (mm)	R r (mm)	R z (mm)	a(∘)	b(∘)	Overhang length (mm)	Flute number	Rake angle (∘)	Flank angle (∘)	Helix angle (∘)
17	2.5	2.5	0	2.5	0	0	40	4	8	10	35
134	3.5	3.5	0	3.5	0	0	40	4	8	10	35

Table 5:

Sub-vector of workpiece material.

ID	Trademark	Rockwell hardness (HRC)	Elastic modulus (GPa)	Shear modulus (GPa)	Poisson ratio	Density (g/cm3)	Yield strength (MPa)	Tensile strength (MPa)	Thermal conductivity (W/m2K)
1	TC4	35	110	42	0.33	4.43	824	906	6.8
2	TC17	40	112	44	0.32	4.68	1030	1120	8.2
3	GH4169	37	199.9	77.2	0.3	8.24	550	965	13.4

Table 6:

Sub-vector of cooling.

ID	Cooling mode	Pressure (MPa)	Flow rate (L/min)	Temperature (°C)
27	1 (coolant)	0.6	40	25

Table 7:

Sub-vector of part feature.

ID	Part category	Local feature
11	3 (blade)	6 (tenon)
13	1 (open blisk)	2 (blade body)

Table 8:

Sub-vector of machining procedure and method.

ID	Machining procedure	Machining method
13	3 (finishing)	6 (clean-up)
15	1 (roughing)	3 (cylindrical milling)
17	3 (finishing)	5 (point milling)

These condition subvectors are combined into various process condition vectors; these process condition vectors are listed in Table 9.

Table 9:

Process condition vector.

ID	Machine tool ID	Cutting tool ID	Workpiece material ID	Cooling ID	Part feature ID	Machining procedure and method ID
10	3	17	3	27	11	17
14	3	17	1	27	13	17
31	5	134	3	27	11	13
38	3	17	2	27	11	15
43	3	17	3	27	13	13

The process parameter vectors corresponding to these process condition vectors in process parameter database are listed in Table 10.

Table 10:

Process parameter vector.

Condition ID	Spindle speed of revolution (rpm)	Cutting depth (mm)	Cutting width (mm)	Feed rate (mm/min)
10	10000	0.15	0.2	3000
14	8000	0.3	0.2	3000
31	6000	0.15	0.2	2500
38	8000	0.25	0.25	5000
43	8000	0.15	0.2	2500

A group of actual process conditions are taken out; this method is adopted to determine the process parameters of these actual process conditions. In order to express process condition succinctly, the process condition vectors are combined from the existing condition subvectors in the examples. For the situation of the condition subvectors of actual process condition is not in the process parameter database, the process and method of calculation are the same as these examples. The actual process conditions are listed in Table 11.

Table 11:

Actual process condition vector.

ID	Machine tool ID	Cutting tool ID	Workpiece material ID	Cooling ID	Part feature ID	Machining procedure and method ID
1	3	134	2	27	11	13
2	5	17	1	27	11	17
3	3	134	3	27	11	13
4	5	134	3	27	13	17
5	3	17	2	27	11	15
6	5	17	1	27	13	15

The matching degrees of various actual process conditions and process conditions already in the process parameter database are calculated using this method, and the results are listed in Table 12. In the table, the rows represent the actual process conditions and the columns represent the process conditions already in the process parameter database.

Table 12:

Matching degrees of actual conditions and existing conditions.

Condition ID	10	14	31	38	43
1	0.1594	0.1458	0.3829	0.1506	0.2344
2	0.5075	0.2836	0.1657	0.1464	0.1431
3	0.1739	0.1438	1.3688	0.1409	0.2896
4	0.2834	0.5064	0.1491	0.1262	0.1725
5	0.1409	0.1313	0.1401	+ ∞	0.1267
6	0.1229	0.1497	0.1298	0.2628	0.1444

Wherein, the matching degree of + ∞ means that the matching degree is infinite and the two process conditions are completely identical.

The matching degree of 0.3 is taken as the threshold of matching degree, and if the matching degree of two condition vectors is lower than 0.3, there is no correlation between the two condition vectors and the process parameters of the two condition vectors can not be cross-reference. When the matching degree is greater than 0.3, the greater the matching degree is, the more similar the process conditions are, according to the definition of the matching degree of process condition. Then the process parameters of process condition vector with the greatest matching degree in the process parameter database should be taken as the process parameters of the actual process condition.

The matched process condition and process parameters of various actual process conditions are obtained according to the results in Table 12, and the matched results and process parameters are listed in Table 13.

Table 13:

Matched results and process parameters of actual conditions.

Actualcondition	Matchingcondition	Spindle speed of revolution (rpm)	Cutting depth (mm)	Cutting width (mm)	Feed rate (mm/min)
1	31	6000	0.15	0.2	2500
2	10	10000	0.15	0.2	3000
3	31	6000	0.15	0.2	2500
4	14	8000	0.3	0.2	3000
5	38	8000	0.25	0.25	5000
6	None

All the matching degrees of actual process condition 6 and the process conditions in the process parameter database are less than 0.3, so there is no process condition matched to this process condition. Therefore, the process parameters of the actual process condition 6 cannot be elected in the process parameter database.

The typical aero-engine part CNC machining process database system is designed and developed using the method of condition division and matching proposed in this paper. The database system contains the sub-databases of machine tool, cutting tool, workpiece material, cooling, part feature, machining procedure and method, and process parameter. The process data accumulated in practical production are arranged into the database system for the use in selection of process parameters. In this system, enter the actual process condition in the parameter query interface, then the corresponding process parameters could be checked out.

6. Conclusion

In this study, the following useful conclusions can be drawn.

1. The division and vector expression of the process condition were presented in this paper and they could be used to describe the CNC machining process condition of complex structure and difficult-to-cut material parts. The milling process condition of aero-engine part was expressed accurately through this method.

2. The accumulated process data were arranged according to the condition vector and process parameter vector, then the mapping of the existing process conditions to the process parameters was established. A large amount of process data and their mappings were stored in the process parameter database, as the basic data of the optimal selection of process parameters.

3. On the basis of the process condition vector, process condition matching degree was defined and its calculation was presented. Matching degree was used to represent the similarity of actual process condition and existing process condition. The greater the matching degree was, the more similar the process conditions were and the more reference value the process parameters had. The process condition matching degree provided a theoretical basis for the intelligent matching of CNC machining process condition.

4. The optimal selection of process parameters based on process condition matching was proposed. This method was based on the process parameter database, aimed at the CNC milling of the complex structure and difficult-to-cut material aero-engine parts, and achieved the optimal selection of process parameters. Finally, the feasibility and effectiveness of this method were verified through a group of instances.

This work investigated various factors in machining process comprehensively, utilized the process data that evolved naturally and accumulated in practical production adequately, and avoided the deviation caused by simplifying optimization model in traditional parameter selection effectively. The succession of the process parameters of the same category part was enhanced and the actual availability and accuracy of the optimal parameters were improved, for the complex parts. An effective program was provided for the automated selection of the optimal process parameters and the design and development of the typical aero-engine part CNC machining process database system.

Conflict of Interests

The authors declare no conflict of interests.