Influence of uncertain parameters on machining distortion of thin-walled parts

Thin-walled parts refer to lightweight structural parts comprised of thin plates and stiffeners. During the machining process of thin-walled parts, machining distortion often occurs due to uncertain factors such as varying stiffness, cutting force, cutting temperature, residual stress, and other factors. This paper studied the minimization of the failure probability of machining distortion by controlling the uncertainties of inputs. For this, a fuzzy inference model for the machining system was proposed to determine the effects of uncertain factors on the machining distortion errors, which was composed of rule frame and result frame. In the rule frame, machining parameters, outline size, and wall thickness were used as inputs. In the result frame, linear stiffness, cutter path, as well as cutting force were taken as the input parameters. The values of machining distortion were the output, taken into a threshold function. Comprehensive matching was defined to measure the importance of uncertain inputs to outputs. Machining distortion will exceed the specification (failure) with the increase in comprehensive matching. Therefore, the comprehensive matching index evaluates the effects of the uncertainties on the machining distortion and quantifies the effects of given uncertain parameters. Two engineering examples were employed to illustrate the accuracy and efficiency of the proposed approach. It revealed that the comprehensive matching of cutting force to the failure probability of machining distortion was the maximum, 0.040, which was 12 to 13 times greater than that of linear stiffness or cutter path.


Introduction
Thin-walled components refer to lightweight structural parts composed of various thin plates and stiffeners, where the wall thickness is less than 2.5 mm or the ratio between wall thickness and outline size is less than 1:10. Uncertainties occurring in the machining process of thin-walled components will lead to uncertain performance, such as dimensional accuracy and surface finish quality. Some researchers proposed that processing technology optimization, auxiliary support, high-speed machining, and numerical control compensation technology can control the machining distortion [1,2]. To evaluate the effects of uncertain factors on the 1 3 machining distortion, uncertainty analysis has been successfully and widely applied in engineering.
In view of the processing technology optimization, the research on the machining distortion of thin-walled components is carried out from the aspects of machining parameters optimization, tool path, and clamping layout [3].
Machining parameter optimization can appropriately reduce the load of cutting force, so as to achieve the purpose of controlling the machining distortion. Sridhar et al. [4] obtained the combination of machining parameters with minimal machining distortion based on the orthogonal test. Xue et al. [5] proposed a synchronous optimization algorithm based on finite element method and genetic algorithm. Minimum distortion was used as the objective function, the results showed that the optimal cutting speed was 97.3 m/ min and the feed rate was 0.142 mm/z. Hu et al. [6] analyzed the influence of cutting speed, depth of cut, width of cut, and feed rate on the machining distortion through finite element orthogonal advantage analysis method. Cong et al. [7] proposed distortion simulation and prediction method according to the mapping mechanism of machining conditions on residual stress. The optimization results showed that feed rate was 0.0599 mm/z, cutting speed was 72.5627 m/min, and depth of cut was 0.109 mm. Additionally, Vipindas et al. [8] investigated the single effect and interactive effects of cutting speed, feed rate, and depth of cut on surface roughness and top burr formation during micro end milling of Ti-6Al-4 V.
Cutter path has indirect effects on the machining distortion through the influence release sequence of residual stress and stiffness of the workpiece. Li et al. [9] changed the problem of tool path optimization to compensate the machining distortion into mixed integer linear programming. Branch and bound method was used to solve the problem and realize the optimal positioning of tool path. Wang et al. [10] studied the influence of rib structure on machining distortion of thin-walled components through simulation model of milling process of such components. They found that the distortion increased obviously with the increase of rib spacing. Ma et al. [11] investigated the influence of machining path on the machining distortion in view of the "Jiugongge" type cavity thin-walled components. It was concluded that milling with "outside" and "inside" style was the best. Tan et al. [12] proposed that proper selection of the cutting orientation is essential in achieving the better-quality level. The results showed that horizontal downward orientation led to the highest cutting forces and vertical orientation led to the best tool life.
Due to the low rigidity of thin-walled components, the shape of the workpiece will be affected by the clamping force of the fixture. If additional stress is generated due to improper selection of clamping support points, the thinwalled components will be obviously distorted. Fei et al. [13] proposed that machining distortion can be controlled through adding supporting fixture elements on the back of tool-workpiece. Rex et al. [14], Li et al. [15], and Milad et al. [16] achieved multi-objective optimization of fixture layout and clamping force using genetic algorithm, in which case, the distortion value was reduced and distortion uniformity was also improved.
At present, the distortion problems of thin-walled components were solved through processing technology optimization, clamping layout, auxiliary support technology, high-speed cutting technology, and numerical control compensation technology. However, the ideal effect was only achieved in specific components. Due to the complex and changeable characteristics of thin-walled components, uncertainties play a significant role in the milling process. Most studies did not consider the effects of the uncertainties of these factors on the error distribution of the machining distortion. According to the researches, stiffness, cutter path, and cutting force are of great uncertain. Therefore, this study explores the effects of the uncertainties of linear stiffness, cutter path, and cutting force on the machining distortion and defines failure probability-based importance measures for these uncertain factors. In the paper, based on the given values, the failure probability of machining distortion is taken as the output and the sensitivities of uncertain variables to the machining distortion are measured.
The structure of this paper is organized as follows. In Section 2, a new approach to evaluate the effects of uncertain variables on the machining distortion is described. Furthermore, the definition of the failure probability of machining distortion and the establishment of a fuzzy process for machining distortion are described in detail. In Section 3, several examples are presented to illustrate the accuracy and efficiency of the proposed method. Main conclusions are drawn in Section 4.
2 Importance measure of input parameters on the failure probability of machining distortion

Definition of the failure probability of machining distortion
Machining distortion can be described as a function [17], as presented in Eq. (1): where Y is the machining distortion and X 1 to X 4 represent uncertain input variables. Once the machining distortion becomes large enough to cause unacceptable workpiece dimensional errors, the machining distortion can be regarded as a failure. Momentindependent importance measure analysis can reflect the influence of uncertainties of inputs on statistical characteristics of response output [18]. Based on this method, importance measures of uncertain input variables on the failure probability of machining distortion are defined as: where δ i reflects how uncertainty in the output can be apportioned to different sources of uncertainty in the model input; I F represents the indicator function in the failure area, as shown in Eq. (3) and Fig. 1; E(I F (X)) represents the failure probability of machining distortion under the influence of X i ; E(I F (X|X i )) represents the condition failure probability of machining distortion under the influence of X i . The indicator function I F is defined as: where a and b are threshold values for acceptable distortion and g is the actual distortion. If g is from a to b, then the machining distortion is feasible. Once g is out of the range, the value of machining distortion exceeds the specification, i.e., failure.
Based on the index in Eq. (2), the effects of input variables on the failure probability of machining distortion are analyzed and the failure probability can be minimized through controlling the uncertainties of input variables. By means of the three-point estimate method [19], the input variables with a high importance to the failure probability of machining distortion are obtained. Firstly, three estimate points (X i (1) , X i (2) , and X i (3) ) are extracted from the input variables X i ; secondly, standard locations and weights of three estimate points can be obtained by solving the following Eqs. (4)-(9); Thirdly, the estimate values of δ i are computed using Eq. (10).
where X i (1) , X i (2) , and X i (3) represent estimate points of X i ; l X (1) i , l X (2) i , and l X (3) i are the standard locations of the three estimate points; P X i (1) , P X i (2) , and P X i (3) are the weights of the three estimate points; 1X i , 2X i , 3X i , and 4X i are mean value, standard deviation, kurtosis, and skewness of X i , respectively; δ i represents the importance measure for each input variable.

AND-OR-TREE for the machining system
AND-OR-TREE decomposes complex origin problems into several subproblems and then solves uncertainties in subproblems [20]. Machining processes of thin-walled components are filled with uncertainties and fuzzy values. Throughout the machining process, external forces, environment, and internal state are time-variant. Moreover, machining qualities are uncertain due to the time-variant characteristics. Hence, an AND-OR-TREE is utilized to model the behavior of machining systems which includes uncertain input and fuzzy output, as shown in Fig. 2. The

Failure area
Expected distortion distortion error of the finished surface is caused by varying stiffness, length of workpiece, cutter path, and cutting force. Simultaneously, outline width, wall thickness, depth of cut, and width of cut have significant impacts on workpiece stiffness [21]. Cutting speed, feed rate, depth of cut, and width of cut influence the cutting force [21]. A fuzzy inference frame is developed to solve the problems in the AND-OR-TREE, presented in Fig. 3. In the study, the fuzzy inference frame is used to study the impacts of workpiece geometry, machining parameters, and cutter path on the machining distortion, which is divided into two modules: rule frame (IF-THEN) and result frame (IF-THEN). The rule frame is composed of fuzzy conditions, uncertain evidences, rule importance, evidence importance, and certainty factors [22]. Evidences directly influence the fuzzy conditions, which represent attribute characteristics of each rule frame. Each evidence in the rule frame is assigned a real number between 0 and 1 that reflects the impact level of each evidence on the fuzzy conditions, called evidence importance. In addition, rule importance is defined to evaluate the effects of rules on the results. The certainty factor is a numerical value between − 1 and + 1, defined as a measure of belief and disbelief of the evidence [23]. The established inference frames are expressed in Tables 1  and 2. The result frame consists of fuzzy conditions, fuzzy outputs, and rule importance.

Failure area
In the study, the evidences (k 1 to k 6 ) have negligible effects on fuzzy conditions 2 (X 2 length) and 3 (X 3 cutter path); therefore, the effects of these evidences on the fuzzy condition 1 (X 1 stiffness) and fuzzy condition 4 (X 4 cutting force) were mainly investigated.

AND AND
As illustrated in Fig. 3, each fuzzy inference structure includes input layer and fuzzy output layer. Input and output membership can be expressed as: Rule frame: RuleID_1: therefore, the analysis of the stage is discussed in detail in the subsequent section.

Fuzzy inference model
The detailed procedure for quantifying the influence of uncertain factors on the machining distortion is illustrated in Fig. 5. The analysis is carried out as follows: Step 1: uncertain factors in the machining system are collected.
Step 2: in the corresponding knowledge base, uncertain input and fuzzy output are determined through establishing the AND-OR-TREE for the machining system.
Step 3: uncertain factors are classified into two categories: evidences and conditions according to the inference frame. Constraints of evidences and conditions are given.
Step 4: whether the attribute features of fuzzy conditions match with evidences is evaluated. The matching degree reflects the impacts of evidences on the fuzzy conditions. The matching degree of evidences to associated conditions is an element in the fuzzy matrix. Evidences with lower matching degree will be filtered out.
Step 5: the importance measure for the influence of each rule on the machining distortion is implemented. Further details of the calculation of importance measure will be discussed in the next section.
Step 6: based on matching degree of evidences to all the fuzzy conditions, certainty factors of evidences and rule importance, comprehensive matching is calculated. Whether the value of comprehensive matching is greater than s is evaluated. Notably, s is a threshold value [25]. If no, go to Step 7; if yes, go to Step 8. Further details of the calculation of matching degree will be provided in Section 2.3.3.
Step 7: whether the matching degree, importance measure, and comprehensive matching are in accord with the actual RuleID_4: Result frame: where "?" represents a threshold value, which depends on the research object.

Detailed procedure for fuzzy inference process
The fuzzy inference process for machining distortion is depicted in Fig. 4. The process consists of two stages. In stage I, thin-walled components are machined with specified input parameters and then distortion data is collected. Geometrical parameters of the designed workpiece, such as length, width, and thickness and machining parameters, such as cutting speed, feed rate, depth of cut, width of cut, and cutter path are input into the machine system. Moreover, a dynamometer is used to monitor the cutting force. In stage II, statistical characteristics such as standard deviation, mean value, minimum values, maximum values, kurtosis [24], and skewness [24] of input parameters are summarized. In this stage, the fuzzy inference model for machining distortion is developed to describe the impacts of uncertain inputs on the fuzzy output. Noteworthy, stage II is an important stage; 1 3 machining system is recognized through the comparison with required thresholds.
Step 8: measures for minimizing the machining distortion are proposed.

Calculation of comprehensive matching
Comprehensive matching describes the matching degree of uncertain inputs induced distortion to the failure area of machining distortion. In other words, the greater the comprehensive matching value, the closer the machining distortion caused by uncertain factors is to the failure area. The main procedure for calculating comprehensive matching is illustrated as follows: Step 1: a local normalized fuzzy distance between various uncertain evidences is calculated as: where k i and k j represent uncertain evidences; μ ki and μ kj are membership functions. The normal fuzzy function [25] is used as the membership function in the study as shown: where X i represents fuzzy conditions; σ 2 is the variance of the evidence; a is an empirical value.
Step 2: the matching degree of evidences to associated conditions is calculated in Eq. (16).
where match � reflects the effects of the uncertainties of evidences on the associated fuzzy condition.
Step 3: the matching degree of evidences to all the fuzzy conditions is calculated in Eq. (17).
where match X i , k reflects the impacts of uncertainties of evidences on the fuzzy conditions.
Step 4: based on the matching degree of evidences to all the fuzzy conditions, certainty factors of evidences and rule importance, the comprehensive matching degree is obtained. If fuzzy conditions connect through an AND node, the comprehensive matching degree is shown in Eq. (18). If fuzzy conditions connect through an OR node, the comprehensive matching degree is shown in Eq. (19).
where CF i represents the certainty factor of the evidence; ST rule i represents the rule importance.
The comprehensive matching formula for the failure probability of machining distortion is designed as follows: where MD represents machining distortion, X 1 to X 4 represent uncertain inputs.

Experimental procedure
The high-speed milling operations were implemented on a DMC 75 V linear machining center, which has a maximum spindle speed of 18,000 rpm. The sample consisted of 7050-T7451 aluminum alloy machined from a rectangular block of 155 mm (length) × 85 mm (width) × 20 mm (height).
Two flutes flat cemented carbide end mills were employed in this work. The diameter and helix angle of the end mills were 10 mm and 30°, respectively. Figure 6 shows the schematic of the machining experiment. The machining distortions of the thin-walled components were detected using a coordinate measuring machine (CMM), as shown in Fig. 7. Following the distortion measuring method of thin-walled components proposed by Fan et al. [26], the distortion values of the workpiece are measured along two directions (length direction and width direction). A measurement point is set as a base point and a plane containing the measurement point is set as a base plane. Then, the coordinate values of the measurement points of the ideal model are obtained. After machining, the distortion measuring process is carried out on the finished surface and the measurement points are arranged in accordance to the ideal model. The distortion errors of measurement points are obtained by the deviation between measured values of the finished surface and the values of the ideal mode.

Design of experiment
The selection of workpiece geometry, machining parameters, and cutter path affect the distortion of the machined components. As for aircraft structures such as beams, ribs and frames, geometrical parameters have significant impacts on workpiece stiffness [27]; therefore, a T-beam was selected for evaluation in this paper. Among the machining parameters, the following factors significantly influence machining distortion: cutting speed, feed rate, depth of cut, and width of cut. The levels adopted for the factors were defined based on a variety of conditions such as cutter, workpiece material, and cutting strategy, as shown in Table 3. In addition, four cutter path strategies were employed to investigate the distortion error of the machined surface. Milling of surface features was selected for the study. Single-direction milling is a cutter path strategy where the cutter moves in parallel lines across the surface to be machined (see Fig. 8a). The cutter moves across the machined surface, steps over a fixed amount, and moves back to the original position through air before milling across another line. In back-and-forth milling, the cutter draws a zigzag cuter path by moving back and forth across the workpiece (Fig. 8b). This strategy causes the cutter to mill alternatively along the rotational direction and then against it, resulting in up and down milling, respectively. Spiral milling from outwards to inwards is a strategy where the cutter starts at the corner of the feature and then proceeds spirally inwards (Fig. 8c). Spiral milling from inwards to outwards means that the cutter expands from the inner face gradually to the peripheral boundaries of the surface to be machined (Fig. 8d).

Distortion of T-beam
A number of experiments were conducted for evaluating the effects of machining parameters (cutting speed, feed rate, depth of cut, and width of cut) on the distortion errors of a T-beam (Fig. 9). The values of machining parameters can be selected from Table 3. As illustrated in Fig. 10, distortions of the flange top surface along the z-direction and distortions of the web surface along the y-direction were mainly studied. The flange of the T-beam tilts at both ends and pits in the middle. The web surface of the T-beam shows irregular distortion along the y-direction. Maximum distortion of the flange was approximately 0.07 mm (z-direction) and maximum distortion of the web was 0.015 mm (y-direction). Moreover, the maximum bending distortion of the flange was taken as the distortion of the T-beam in the study. As illustrated in Fig. 11a, distortion generally decreases at first and then increases with an increase in cutting speed from 251.2 to 533.8 m/min. With the increase in cutting speed, the cutting temperature rises and the friction coefficient decreases [28]. With the further increase in cutting speed, the dynamic components of the cutting force increase, resulting in an increase in total cutting force, in which case distortion starts to increase [28]. As depicted in Fig. 11b and c, distortion increases with an increase in feed rate and depth of cut. Higher feed rate and depth of cut cause high machining loads due to a high rate of material removal which increases the friction between cutter and workpiece interfaces [28]. As visible in Fig. 11d, width of cut has minimal effects on the distortion (Fig. 11).

Distortion of surface features
Milling of surface features in machining thin-walled components can be accomplished by employing the four cutter path strategies mentioned in Section 3.2. Machining parameters for the surface features are shown in Table 4. Figure 12 shows the geometrical parameters of the machined surface feature while Fig. 13 shows the machined samples. Based on the distortion results obtained from the experiments depicted in Fig. 13 and summarized numerically in Table 5, the effect of the cutter path from spiral milling from inwards to outwards on the distortion errors is the smallest among the four strategies. The cutter path from inwards to outwards makes the residual stress distribute evenly and part of the residual stress can be offset; therefore, the machining distortion is reduced.

Uncertain parameters data set
Based upon the analysis in Section 2, stiffness, length, cutter path, and cutting force are taken as fuzzy conditions. Width, thickness, depth of cut, width of cut, cutting speed, and feed rate are taken as uncertain evidence. Machining distortion is chosen as fuzzy output. The described T-beam and surface feature setups are used as the test cases. Furthermore, a distortion error greater than 0.03 mm was defined as a failure in the study.

Case 1 T-beam.
In the first case, the impact of each evidence on the partial fuzzy conditions was studied and then the effects of fuzzy conditions on the machining distortion were further investigated. Based on Eq. (15), membership functions for each evidence are illustrated in Eq. (21): Impact of uncertainty of each evidence on the fuzzy condition X 1 (stiffness): • Width and thickness of workpiece: where k 1 and k 2 represent width and thickness, respectively, whose constraints are determined according to the theory of thin plates [29]. X 1 is stiffness. X i is the membership function. If the value of X i is equal to 0, the uncertainty of evidence will not influence the fuzzy condition X i .

Depth of cut:
where k 3 represents the depth of cut, the range of which is determined based on a typical finishing operation.
Width of cut: where k 4 represents the width of cut, the range of which is also determined based on a typical finishing operation.
Impact of uncertainty of each evidence on the fuzzy condition X 4 (cutting force): Depth of cut and width of cut: These two membership functions are shown in Eqs. (22) and (23).
Cutting speed: where k 5 represents cutting speed, the constraints of which are determined by the selection principle of machining parameters for thin-walled components, i.e., high cutting speed and medium feed rate.
Feed rate: where k 6 represents the feed rate. The constraints of feed rate and cutting speed are also defined according to the selection principle of machining parameters for thin-walled components.
Case 2: surface feature In the second case, the effects of cutter path (fuzzy condition) on the machining distortion were directly studied. Therefore, membership functions were not established for the case.

Fuzzy inference process
The fuzzy inference process is used to measure the effects of distributed parameters of uncertain input on the machining distortion. Therefore, rule frames and result frames for two cases were established. In case 1 (T-beam), the relationships between the inputs (fuzzy conditions and evidences) and the output (machining distortion) were studied using RuleID_1&2, RuleID_4, ResultID_1&2, and Result_4. Linear stiffness is defined to demonstrate the impact of rules (RuleID_1 and RuleID_2) on the output (see Eq. (26)). In case 2 (surface feature), RuleID_3 and ResultID_3 were extracted to determine the effects of inputs (fuzzy condition) on the output. Detailed inference processes are listed in the next paragraphs.
where EI is instantaneous stiffness, L is workpiece length, and K is linear stiffness. Step 1: certainty factors of evidences are calculated.
Step 2: inference structures for machining distortion are checked, i.e., noisy-AND and noisy-OR models.
Step 3: based upon the certainty factors of evidences, evidence importance, and inference structures, rule importance is determined.
Step 4: comprehensive matching between fuzzy conditions and outputs is obtained.

Case 2
The fuzzy inference process for surface feature is illustrated in Fig. 15. Inference procedures in the second case are similar to the first case.

Comprehensive matching
Comprehensive matching is used to recognize whether the fuzzy conditions influence machining distortion. When the value of comprehensive matching is greater than a threshold value [30,31], the rule composed of fuzzy conditions and evidences can be adopted to reflect the impacts of uncertain factors on the machining distortion. The calculation process for comprehensive matching can be found in Section 2.3.3.

Certainty factor of evidence
The certainty factor of evidence (Table 6) is a measure of uncertainty associated with a hypothesis, expressing the degree to which the observation of evidence influences the confidence in the hypothesis, as shown in the following: where h is ratio; E(k i ) and √ Var(k i ) are mean value and standard deviation of the input variables, respectively; 1 − 1 h 2 is taken as the certainty factor [32].
Rule importance Rule importance evaluates the effects of uncertainty of each rule frame on the result throughout the fuzzy inference process. In Section 2.1, δ i is used as an indicator to measure the effects of uncertain input on the output; therefore, rule importance can be calculated from the procedures in Eqs. (4)- (10). As shown in Fig. 16, among four rule frames, RuleID_4 (i.e., cutting force) is the biggest influential factor. The reason why the cutting force strongly influences the machining distortion is not only due to machining parameters but also the stiffness of the workpiece. In the radial direction, the displacement of the workpiece causes an increase or decrease in the actual radial depth of cut, which further influences the cutting force. In the steady cutting process, there is a certain relationship between the cutting force, the radial depth of cut, and the distortion caused by the vibration of the workpiece. In addition, for different cutter paths, different overcutting quantities are brought about owing to a dissimilarity of instantaneous stiffness in different positions of the workpiece. Moreover, different cutter path will cause totally different residual stress distribution [33]. This is the reason why RuleID_3 (i.e., cutter path) affects the machining distortion.

Comprehensive matching
Based on the fuzzy distance between evidences and rule importance, comprehensive matching of each fuzzy condition to the machining distortion is evaluated. The detailed procedures for solving the comprehensive matching are shown in Section 2.3.3. For the two cases in the study, fuzzy distance and comprehensive matching are presented in Fig. 17. Comprehensive matching of each fuzzy condition to the machining distortion is illustrated in Table 7. The comprehensive matching of X 4 to MD is the maximum, which illustrates that the uncertainty of X 4 significantly influences the failure probability of machining distortion. Moreover, the comprehensive matching of X 4 to MD is 13 times greater than that of X 1 and X 2 to MD and 12 times greater than that of X 3 to MD. In other words, the effects of uncertainty of cutting force on the failure probability of machining distortion is 12 to 13 times greater than that of linear stiffness or cutter path. Furthermore, it is observed that the total effects of uncertainties of linear stiffness, cutter path, and cutting force on the failure probability of machining distortion are 0.044. Therefore, when the geometrical parameters of the workpiece are determined, failure probability of machining distortion can be decreased through minimizing the  uncertainty of cutting force and choosing an optimal cutter path. Based upon the analysis in Section 4.1, the distortion can be decreased through adopting low depth of cut (0.5 mm), high width of cut (2 mm), high cutting speed (475 m/min), and medium feed rate (0.15 mm/z) as the finishing parameters. Furthermore, the machining distortion can also be reduced by using the cutter path from inwards to outwards.   The fuzzy distance in RuleID_1&2 means the distance between k 1 , k 2 , k 3 , and k 4 ; the fuzzy distance in RuleID_4 represents the distance between k 3 , k 4 , k 5, and k 6 . λ match ' represents the matching degree of evidences to the associated condition, for example, λ match1&2 ' reflects the effects of evidences (k 1 , k 2 , k 3 , and k 4 ) on the fuzzy condition (X 1 and X 2 ) and λ match4 ' reflects the effects of uncertainties of evidences (k 3 , k 4 , k 5 , and k 6 ) on the fuzzy condition (X 4 ).

Model comparison
As with the previous researches, comparison between fuzzy inference method and other uncertainty analysis models is carried out. Uncertainty analysis models include DRSS (Distributed receptorbased sourceappointment statistical) model, BRRT (Bayesian recursive regression tree) model, and EILP (enhanced-interval linear programming) model [34]. As illustrated in Table 8, fuzzy inference model proposed in the paper is the preferred solution. In addition, the proposed model can also be used in stainless steel, titanium alloy, etc.

Conclusions
In this paper, a fuzzy inference model for machining distortion was developed to estimate the effects of uncertain factors on the failure probability of machining distortion. The main conclusions drawn are as follows: (1) The failure probability-based importance measure was defined, which measured the effects of input variables on the failure probability of machining distortion. (2) An AND-OR-TREE for machining systems was established through association between uncertain inputs and fuzzy outputs, which was composed of outputs, fuzzy conditions, and uncertain evidences. IF-THEN rules were introduced to conduct the uncertainty analysis. (3) A fuzzy inference model for machining distortion was developed, which was divided into rule frame and result frame. In the rule frame (evidences and fuzzy conditions), outline width, wall thickness, depth of cut, width of cut, cutting speed, and feed rate were taken as the inputs, stiffness, length, cutter path, and cutting force were regarded as the outputs. In the result frame (fuzzy conditions and output), the stiffness, length, cutter path, and cutting force were regarded as inputs and machining distortion were as the output. Additionally, comprehensive matching was used as the importance measure indicator. (4) T-beam and surface feature were employed to illustrate the efficiency of the proposed method. The comprehensive matching of cutting force to machining distortion is the maximum, which is 13 times greater than that of linear stiffness to machining distortion and 12 times greater than that of cutter path to machining distortion. In other words, cutting force had the highest effect on the machining distortion, 0.040. This paper only studied the effects of uncertainties of linear stiffness, cutter path, and cutting force on the machining distortion. In the further studies, the influence of other uncertain factors (e.g., cutting temperature and clamping units) in the machining system on the machining output can be investigated.
Funding This work is supported by the Shandong Natural Science Foundation (Grant No. ZR2022QE043) and Scientific research project of young outstanding talents of Qingdao university (Grant No. DC2200000908).
Data availability All data generated or analyzed during this study are included in this manuscript.

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