Xioyue LI, Ling LI,*, Yinfei YANG, Guolong ZHAO, Ning HE,Xiocen DING, Yowen SHI, Longxin FAN, Hui LAN,Muhmmd JAMIL

a National Key Laboratory of Science and Technology on Helicopter Transmission, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China

b AVIC Jiangxi Hongdu Aviation Industry Group Company Ltd, Nanchang 330000, China

KEYWORDS Finishing allowance;Linear programming;Machining deformation;Residual stresses;Simplex algorithm

Abstract Owing to reliability and high strength-to-weight ratio,large thin-walled components are widely used in the aviation and aerospace industry. Due to the complex features and sequence involved in the machining process of large thin-walled components,machining deformation of component is easy to exceed the specification.In order to address the problem,it is important to retain the appropriate finishing allowance. To find the overall machining deformation, finishing allowance-induced deformation (web finishing allowance, sidewall finishing allowance) and initial residual stress-induced deformation were considered as major factors. Meanwhile, machined surface residual stress-induced deformation, clamping stress-induced deformation, thermal deformation, gravity-induced deformation and inertial force-induced deformation were neglected in the optimization model. Six-peak Gaussian function was introduced to fit the initial residual stress.Based upon the obtained function of initial residual stress,a deformation prediction model between initial residual stress and finishing allowance was established to attain the finishing allowanceinduced deformation. In addition, linear programming optimization model based on the simplex algorithm was developed to optimize the overall machining deformation. Results have concluded that the overall machining deformation reached the minimum value when sidewall finishing allowance and web finishing allowance varied between 1 and 2 mm. Additionally, web finishing allowance-induced deformation and sidewall finishing allowance-induced deformation were 1.05 mm and 0.7 mm. Furthermore, the machining deformation decreased to 0.3-0.38 mm with the application of optimized finishing allowance allocation strategy,which made 39-56%reduction of the overall machining deformation compared to that in conventional method.

1. Introduction

Large thin-walled components with light weight and high reliability are widely used in the aerospace fields such as the flange in wing, girder, web and bulkhead in fuselage.1In order to ensure the machining accuracy of such kind of components,it is necessary to retain the appropriate finishing allowance.However, improper allocation of the finishing allowance leads to oversize or undersize components,or even the final components may be scrapped.2

During the machining of large thin-walled components,it is difficult to satisfy the design requirements owing to lowstiffness,3low-rigidity4and machining deformation caused by machined surface residual stress, clamping conditions, heat,gravity, inertial force, initial residual stress and improper finishing allowance. Thus, increasing stiffness and controlling machining deformation are key issues in the NC machining.A lot of research have been done to address the problems.

The generation of machined surface residual stress involves various factors such as cutting parameters, machine tool performance, blank properties, interaction between components structure and machining environment, and so on. Li et al.5mentioned that less machined surface residual stress generated by using different cutting depths at different cutting stages.Huang et al.6analyzed the coupling effects of machined surface residual stress and initial residual stress on the deformation of an aluminium alloy plate. Meng et al.7predicted the deformation caused by the machined surface residual stress in the components with different stiffness. Madariaga et al.8found that the machining deformation is determined by the magnitude and location of the residual stress profile induced by the cutting conditions. Huang et al.9obtained the required distribution of the residual stress by utilizing known residual stress to calculate the machining parameters.Jiang et al.10conducted the interference fit hi-lock bolt insertion experiment.They found that the tensile hoop stress was produced mainly on the hole wall,and transformed into compressive hoop stress when interference fit size was large. Although the redistribution and the release of the machined surface residual stress resulted in the deformation, machined surface residual stress was still not the major factor causing deformation since the influencing depth of machined surface residual stress was less than 0.2 mm.11

Clamping conditions such as clamping sequence, fixture layout and clamping force also affect the machining deformation. Their essence is to maximize the components stiffness to realize the deformation resistance. Kang et al.12optimized the clamping force based on the three-dimensional stability model of fixture system.Lu and Zhao.13proposed a method of fixture layout optimization for sheet metal parts based on 4-2-1 positioning scheme. They optimized the positioning points with genetic algorithm and presented a neural network model to predict the deformation. Rex et al.14developed an artificial neural network model to forecast the elastic deformation within regular range of fixture system design parameters.Sundararaman et al.15discovered the relationship between the position of locator and fixture and the maximum deformation through response surface method. Jiao and Xing.16studied a novel method on account of the heuristic algorithm to optimize the machining sequence. Zhou et al.17found a synchronous optimization method of fixture layout and clamping sequence to solve the clamping deformation problem.

Even if chips can take away part of the cutting heat, residual temperature field will still cause thermal deformation. Lu et al.18discovered the behavior of 7075 aluminum alloy plate through several isothermal uniaxial tensile tests. They found a linear relationship between temperature and strain. Rai and Xirouchakis.19presented a transient milling simulation model, which can predict the transient temperature distribution of 2.5-dimension prism parts. Gravity and inertial force will destroy the correct position relationship between the cutter and component, causing the machining tolerance.20

It was noted that clamping conditions,cutting temperature,gravity and inertial force mainly appear in clamping and localization points,which have minor impacts on the overall deflection. However, overall deflection is one of the main factors leading to machining deformation aiming at beams, sheets and frames.21Initial residual stress produces significant influence on the deflection. Huang et al.22found the percentage of the deformation aroused by the initial residual stress is about 90%.Gao et al.23established a semi-analytical deformation prediction model for large thin-walled components. They also analyzed the effects of initial residual stress and equivalent bending stiffness on the machining deformation. Yang et al.24discovered the relationship between fluctuant initial residual stress and deformation. Furthermore, they put forward a deformation evaluation index. Deng and Kiyoshima.25proposed a calculation method based on thermo-elastic-plastic finite element analysis to illustrate the effect of initial residual stress caused by heat treatment on welding residual stress.Deng et al.26also investigated the initial residual stress had no influence on the welding residual stress in the weld zone and its adjacent area, while initial residual stress still existed at the distance more than 40 mm away from the weld centerline. Mehner et al.27discovered the influence of the initial residual stress state on final state after cold rolling by finite element analysis,simulation and X-ray diffraction.Savaria et al.28built a three-dimensional fatigue model considering residual stress,microstructural change and surface roughness to predict the bending life limit.

Finishing allowance has a direct influence on the machining accuracy. On one hand, if finishing allowance is too small, it will be difficult to eliminate the residual shape, position errors and surface defects in the previous process.On the other hand,with the increase of finishing allowance, energy consumption and machining times will increase.More seriously,the component is prone to deformation due to cutting off a large amount of finishing allowance.29Therefore, it is necessary to control the magnitude of finishing allowance. Sun et al.30developed a mathematical model to deal with the numerous constraints in finishing allowance optimization. It is obtained by integrating multipliers approach and the Broyden-Fletcher-Goldfarb-Shanno algorithm. Li et al.31extracted planar features from three-dimensional point clouds and developed a new method for evaluating finishing allowance of two-step rough precision casting. Wang et al.32discovered the basic principle of threedimensional digital matching and alignment of geometric components combined with the finishing allowance modeling and analysis framework of large thin-walled components. Jiang et al.33found that reasonable allowance allocation to the related machining surface can improve the stiffness of the designed part. Zhang et al.34studied the optimization model of finishing allowance by means of the capability maturity model. The proposed model ensured enough stock allowance which was suitable for the single components and the integrated components. Wu and Dai.35analyzed the influence of the unit finishing allowance on manufacturing defects through establishing an evaluation model about machining defect risk.

Although the abovementioned researches focused on the relationship between machining deformation and the aforementioned factors through FEM, theoretical model and machining experiment. These research mainly focused on the combined effects of theses parameters on the machining deformation. Specific optimization strategy has not been proposed.As mentioned earlier, initial residual stress has significant influence on the machining deformation. Machined surface residual stress, heat, gravity and inertial force have a few effects on the machining deformation. Thereby, the machined surface residual stress, clamping conditions, thermal, gravity and inertial force were ignored. A novel model for finishing allowance optimization based on the simplex algorithm was presented in this paper.Initial residual stress-induced deformation and finishing allowance-induced deformation were taken as major factors. The single-sided component was taken as the research project, and then the deformation prediction model between initial residual stress and finishing allowance was firstly established in Section 2. Meanwhile, simulation analysis was carried out. In Section 3, combined effects of these factors on the overall machining deformation were discussed through linear optimization model based on the simplex algorithm. Then, the optimized finishing allowance allocation strategy is depicted. Finally, the feasibility of this method is proved by the actual machining experiments.

2. Approach of finishing allowance modeling

2.1. Initial residual stress distribution

Fig. 1 Direction of initial residual stress.

Fig. 2 Distribution of initial residual stress.

In order to obtain the initial residual stress, the crack compliance method is adopted.22The distribution laws of initial residual stress of blanks from same batch of 7050-T7451 aluminum alloy plate are similar.36Therefore, three typical blank are selected to measure their initial residual stress.Initial dimensions of all blanks (A1, A2, A3) are 1300 mm×400 mm×70 mm (length (x direction)×width (y direction)×thickness (z direction)), see Fig. 1. In this study,the deformation of the component along the x direction is only considered, therefore, we only investigate the initial residual stress along the x direction (σx), as shown in Fig. 1. designed component Their initial residual stress is perpendicular to the plane of yz and parallel to x-axis. Fig. 2 presents the initial residual stress profiles. The initial residual stresses are distributed along the neutral surface symmetrically.And the profiles of the initial residual stress present‘M’or‘W’shape of the periodic vibration with tensile stress and compressive stress.The initial residual stress changes from compressive stress into tensile stress to compressive stress.Except for the region I,the most values of residual stresses are -20-20 MPa. In addition,the surface initial residual stresses increase in a considerably short thickness, which induces a high stress gradient, thereby,the region I has a significant effect on the machining deformation.The maximum absolute values of compressive stresses are 70 MPa,80 MPa or 90 MPa respectively.The larger values are easy to generate in the upper surface and base surface. The smaller absolute values of tensile stresses are 13 MPa,15 MPa, or 16 MPa respectively. Except for blank A1, the smaller values are all in the distance of 10 mm away from the neutral surface.Based upon the features of the initial residual stress,multi-peak Gaussian function is introduced to fit the initial residual stress in Origin software.36The expressions of fitting results and coefficients of each blank are shown in Eq. (1) and Table 1, respectively. Fitting accuracies of three profiles are 99.894%, 99.164% and 99.380% respectively,which proves the feasibility of Gaussian function.

Table 1 Coefficients of fitting curves.

where σnrepresents the value of initial residual stress along the thickness, z is the blank thickness.n is the blank.a0,ai,biand ciare the coefficients of six-peak Gaussian curve.

2.2. Analytical model between initial residual stress, finishing allowance and machining deformation

In the milling of single-sided component,the material removal process is asymmetrical, therefore, final dimension of singlesided component is easy to be out of machining tolerance.Single-sided component is illustrated in Fig. 3, the effects of finishing allowance on the machining deformation under the influence of initial residual stress are discussed.The parameters of the designed component are depicted in Table 2.

In the paper,the single-sided component is regarded as uniform beam, the section diagram of single-sided component is shown in Fig.4.Along the length direction of the component,the cross-section is U-shaped in general. ω is bending deformation. The main machining deformation of singlesided component is along the length direction (Fig. 5), therefore, only the bending deformation along the length direction is considered.37

Fig. 3 Single-sided component.

Table 2 Component dimension.

Fig. 4 Section diagram of single-sided component.

Fig. 5 Diagram of deformation direction.

The bending deformation of single-sided component in this paper is simplified as the bending deflection of a single beam.Maximum deflection represents the machining deformation,see Eqs. (2)-(5).

where ω is bending deformation, M1and M2are additional bending moments when the sidewall finishing allowance and web finishing allowance are removed. σnis chosen from Section 2.1. While H is original blank thickness, taken as 70 mm in this paper.zcis the centroid of the component.E is modulus of elasticity. I is moment of inertia. t is called web base allowance (the distance between the bottom of web and bottom of blank38), taken as 7 mm in the paper.

2.3. Simulation analysis

To validate the effectiveness of the analytical model, the finite element method(FEM)is used to simulate the machining process of the single-sided component with different finishing allowance. Based upon the abovementioned analysis,machined surface residual stress is not the main factor influencing the machining deformation.Thus,only the initial residual is introduced into the finite element model. ABAQUS software is used to simulate the actual machining.39The material removal process in each step is accomplished by the‘‘birth and death unit” technology.40Through editing the ABAQUS keyword, the units chosen to be died are invalidated, which are removed during the machining process. The process flow is introduced in Fig. 6.

The ranges of sidewall finishing allowance and web finishing allowance vary from 1 mm to 5 mm.The influence of single factor and combined effects of two factors on the machining deformation are discussed in detail. It is worthy to mention that web finishing allowance is fixed as zero when the effect of sidewall finishing allowance on the machining deformation is under consideration. Likewise, the effect of web finishing allowance on the machining deformation is studied, and the sidewall finishing allowance is set to zero.The simulation analysis procedures are summarized as follows:

(i) Import blank and designed component in ABAQUS software and define the mechanical properties.

(ii) Add the initial residual stress shown in Fig. 2 into the blank.

(iii) Remove the material.

Fig. 6 Machining process.

3. Results and discussions

3.1. Relationship between finishing allowance and machining deformation

Fig. 7 Relationship between sidewall finishing allowance and deformation (web finishing allowance was fixed as zero).

Fig. 8 Relationship between web finishing allowance and deformation (sidewall finishing allowance was taken as zero).

Figs. 7 and 8 shows the machining deformation results of the analytical model and FEM. Fig. 7 presents the relationship between sidewall finishing allowance and machining deformation. A positive linear correlation between the sidewall finishing allowance and deformation is identified. In addition, the machining deformation increases gradually with the growth of sidewall finishing allowance. The maximum absolute value of sidewall finishing allowance-induced deformation is around 0.7 mm. As indicated in Fig. 8, the machining deformation raises with the increment of web finishing allowance.The maximum value of web-induced deformation is about 1.9 mm.Compared with the FEM results,the maximum relative errors of the analytical model are 4.5% (the component is machined from blank A1), 12% (the component is machined from blank A2), 8.6% (the component is machined from blank A3) in Fig. 7, respectively. Also, the maximum relative errors are 7.8% (the component is machined from blank A1), 17% (the component is machined from blank A2), 14% (the component is machined from blank A3) in Fig. 8, respectively. In spite of certain errors,the results of analytical model are in accordance well with those of FEM results.From Figs.9 and 10,the proportion of the sidewall finishing allowance-induced deformation accounting for the superimposition of the sidewall finishing allowance-induced deformation and web finishing allowance induced-deformation is 19%-40%. The proportion of the web-induced deformation is 60%-81%. Thereafter,the ratio of web-induced deformation to sidewall-induced deformation is about 1.5-4.Fig.11 is the machining deformation under the combined effects of sidewall and web. From Figs. 12 and 13, web plays a significant role in the machining deformation and sidewall can restrain the machining deformation.The deformation of the component machined from blank A1is also larger than that of other components. The reason is that initial residual stress of blank A1has a higher fluctuant amplitude.

Fig. 9 Proportion of sidewall finishing allowance-induced deformation.

Fig. 10 Proportion of web finishing allowance-induced deformation.

Fig. 11 Combined effects of sidewall finishing allowance and web finishing allowance on the machining deformation.

Fig. 12 Sidewall-induced deformation.

Fig. 13 Web-induced deformation.

3.2. Optimization of finishing allowance based on the simplex algorithm

In actual machining, the machining deformation consists of web finishing allowance-induced deformation, sidewall finishing allowance-induced deformation, initial residual stressinduced deformation, machined surface residual stressinduced deformation, clamping stress-induced deformation,thermal deformation,gravity and inertial force-induced deformation. However, based on the aforementioned analysis, it is concluded that the finishing allowance and initial residual stress have significant impacts on the machining deformation.Machined surface residual stress,clamping stress,heat,gravity and inertial force have minor effects on the machining deformation.Thereby,the influence of these factors(machined surface residual stress, clamping stress, heat, gravity and inertial force) on the machining deformation were neglected. The effects of finishing allowance and initial residual stress on the machining deformation are analyzed.Hence,machining deformation is regarded as dependent variable. As for the independent variables,deformation caused by web finishing allowance,sidewall finishing allowance and initial residual stress are conducted as major variables. Finally, the finish allowance allocation strategy is obtained through optimization model.

Linear programming is used to optimize the finishing allowance. Minimizing the machining deformation is the ultimate goal, which is influenced by the multiple variables. Optimization model of machining deformation of single-sided component is shown in Eq. (6). Where w is machining deformation,x2is web finishing allowance-induced deformation. Since the values of sidewall finishing allowance-induced deformation are always negative, the absolute value of x1, -x1is sidewall finishing allowance-induced deformation.Machining deformation w is objective function. Constraint conditions are defined based on the last Section 3.1.While x1and x2are the decision variables. The numbers on the right of the constraint inequalities are called the constraint constants. As is presented in Fig. 14, optimal solution and optimal value of objective function in feasible region can be found by computational tool.41From Eq. (6), there are two major variables and no linear equation, therefore, simplex algorithm is adopted to solve the problem.42The whole solution procedures are described as follows:

Step 1: Decision variables, objective function and constraint conditions are defined.

Step 2: The mathematical models of linear programming should be converted into standard forms, including the standardization of objective function, standardization of decision variables, standardization of constraint constants and standardization of constrained inequalities. In the paper, decision variables and constraint constants have been standard forms, so it is necessary to standardize the objective function and constrained inequalities.

(i) Standardization of objective function:min w=x2-x1, if y =-w, that max y =x1-x2.

(ii) Standardization of constrained inequalities.

Fig. 14 Equivalent linear programming model.

When the constraint condition is an inequality with the form of ‘‘≤”, a new nonnegative variable is added to the left of the inequality,in the meanwhile, sign of inequality is translated into equality sign. If the constraint condition is an inequality with the form of‘‘≥”,a new nonnegative remaining variable is subtracted to the left of the inequality. Both added variable and subtracted variable, they are called slack variables.Eq.(6)is transformed into Eq.(7).The introduced slack variables x3, x4, x7are executed as basic variables. Original variables x1,x2are non-basic variables.Because the coefficient of x5is negative in the constraint equality,x5cannot be used as a basic variable. The third constraint equality lacks a basic variable. As a result, it is necessary to introduce one artificial variable. The value of artificial variable must be zero, which has no effect on the basic feasible solution of the original problem. In order to reduce the value of artificial variable to zero,big M method is used.43The big M method gives a sufficiently large coefficient ‘‘M” to the artificial variable in the objective function,which is uneconomical,so we should replace the artificial variable from the basic variables with non-artificial variables as soon as possible. ‘‘M” is a sufficiently large positive number. Finally, the mathematical model of the finishing allowance optimization is illustrated in Eq. (8).

According to the Eq. (8), the expression of basic feasible solution is proposed in Eq.(9).If the value of artificial variable becomes zero,only the values of x1and x2will have significant impacts on the optimal value of the objective function. It is accord with the machining process. x3, x4, x5, x6, x7were ignored. Optimal solutions of x1and x2are analyzed to minimize the deformation. In the complex machining process, x3,x4,x5and x7are deemed as inertial force-induced deformation,machined surface residual stress-induced deformation,thermal deformation and clamping stress-induced deformation. x6is introduced artificial variable whose value becomes zero in the end, thus x6is regarded as gravity-induced deformation.As mentioned above, -x1is sidewall finishing allowanceinduced deformation and x2is web finishing allowanceinduced deformation.The values range of x1and x2are determined by the structure of web and sidewall together with initial residual stress. Actually, the linear programming optimization model returns an original machining system under the effects of multiple factors.

Fig. 15 Framework of solution process of the simplex algorithm.

Step 3: Optimal solutions are found.

(i) The initial basic feasible solution is found.

(ii) Whether the basic feasible solution is the optimal solution is determined, if so, the solution process is over,

otherwise we should go to the next step.

(iii) The direction of the improved solution is determined.

(iv) A new basic feasible solution to optimize the value of objective function is obtained.

Aiming at the problem in the paper,a specific analysis process is proposed in Fig. 15. The computational process is iterated three times until the value of artificial variable is zero.Table 3 are the solution results. Where Cjrepresents the coefficients of variables in the objective function,CBis a line vector of value coefficient of the basic feasible solution,XBshows column vectors composed of basic variables, b expresses the constant on the right of the constraint equation, σjindicates optimality test of each variable, shown in Eq. (10).44During the first solution, basic variables are x3, x4, x6and x7. Nonbasic variables are x1, x2and x5. Initial basic feasible solution is X(0)=(0,0,0.7,1.9,0,0,0).The value of object function is y(0)=0. There is a positive value existing in the line of optimality test and the value of artificial variable is not zero.The current basic feasible solution is not optimal solution.So, the calculation process continues. x1makes the fastest increase of the objective function, which helps x1chosen to be entering variable and x3become leaving variable. During the second solution, x1, x4, x6and x7are basic variables, x3,x2and x5are non-basic variables. Basic feasible solution is X(1)=(0.7,0,0,1.9,0,1.05,2.8).The value of object function is y(1)=0.7.However,present solution is not optimal solution yet,the solution process still needs to be calculated.During the third solution, x2becomes entering variable and x6turns into leaving variable. X(2)=(0.7, 1.05, 0, 0.85, 0, 0, 1.75) is basic feasible solution and the value of object function is y(2)=-0.35. Currently, the value of artificial variable is zero,therefore,the solution process is over.X(2)and y(2)are optimal solution and optimal value of linear programming optimization model respectively. Minimal value of y is 0.35. As for the single-sided component,the minimal value of deformation caused by finishing allowance and initial residual stress is 0.35 mm. Web-induced deformation and sidewall-induced deformation are 1.05 mm and 0.7 mm respectively. Returning back to Fig.6(a) and (b), optimized parameters of finishing allowance are put forward. Finishing allowance of the web and sidewall vary from 1 mm to 2 mm.

Table 3 Results of simplex algorithm.

Table 4 Detailed machining parameters.

Fig. 16 Designed component machined through optimized finishing allowance alocation strategy.

3.3. Machining experiment

In contrast with the deformation results of components in conventional method, machining experiment using optimized finishing allowance allocation strategy is implemented to validate the feasibility of analytical model and optimization model of finishing allowance. Table 4 presents the detailed machining parameters. According to the abovementioned machining process in Fig. 6, the single-sided component is acquired. Five-axis high speed machining center of DMGDMU 80P is used to machine the component. Blanks B1, B2,B3C1, C2, C3and blanks A1, A2, A3mentioned in Section 2 are from same batch. Thereafter, their initial residual stresses are similar. Blanks B1, B2, B3and blanks C1, C2, C3are adopted to machine the designed component in Fig. 3. The deformation results of the components are shown in Figs. 16 and 17. Keyence LK-G3 laser rangefinder scans characteristic margin of designed component. z coordinates of one point at an interval of 100 mm along the x direction, which are as the deformation value. As depicted in Fig. 17, the machining deformation of the component machined through optimized finishing allowance allocation strategy are 0.3-0.38 mm,which made 39%-56%reduction of the machining deformation compared to that in conventional method.

Fig. 17 Comparison between conventional method and optimized finishing allowance allocation strategy.

4. Conclusions

The paper investigated the combined effect of finishing allowance, initial residual stress on the overall machining deformation based on the simplex algorithm. Optimized finishing allowance allocation strategy was proposed. The following conclusions are summarized:

(1) The profile of initial residual stress presented ‘‘M” or‘‘W” shape. The larger magnitudes of initial residual stress appeared in the upper and base surface.And most smaller magnitudes generated in the distance of 10 mm away from the neutral surface. Furthermore, the distribution of initial residual stress was fitted through sixpeak Gaussian distribution.

(2) The influence rules of web finishing allowance and sidewall finishing allowance under the effects of initial residual stress on the overall machining deformation were analyzed. There were positive correlations between machining deformation and finishing allowance.

(3) Web finishing allowance-induced-deformation and sidewall finishing allowance-induced deformation were taken as major factors.And other parameters(machined surface residual stress, heat, gravity and inertial force)induced deformation were neglected in the optimization model. Optimized finishing allowance strategy was proposed by means of simplex algorithm. Web finishing allowance-induced deformation and sidewall finishing allowance-induced deformation were 1.05 mm and 0.7 mm. Optimal web finishing allowance and sidewall finishing allowance varied from 1 mm to 2 mm. The machining deformation decreased to 0.3-0.38 mm,which made 39%-56% reduction compared to that in conventional method.

Acknowledgements

This study was co-supported by the National Natural Science Foundation of China (No.51405226) and Postgraduate Research & Practice Innovation Program of Jiangsu Province of China (No. KYCX19_0165).