Deformation error compensation in 5-Axis milling operations of turbine blades

The precision and performance of machined flexible parts are under influence of deformation errors during end milling operations. Thus, prediction and compensation of deformation errors during milling operations of flexible parts can provide a key tool in accuracy enhancement of part production. In this study, an improved virtual machining system is proposed in order to assess and compensate deformation errors caused by cutting temperature and forces in 5-axis milling operations of flexible parts. The improved Johnson–Cook model is utilized to investigate the cumulative impact of strain rate and deformation temperatures on flow stress during milling operations of turbine blade. To estimate deformation errors caused by cutting forces and temperature on the workpiece and cutting tool, the finite element analysis is then applied. As a result, volumetric vectors of deformation error at each cutting location along the machining pathways are then generated in order to be compensated within new compensated machining tool paths. Thus, the deformation error created by cutting forces and temperature on the workpiece and cutting tool are compensated in order to enhance accuracy during 5-axis milling operation of flexible turbine blades. Experiments are carried out using a 5-axis CNC machine tool and errors are quantified using a CMM to verify the developed strategy in the study. As a consequence, precision of machining operations on flexible turbine blades can be enhanced by employing the developed virtual machining system in the study.


Introduction
In order to achieve tight tolerances of complex parts, such as blisks, turbine blades, and impellers which are commonly utilized in aviation industries, accuracy of machining operations should be evaluated and enhanced. Due to low rigidity of turbine blades during 5-axis machining operations, errors are created which can decrease accuracy as well as quality of machined components. The deformation errors caused by cutting temperature and forces in low-rigidity and flexible parts milling operations are the main cause of inaccuracy in the machined components in the aerospace and aeronautics sector.
Cutting temperatures and forces can cause inaccuracy in machined thin-walled components, leading in deformation error of final products. Previous research works [1][2][3][4] indicate that the deformation error caused by the cutting forces and cutting temperatures is the prevailing issue in achieving high-precision machining of thin-walled components. As a result, monitoring and compensating the machining deformation error is critical and has received a lot of attention from scientists in different research works [5][6][7][8]. Virtual machining systems are being developed to increase precision and productivity of the part production process. The production method and effective parameters can be modeled and modified in virtual environments. As a result, using the virtual machining method, more added values can be obtained by applying the modified parameters of production process in virtual environments [9,10]. To reduce cutting forces and tool deformation errors during milling operations, Law et al. [11] increased the accuracy of machined parts by reducing the tangential cut depth in corner milling processes.
To enhance the precision of extremely flexible aerospace components in peripheral end milling operations, simulation and reduction of static form defects in plate milling operations is presented by Budak and Altintas [12]. Wang et al. [13] presented a cutting pattern modification strategy for reducing component displacement in thin-wall milling, which minimizes the axial depth of cut to minimize component displacement errors. To properly anticipate and evaluate the deformation error in impeller blades 5-axis milling operations, Wang et al. [14] proposed an advanced model of deformation error in thin-wall surface components during the milling metal cutting using finite element approach. To increase cutting precision of thin-walled parts, Du et al. [15] developed circumferential machining force impacted error correction using an analytically model approach and APDL displacement calculation. In order to detect and compensate the volumetric errors in the machining operations of thin-walled parts, error compensation for machining of large thin-walled part with sculptured surface based on on-machine measurement is presented by Huang et al. [10]. To enhance the accuracy of machined parts, Cho et al. [16] developed an enhanced error detection and compensation methodology in order to collect inspection data using the On-Machine Measurement device employing a spindle-mounted touchtrigger probe. To analyze and compensate the deformation error of thin-walled web parts, an advanced on-machine observations compensation system for machining operations of thin-web parts is presented by Ge et al. [17]. In order to achieve the tight tolerances during machining operations of flexible parts, an advanced iterative measurement and experiment processes are needed to be used in terms of accuracy enhancement of final products [18]. Multiple sensors are applied to the machining process in real-time compensation methods to track signals and substantial improvements to production variables and cutting tool orientations [19]. As a consequence of the online cutting process monitoring system, the deflection error can be reduced [20]. To reduce deformation errors in end milling operations, Wang et al. [21] proposed real-time deformation correction in large thin-walled components machining. Liu et al. [4] proposed a technique for compensating for real-time machining defects based on edge characteristics for cutting force generated deformations in flank milling in order to improve machined component precision. In-process compensation methodology of deformations error by adding the piezo-electric actuator is developed by Diez et al. [22] to improve the precision of flexible part end milling operations. To enhance the precision of CNC machine tools, Zhou et al. [23] provide modeling and adjustment of thermal properties of leadscrew for machine tool depending on CNC system actual statistics data. A closed-loop error compensation method for robotic flank milling is presented by Xiong et al. [24] to increase surface accuracy of produced parts using milling operations. Thermal error compensation modeling for CNC machine tool worktables is proposed by Wei et al. [25] to provide high prediction accuracy and stability in worktables of CNC machine tool during machining operations. To improve the error compensation effect for low-stiffness structure during milling operations, machining error compensation for thin-walled parts considering time-varying cutting condition is proposed by Zhao et al. [26]. Soori and Asmael [27] developed optimal machining settings to reduce displacement inaccuracy in thin-walled rotor blades 5-axis milling operations. To minimize surface integrity and residual stress during grinding operations of Inconel 718, optimized machining parameters using the Taguchi optimization approach is presented by Soori and Arezoo [28]. To execute the technology, modern sensors need to be inserted in the machining process in order to directly measure and alter machining parameters, which can increase the cost of milling accuracy enhancement [29]. In addition, repeated 'float' cuts are frequently employed in industry to gradually erase surface dimensional inaccuracies caused by machining processes deformation [30]. All of the presented strategies in error detection and compensation would increase machining time and costs, making it hard to fully utilize the machine tool. One of the most extensively used strategies in compensation of dimensional surface and deformation errors during milling operations of flexible parts is tool path modification methodologies [31][32][33]. Soori et al. described virtual machining approaches and methodologies for evaluating and improving machining operations in digital settings [34][35][36][37][38][39]. Soori and Arezoo [40] proposed a review in machining caused residual stress to assess and decrease residual stress throughout metal cutting operations. Soori et al. [41] presented an enhanced virtual machining approach for improving surface characteristics of turbine blades during five-axis end milling operations. Altintas and Merdol [42] described a virtual machining system and application in order to achieve optimal milling conditions. Altintas et al. [43] describe a virtual adjustment of deflection error utilizing a cutting tool pathways modification method to improve accuracy in ball end milling of flexible turbine blades. However, the impacts of cutting temperature on deflection error throughout flexible turbine blade milling operations are not examined in the study. Habibi et al. [32] adjusted the location of cutting tool pathways throughout milling operations of free form surfaces in order to improve fiveaxis ball end milling precision.
According to recently published research works, the area of deformation error compensation due to cutting temperature and forces through using virtual machining technologies in 5 axis CNC end milling of flexible turbine blades has not been investigated. Error prediction and compensation methodology is developed in this study in order to predict and compensate deformation errors due to cutting forces as well as cutting temperatures during flexible turbine blade milling operations. Accurate prediction of tool and workpiece deformation errors can be calculated by applying simulated cutting temperature and forces to the tool's cantilever model and the workpiece's Finite Element Method (FEM) model. As a result, it is possible to compensate the deformation errors during machining operations without repeating machining experiments using developed virtual machining system in the study. A virtual machining system is proposed in this work for predicting and compensating deformation error caused by cutting tool and workpiece deformation throughout turbine blade milling operations. Using an adapted Johnson-Cook model, the impact of strain intensity and deformation temperatures on flow stress throughout turbine blade end milling is explored. As a result, cutting temperatures along machining paths are analyzed by applying the FEM in order to be used in deformation error calculation of flexible blades. Also, the tool deflection errors along machining paths are obtained in order to calculate the deformation error in the machined turbine blade. Then, the FEM is implemented in order to obtain the deformation error as a result of cutting temperatures and forces along machining paths. Volumetric vectors of deformation error at each cutting tool location along the machining pathways are calculated by considering the cutting tool and workpiece defection errors in order to compensate the deformation error in the machined turbine blade. Finally, new compensated machining paths regarding the calculated volumetric error of deformation error at each cutting tool location along the machining pathways are generated to increase accuracy in 5-axis milling operations of flexible turbine blades. To validate the methodology developed in the analysis, experimental machining operations of the flexible turbine blade are implemented by the 5-axis CNC machine tool. To obtain the errors in machined parts, the CMM machine tool is then used. As a consequence, the precision of the 5-axis milling process of flexible turbine blades can be improved by employing the virtual machining system designed in the study.

Cutting force model in 5-Axis milling operations
Song et al. [44] established a cutting force approach to simulate cutting forces during 5-Axis CNC milling operations. Geometry of 5-axis milling machines, global coordinate system (GCS) as fixed coordinate system, feed cross-feed normal system (FCN) as moving coordinate, and tool coordinate system (TCS) with lead angle (α) (a) and tilt angle (β) (b)are shown in Fig. 1.
The transformation matrix R from FCN to TCS can be obtained using the rotation matrix [44], where α and β are lead and tilt angles, respectively, as shown in Fig. 1. Therefore, undeformed chip thickness can be calculated in TCS as [44], where f t is feed per tooth, n TCS is the surface normal vector of milling cutter in TCS, and F TCS is the feed direction vector in TCS and can be evaluated by where R is obtained by Eq. (1). During 5-axis end milling, the cutting tool and workpiece interaction is shown in Fig. 2.
The cutter tool and workpiece interaction regions are condensed to a few fundamental tool tip in order to create a mathematical simulation of end milling cutting forces. As a consequence, the dz element can be written in the following format, where Z max and Z min are the cutting tool's maximum and minimum z coordinates, as well as engagement of workpiece, respectively, and N z is the discretization number of regarding engaged areas. So, the radial, tangential and axial cutting forces in the lth time interval, m th element on the j th cutting tooth can be presented by the differentiated forms as, where K rc , K tc , and K ac are the cutting force coefficients that are donated by the shearing action in radial, tangential, and axial directions, respectively. Also, K te , K re , and K ae are the coefficients of edge force, which are derived from the experimental results. The selected element's uncut chip thickness in the cutting tool's normal direction is w . During the chip generation process, the difference in chip thickness is measured as db which can be calculated as db = dz∕sin where is axial immersion angle. So, the cutting edge's differential cutting length can be expressed as, where R(z) is the local radius of cutting tool along Z axis during milling operations. The rectangular window function of cutting tooth j throughout cutting operations may be represented as, where the cutting tool edges' start and exit angles regrading to the Z coordinate are as st (z) and ex (z) respectively. As a consequence, the following equation can be used to determine differential cutting forces, where (z) is axial immersion angle which is the angle between cutting tool axis and edge normal and j (z, t) is radial immersion angle.
Ultimately, the contribution of differential cutting forces was calculated by the N z discretized elements and the The cutter tool and workpiece interaction during 5-axis end milling engaged flute of cutter, The cutting tool's total cutting force can be determined as, where N t is the cutting-edge numbers which are engaged in the cutting.

Johnson-Cook model
The Johnson-Cook model is employed in the calculation of the pressure distribution of a material as a mix of strain influences, thermal properties, and rate of strain due to theoretical flexibility and precision of model. The three variables describe the effect of hardness due to stress, rate of strain, and heat relaxing on the stress of the component's flow throughout deformation. Due to the obvious method's versatility in FEM analysis, it is used to assess the deformation inclinations of different materials.
where ε is equivalent plastic strain, ε˙ and ε˙0 are the equivalent and basis plastic strain rates, T, T m , and T 0 are the cutting zone temperature, melting and experimental room temperatures, respectively. The m is index of softening with heat, while the hardening index of the strain is N. A, B, and C are the material's rate of elastic modulus, strain, and strain responsiveness, respectively. (10) According to the Johnson-Cook model, the three influencing elements of strain, strain amplitude, and temperature are completely independent of each other, limiting any one of them from having a cumulative effect. Such strain rate dependency is difficult to anticipate using the usual J-C constitutive model. The updated Johnson-Cook model studies the interplay between deformation temperature and strain rate on flow stress, significantly improving the model's prediction accuracy over the original Johnson-Cook model [46].
Lin et al. [47] propose an updated Johnson-Cook model to solve the Johnson-Cook model's limitations as Eq. (12),

Analysis of deflection errors in thin-wall workpieces during machining processes
Dimensional inaccuracy of machined surfaces refers to the discrepancy between the real machined surface and theoretical surface of component as CAD model. In thin-walled Fig. 3 The impact of cutting temperature and forces on surface dimensional inaccuracies component machining, deformation error is also caused by cutting temperature and forces. Cutting pressures and temperatures cause the workpiece to deform throughout metal cutting, resulting in geometrical error as well as inaccuracy. The impacts of thermal errors and forces-induced errors are depicted in Fig. 3. The surface dimensional inaccuracy can be presented as Eq. (14) [49], where t,p and f ,p are deflection error for the Point P was established by normal projections of the cutting temperature and cutting forces, respectively.
The nominal radial depth of cut, indicated by RN, is the spacing between the original surface to be machined and the intended machined surface. To guarantee that the surface dimensional inaccuracy does not exceed the tolerance throughout the milling process, R A is often specified to be different from R N ≠ R A . So, the surface dimensional inaccuracy can be calculated as [49], (14) e p = t,p + f ,p Note that R N and R A are the cut's nominal and specified radial depth of cut, respectively.

Tool deflection error
Modeling the cutting tool to the elastic supports on a cantilever beam can be used to compute the cutting tool's deflection error which is illustrated in Fig. 4 [50].
The contact point CC of the cutting tool is considered in the deflection error prediction in the Xc, Yc and Zc axes. The deflection error in the Z-axis can be neglected since the cutting tool is highly rigid in the direction during milling operations. Thus, deflection error of cutting tool by considering the cutting forces in X and Y directions can be presented as [50], where δ is the deflection of cutter; F is the applied force during machining; D is the overhang of tool; is the cutting depth; E is Young's ap modulus; I is the area moment of inertia of the cutter; Z is the deflection position. As a result, the cutter's deflection error in the X and Y axes can be computed as [50], where X,Y is the deflection error of tool in X and Y axes; F, D are cutting force and tool overhang, respectively. a p is the depth of cutting; E is Young's modulus; I is the area moment of inertia of the cutter; Z is the position of the deflection.

Analysis of thin-wall workpiece deformation errors
The typical deviation of the actual machined surface from the ideal machined surface in machining operations is surface dimensional inaccuracy. Cutting pressures and cutting temperatures cause a deflection inaccuracy when machining thin-walled workpieces. The Cutting forces and generated heat in the cutting zone displace the workpiece into a new position when the cutting tool is engaged to (15) Fig. 4 Cutting toll deflection error due to cutting forces [50] the thin wall workpiece. Figure 5 shows the amount of deformation error generated by cutting tool and workpiece deflections in a machined thin-walled component, where t and w are cutting tool and workpiece deformations, respectively [51].
The created deformation errors in the machined parts are different throughout machining surfaces due to differences between cutting forces and workpiece stiffness at each position of cutting tool along machining paths. A cantilever beam can be used in order to simulate and forecast the deflection inaccuracy of a cutting tool. Workpiece's deformation error can be accurately calculated by using a finite element model. Metal is removed from a thin-walled component and the workpiece thickness are decreasing which can reduce the stiffness of the workpiece along the cutting tool's path during the milling operations. So, the workpiece stiffness during milling operations is achieved by suspending the contributions of the removed material from the original workpiece by using an efficient structural stiffness correction parameters. Thus, the workpiece deformation error w,j (W;Z, t) is calculated by combining the stiffness of the in process workpiece and the dispersed cutting forces to derive the equations of static equilibrium for the investigated machining system, as illustrated in Eq. (18). [51].
where u T w,j (W;Z, t) is the statics displacement of workpiece, and n j (W;Z, t) is the unit normal of envelope surface. Workpiece and cutter deformation errors are calculated independently. The overall deformation error is then calculated by adding the individual components at a particular time. As a consequence, we can calculate the overall deformation error as [51], where t,j (W;Z, t)and w,j (W;Z, t) are deformation error of cutting tool and machined part.
Ultimately, in order to locate the appropriate machined surface, the whole deformation errors are imprinted on the flute's envelope surface formed by the j th flute [51]. As a result, the cutting tool's deformation error can be determined as Eq. (20) [51].
where S j (W;Z, t) is the surface of sphere centers of cutting tool on the workpiece along the Z axis and feed direction. Each flute of the cutting tool can provide a unique computed 5 The machined thin-walled part's deformation inaccuracy [51] Fig. 6 The volumetric error vector generation due to cutting temperatures and forces during 5-axis milling operations of turbine blade machined surface due to unique amount for each flute's sphere radius function as r j (z) and deformation error e j (W;Z, t).

Deformation error compensation methodology
In order to derive volumetric errors at each position of cutting tool along machining paths, error vectors related to cutting temperatures and forces are constructed Using the idea of closed loop at during the turbine blade's 5-axis milling operations [52]. Figure 6 depicts the closed loop volumetric error vector resulting of cutting forces and cutting temperatures during turbine blade 5-axis milling operations. Throughout milling operations, the point O is the point of contact between the workpiece and the cutter. As a result, the real and compensated volumetric error vectors of cutting forces and cutting temperatures for the 20 sample points in 5-Axis operations of turbine blade can be shown in Fig. 7.
In order to compensate the volumetric error vectors of contact points O along cutting tool paths during chip formation process, the algorithm of iterative compensation is used. In this method, the surface error of machined part is moving step by step to the nominal dimensions of the part in order to compensate the volumetric error vectors during milling operations. The volumetric error vector at the point O is obtained as V 1 . Then, the point O will be moved to the Point O ′ 1 as the The iterative method of surface error compensation at the contact point of O is shown in Fig. 8. Figure 9 shows a flowchart of the surface error compensation using the iterative technique at the contact point of O.

Virtual machining system
The virtual machining method is developed in this study using the programming language of Visual Basic. The system can receive the nominal machining paths, the geometry and material characteristics of the milling cutting tool, as well as the CAD model of sample component. Based on cutting tool attributes and machining operation parameters, the generated virtual machining system can calculate cutting  forces for each point of cutting tool throughout machining pathways. By using the calculated cutting forces along machining paths, the amounts of ε as equivalent plastic strain, ε˙ and ε˙0 as the equivalent and basis plastic strain rates are calculated. As a result, the modified model of Johnson-Cook for the Al-7075 alloy as Eq. (13) is used to obtain the information of cutting temperature (T) during milling operations. To calculate the cutting temperature during the chip generation process, the developed virtual machining system is coupled to the FEM analysis software of Abaqus R2016X. The CAD model of the product is then meshgenerated, allowing it to be examined using finite element methodology. As a result, the FEM technique can properly forecast the cutting temperature during the chip production process. Figure 10 depicts the workflow and approach used by the virtual machining system to determine cutting temperature and force during machining operations.
The cutting temperature and force data are then submitted to the Abaqus R2016X FEM analyzer, which calculates the machined turbine blade's deformation error. To assess the temperature and cutting force-induced deformation errors during end milling, the mesh is then added to a thin-walled CAD model of turbine blade. To compute node displacement, the expected cutting temperature and forces at each location of the cutting tool during machining pathways are integrated to each part model's mesh node. So, the deformation error due to cutting temperature and forces can be calculated for each cutting tool location. Figure 11 depicts the method and approach for computing cutting force and evaluating deformation error using the advanced virtual machining system.
Then, volumetric error vectors of the deformation error at each cutting tool location along the machining paths are calculated. To compensate the obtained volumetric error vector at each position of cutting tool along the machining paths, the iterative method which is described in the Sect. 7 is used. Then, new machining paths regarding the calculated volumetric vectors of deformation error at each cutting tool location along the machining pathways are generated in order to compensate the deformation error in the machined turbine blade. Figure 12 shows the deformation error compensation methodology in the study.
The flowchart of the study in deformation error compensation during 5-Axis milling operations of turbine blades is shown in Fig. 13.

Validation
In order to assess the proposed techniques in the study, the turbine blade is manufactured using a 5-axis CNC milling machine tool Kondia HM 1060. AL 7075 is the material used for the turbine blades. Dimensions of sample turbine blade in the experiments are as 170 mm length and 143.36 mm width. Also, the average of sample turbine blade thickness is 2.15 mm. The Masterccam software is used to obtain cutting tool pathways during 5-Axis CNC milling operations of sample turbine blades. The turbine blades are then manufactured using a 5-axis Kondia HM 1060 machine tool. The experiment was conducted using a carbide ball nose end mill with an 8 mm diameter, helix angel 30°, flute number 4, overall length 60 mm, and flute length 35 mm. The spindle speed is 200 m/min and the feed rate is 200 mm/  Figure 14 depicts the milling operations of sample turbine blade.
The proposed cutting force model of a ball nose end mill throughout 5-axis CNC machining processes is utilized to estimate cutting forces [44]. To obtain the cutting forces coefficients for Al 7075 materials, the mean cutting forces of twenty slot milling experiments with 1.5 mm cutting depth are recorded using the Kistler 9139AA dynamometer. The feed rate 100 mm/min and feed per tooth 0.5 mm, and the spindle rotates 3000 rpm are selected for the machining Fig. 16 The process of surface generation from the measured data of machined turbine blade    Table 1 The measured and predicted deformation errors of 10 selected points of the machined turbine blade without and with compensation methodology for the L5 line in Fig. 18 No Before The CMM machine is used to measure the deformation inaccuracy of machined turbine blades. The used probe in the measuring process is Renishaw SP25M while its repeated accuracy in directions of touching is 1 μm. The inspection procedure of machined turbine blade by using the CMM machine is shown in Fig. 15.
Loft through the section curves is passed in order to generate a NURBS surface from the scan data. As a result, the machined surfaces of turbine blade is generated in virtual environments to be used in the deformation error calculation. The process of surface generation from the measured data of machined turbine blade is shown in Fig. 16.
The amount of error between the CAD model of turbine blade and fitted NURBS surfaces from the measured data of the CMM machine are calculated in each section of generated surface as the deformation error of machined turbine blade. The calculated turbine blade deformation errors are shown in Figs. 17 and 18.
The Abaqus program is utilized to estimate and evaluate the deformation errors of machined turbine blades caused by cutting temperature and forces. Figure 19 depicts the predicted deformation error of the produced turbine blade using the FEM approach.
As a consequence, the measured and predicted deformation errors in the machined turbine blade without and with compensation methodology for the L5 line in Fig. 18 is presented in Fig. 20.
The measured and predicted deformation errors of 10 selected points of the machined turbine blade without and with compensation methodology for the L5 line in Fig. 18 is shown in Table 1.

Conclusion
The machined thin-walled parts are deflected during the chip forming process of machining operations caused by cutting temperature and forces. In order to develop the thinwalled parts milling operations, the errors can be modeled and studied in virtual environments. In the research work, deformation error compensation methodology caused by cutting temperature and forces is presented by applying virtual machining system. By assessing the combined influence of tensile rate and deformation temperature on flow stress, the modified Johnson-Cook Model is employed in the study (22) K tc = 810.42, K te = 16.93 K rc = 406.1, K re = 17.68 K ac = 241.28, K ae = 0.74 to investigate and examine the chip formation process. Then, in order to obtain the cutting temperatures during chip formation process, the FEM method is used. As a result, the deformation errors caused by cutting temperature and forces to the workpiece and cutting tool along machining paths are calculated by using the FEM method. Finally, volumetric vectors of deformation error at each cutting tool location along the machining pathways are generated in order to be compensated utilizing the study's established system. Ultimately, cutting operations on the flexible turbine blade are conducted utilizing a 5-axis CNC machine tool and the deformation errors are then measured using CMM equipment in order to validate the established procedure in the study. The following summarizes the findings from the study, 1. A 94.3% compatibility is obtained in comparison with the results of experimental and virtual machining system for the flexible turbine blade deformation error. 2. A 71.48% reduction in the measured errors of the machined flexible turbine bale is obtained by the modified machining paths in the error compensation algorithm. 3. The force-induced error vectors are bigger than cutting temperature error vectors at each position of cutting tool along machining paths due to effects of cutting forces to the flexible turbine blades during 5-axis milling operations. 4. The amount of deformation error due to the cutting tool as well as flexible workpiece deflection can be accurately predicted and minimized. As a result, accuracy of machined parts using the developed virtual machining system in the study can be enhanced. 5. The precision of the machined flexible turbine blades can be increase by using the modified machining codes in the developed virtual machining system. 6. To enhance the accuracy of machined parts, The influence of cutting tool configurations, such as radius and helix angle, tool wear, and edges on the thin-walled components deformation error in machining processes can be examined. 7. These are the ideas of author for future research projects.