Research on machining error transmission mechanism and compensation method for near-net-shaped jet engine blades CNC machining process

This study proposes an adaptive CNC machining process based on on-machine measurement to control the machining error of near-net-shaped blades. The multi-source and multi-process machining error transmission model of a near-net-shaped blade is established, and the reduction effect of the machining error transmission chain by the adaptive CNC machining process is qualitatively analyzed based on the machining error transmission flow model. The effects of the adaptive CNC machining process on the positioning benchmark error, machining position error, and machining contouring error are explored through the adaptive CNC machining process experiment. In particular, the ability of the adaptive CNC machining process to cooperatively control the blade position error and the contouring error is discussed in relation to the stiffness of the blade-fixture system. The results show that the adaptive CNC machining process can reasonably reduce the machining errors caused by the positioning benchmark. The final deviation band of the blade body is reduced by 60% based on the adaptive CNC machining process. The adaptive CNC machining process can optimize the contouring error and the position error of the blade tenon root under the enough premise of the stiffness of the blade-fixture system. The adaptive CNC machining process has the excellent ability to control machining errors to improve the machining quality of the blade.


Introduction
Blades are some of the key parts for the improvement of the performance and service life of a jet engine [1]. Near-netshaped jet engine blades have excellent performance under high temperature, high pressure, and high speed conditions due to the blade body complex curved surfaces are directly formed by the forming process, and eliminate the material removal of the blade surface. The subsequent CNC machining process for the blade leading edges and trailing edges (LTE), tenon root, and tip is required due to the small curvature radius and the high precision requirements [2]. In the CNC machining process of the blade LTE, tenon root and tip, the positioning, and clamping part is only the bladse body, and the low positioning accuracy and process system stiffness leads to low machining accuracy [3].
The blade machining accuracy is affected by multiple sources of machining error due to the increasingly complex CNC machining processes for complex thin-walled blade structures, such as the blade material characteristics, machine tools, and fixture. Additionally, the complex coupling relationships of these machining error sources may also introduce the final machining error. These machining errors are constantly generated, transmitted, increased, reduced, accumulated, and transmitted to form the dimensional deviation of the final blade. Therefore, it is important to study the machining errors of complex thin-walled blades and further optimize machining error control methods. At present, the research on the machining error control of complex thin-walled parts focuses on forming a systematic research framework based on two aspects. The first research idea is the machining error formation and transfer mechanism of the interrelation between the cutter and the workpiece during the machining process. The essential concept is the research about the geometry, kinematics, and dynamics principle of the CNC machining process, and the key research points are the machining errors between the fitted surface and the ideal machining surface formed by the cutter generating motion for a complex surface blade. E. Budak et al. [4] analyzed the cutting force, structural deformation, and surface accuracy of the machining process to improve machining accuracy through machining process optimization. Weifang Chen et al. [5] proposed a multi-objective fixture layout and clamping force optimization method based on a genetic algorithm to reduce the deformation and increase the consistency of surface deformation. K.P. Padmanaban et al. [6] proposed a method for fixture layout optimization to control the elastic deformation by using an ant colony optimization algorithm. However, the above two methods improve the machining accuracy based on the deformation control from a single fixture optimization method. Abdul Wahid Khan et al. [7] proposed a systematic machining geometric error model, and the machining accuracy was improved through machine tool error identification and analysis. Yan Rong et al. [8] studied the comprehensive stiffness matrix of the overall machining system including the machining parts, machining tools, machining tool spindle, and the cutter, and the stiffness variation laws of the machining system in multiaxis machining of complex curved surfaces were analyzed to improve the machining accuracy through process planning. The above-mentioned studies started with the stiffness of the machining system, with consideration of the stiffness enhancement method and deformation suppression of the entire machining system, which could improve the machining accuracy based on the deformation control for a thin-walled blade weak stiffness machining system. Eduardo Diez et al. [9] proposed a process plan method that could improve the machining accuracy by deformation compensation based on the actual measured cutting force and the estimated deformation. Li-Min Zhou et al. [10] proposed a method for predicting the surface dimensional form errors caused by deflections of both the workpiece and the slender end-mill in the five-axis flank milling of thin-walled parts, and the machining accuracy was improved through deformation prediction during the CNC machining process for the case of the insufficient stiffness of thin-walled parts. Gururaj Bolar et al. [11] established the relationship between the cutting process parameters and the quality of the machining surface, and they improved the machining quality with cutting process parameter combination optimization. Ziling Zhang et al. [12] researched the feedback mechanism between the cutting force and the milling deformation error, and they established a workpiece deformation error model based on an advanced neural network and improved the machining accuracy through a machining accuracy prediction model. Fountas. NA et al. [13] proposed an optimization of the five-axis sculptured surface finishing machining process to reduce the machining error based on decision making and machining error modelling. The above research reduced the machining deformation through the optimization of the machining process and the method of cutting force prediction for weak stiffness systems, and this research provided research methods for the optimization of the adaptive CNC machining process of thin-walled blades.
The second research idea is the spatial accumulation and transmission mechanism of machining errors. These research studies consider the transfer relationship between various machining errors in the CNC machining process. Ma Yan et al. [14] proposed a global optimization numerical analysis algorithm of complex surface-type positioning and machining. The machined surface was adjusted to obtain the maximum coincidence with the projection of the theoretical surface based on this method, and the surface positioning error was reduced with a simulated annealing algorithm. Walid Ghiea et al. [15] proposed a theory of tolerance band characterization based on a small displacement torsor (SDT). The typical tolerance band was characterized as a series of standardized numerical models by the theory, which laid a strict foundation for the error transfer calculation based on the Jacobian matrix form. Jia Feng et al. [16] considered the cumulative error of the multi-step process of blade manufacturing, and the key processes and their effects on the machining errors were analyzed by constructing a workpiece machining error transmission model. SJ Hu et al. [17] proposed a theoretical method for the error flow of mechanical product assembly based on the error transmission characteristics of multiple parts. D.
Ceglarek et al. [18,19]. further developed the state-space equation method to study assembly error variation relationship modeling and the deviation transmission mechanism, and they applied this method to the quality modeling and process quality control of the multi-process assembly process of auto parts. Although the above research methods and ideas are oriented to the multi-product assembly process, the proposed error cumulative effect in the space domain and the transmission effect in the time domain are the essence of multi-source and multi-process error transmission. That is, the transmission characteristics of the errors in the time domain should be considered, and the cumulative effect of the errors in the space domain should also be considered during research for the multiple sources and multi-process machining errors. Considering the research object, the CNC machining of near-net-shaped blade tenon root, tip, and LTE is a typical multi-source multi-process processing process, the machining error transmission, and accumulation methods research will be of guiding significance for this kind of blade machining error control.
The above two research ideas provide methodological guidance for the machining errors of a thin-walled blade. On-machine measurement technology is also an important method to compensate for machining errors with on-machine measurement. Guiassa and Mayer et al. [20] proposed on-machine probing for the correction of the finishing machining process, and the machining error resulting from the deflection and the tool offset error was compensated for solely with data from process-intermittent probing. Bandy HT et al. [21] proposed a methodology for compensating for errors detected by process-intermittent inspection, and this methodology was validated on a prototype system implemented by process-intermittent error compensation software (PIECS) on a turning center. Jian-Hua Yu et al. [22] proposed a new method for controlling machining deformation by eliminating surface errors based on an adaptive two-armed fixture, and adaptive technology has been applied in the design and process integration of a fixture. Nuodi Huang et al. [23] processed an adaptive deformation error compensation method for a large thin-walled part. Therefore, with the continuous development of onmachine measurement technology and the adaptive CNC machining process, the adaptive CNC machining process will be a new idea for blade machining error control. This study focuses on analyzing the effect of adaptive CNC machining processing on the machining error of a near-net-shaped jet engine blade. The sections are arranged as follows. In section 2, the adaptive CNC machining process and the machining errors are theoretically analyzed. In section 3, the conditions and methods of the on-machine measurement, CNC machining are introduced. In section 4, the results and discussion are analyzed, and the adjustment mechanism and the adjustment ability of the adaptive CNC machining for the positioning benchmark error, machining position error, and contouring error are analyzed. Finally, the conclusion is summarized in section 5.
2 Machining errors transmission model of near-net-shaped blade adaptive CNC machining process Figure 1 shows the near-net-shaped blade process characteristics, the blade blank is obtained by the precision forging process, and the contour error of the blade body is in the range of 0.008 to 0.05 mm. The false boss of the blade is firstly used as the benchmark to machine the blade LTE. The blade body is secondly used as the positioning benchmark to precisely machine the blade tenon root and tip. Finally, the precisionmilled blade tenon root is used as the detection benchmark to detect the position error and the contouring error of the blade body and tenon root and tip. Through the above analysis, the blade CNC machining process is a multi-machining procedure process. Figure 2 shows the machining error formation mechanism of the multi-source, multi-machining procedure process. The final machining error is caused by the source errors, including the blade profile error after the precision forging process and the machining error of the false boss, and the introduced machining errors caused by the previous process include the machining error of the blade LTE and tenon root. All of these errors accumulate in the final machining errors of the blade.
In the CNC machining process of the near-net-shaped blade, the specific machining errors are multi-target machining errors (multiple outputs (y)), which mainly include the contouring error and the position error of the blade tenon root. At the same time, the machining error influencing factors that affect the final machining error are also numerous (multiple inputs (x)). These factors mainly include the machine tool errors, errors caused by the tool path position, and errors caused by the instability of the clamping fixture. The single blade machining error (y) with the machining error source (x) satisfies the functional relationship.
The benchmark conversion is also one of the machining error sources of the blade multi-process machining process. Assuming that the benchmark conversion homogeneous matrix between the blade multi-process steps is A i i−1 i ¼ 2; 3; 4 ð Þ , the machining error model is generated by the k-th blade process, and this model accumulates the machining errors in the k-1-th blade process.
The blade multi-source multi-process machining error transfer model based on Formula (1) and (2) is shown in Formula 3. This model reflects the cumulative effect of the machining errors among multiple processes.  The differential equation is obtained: In the entire CNC machining process, the accumulation of various error sources eventually leads to the formation of multi-process processing errors (see Figure 3). Based on Formula (4), in the blade multi-process processing link, the machining error is very sensitive to the error caused by the benchmark conversion. In practice, the various machining errors that affect the blade machining accuracy are interrelated, some machining errors may cancel each other out, and the clamping scheme and the process parameters optimization are used to achieve the reduction and suppression of some machining errors. The adaptive CNC machining process is one of the main methods of partial machining error reduction.
The essence of the adaptive CNC machining process is the transformation between the two coordinates, which are the coordinates of the measuring point obtained by the onmachine measurement, and the coordinates of the theoretical model point. Formula (5) describes the blade adaptive adjustment process.
where q i is the measurement points, and p i is the theoretical model point.
Based on this mathematical principle, first, the CNC machining process of the blade tenon root and tip is a typical fuzzy positioning benchmark machining process due to the contour error of the blade body surface, which is used as the positioning benchmark i in the range of 0.008 to 0.05 mm. The matching function between the theoretical points and the measurement points is set as shown in Formula (6).
where δ is the positioning benchmark error compensation value, and this value is uncertain.
If this value is controlled within the range of 0.008 to 0.05 mm by the adaptive CNC machining process, and this The machining error of the false boss

The machining error of the blade LTE
The adaptive CNC machining process To obtain the milled qualified LTE The second processing step CNC machine the blade tenon root and tip Benchmark: the precisionmilled LTE and the 2nd and 8th section lines of blade The machining error of milled LTE

The machining error of the blade tenon root and tip
The adaptive CNC machining process  value is evenly distributed, at this time, the adaptive CNC machining process based on on-machine measurement will reasonably reduce the error caused by the benchmark positioning error. Second, the position of the blade will deviate during the clamping process. When the clamping force or clamping sequence is not appropriate, the adaptive CNC machining process can compensate for the position deviations caused by the clamping process. The matching function between the theoretical points and the measurement points is set as shown in Formula (7).
Formula (7) describes the mathematical principle of the adaptive CNC machining process to achieve the position compensation of the clamping process, and this objective function is established with the assumption that there is no local deformation in the clamping process. In fact, the fixture designed for this purpose is a rigid-flexible fixture in order to avoid local deformation of the blade as much as possible [3].
Finally, in the CNC machining process, the aim is to control the contour error of the blade tenon root, and the point on the machined blade tenon root side is used as the measurement point to achieve the adaptive CNC machining process of the blade tenon root for the control target of the blade tenon root contour error. The matching function between the theoretical points and the measurement points is set as shown in Formula (8).
The new rotation R 3 (x) and translation matrix t 3 (x) at this time will again affect the tenon root position described in Formula (7). That is, it is necessary to ensure the position error of the tenon root relative to the blade body while controlling the contour error of the blade tenon root. At this time, it is assumed that either the rigidity of the system is insufficient or the cutting force in the roughing stage is too large in the blade CNC machining process, and the new rotation R 3 (x) and translation matrix t 3 (x) will lead to f 2 (x) being far from the best value. It is assumed that the stiffness of the blade during the CNC machining process is sufficient to resist the large cutting force load. Then, the new rotation R 3 (x) and the translation matrix t 3 (x) will have the same trend as that shown in Formula (7). Therefore, considering the influence of the rigidity of the process system, the blade adaptive CNC machining process algorithm for different target objective functions is as shown in Figure 4. Figure 5 shows a flowchart of the adaptive CNC machining process of a specific blade. First, the blade is clamped in the appropriate position of the fixture with the proper clamping force and clamping sequence, which must match the fixture structure. Second, the blade is roughing milled after the clamping. The measurement path and measurement points are first planned based on the blade structure before roughing milling. Then, the measurement data model is obtained and the measurement programming is completed, and the point cloud data of the blade surface is obtained based on the onmachine measurement. The blade measurement model and the theoretical model are established based on the registration algorithm to obtain the CNC machining and reconstruction model of the blade. The specific calculation process is shown in Figure 4. Thirdly, the coordinate system of the blade is adjusted using the rotation matrix and the translation matrix solved for with the registration algorithm, and the machine tool's NC machining program is adjusted at the same time. Finally, the final blade position state is obtained with the adaptive CNC machining process. The on-machine measurement and the adaptive algorithm flow are terminated by judging the uniformity of the machining allowance distribution and the specific value of the machining allowance, and then the next process is begun.
Based on the above analysis, it can be seen that the adaptive CNC machining process cuts off the machining error transfer mode (Formula (2) and Figure 3). The adaptive CNC Fig. 4 Calculation process of adaptive CNC machining process machining process can achieve the control of the blade tenon root position error and the contour error.

Adaptive CNC machining process and positioning benchmark error
During the CNC machining process of the near-net-shaped blade tenon root, the positioning and clamping surface is only the blade body with a contour error of 0.008-0.05 mm. More importantly, these contour errors are random with a large uncertainty within the error band of 0.008 to 0.05 mm, and the positioning benchmark error is introduced when this part is used as the positioning surface. Therefore, the purpose of this experiment is mainly to explore the effect of the adaptive CNC machining process on the reduction of the benchmark positioning error. As shown in Figure 6, the blade concave and convex profiles are used as the measurement surface. Forty-eight measurement points on the blade concave and convex profiles are measured one time on the five-axis machine tool (see Figure 6 (b)) based on the Renishaw probe, and 24 measurement points are in the blade concave (see Figure 6 (c)), and 24 measurement points are in the blade concave.
The measurement experiment site is shown in Figure 7. The special measuring fixture is used to ensure that the measurement process is reliable without displacement and offset. The special fixture is affixed to the five-axis machine tool, and the measurement points and paths must be ensured for the same state when the multiple adjustments of the adaptive CNC machining process are carried out. It should be noted that this special fixture is only used to explore the effect of the adaptive CNC machining process on the reduction of the positioning benchmark errors caused by the initial geometric errors, and the special fixture cannot be used for subsequent tenon root CNC machining.

Adaptive CNC machining process and the blade tenon root machining position error
This experiment is used to explore the influence of the adaptive CNC machining process on the position error of the tenon root during the tenon root CNC machining process, and the position error refers to the position of the machining blade tenon root relative to the blade body, which mainly includes the circular position error and the twist position error. The fixture is a special fixture for the tenon root CNC machining process based on a multi-body rigid-flexible coupling dynamic model design [2]. Figure 8 shows the measurement model of the blade according to the fixture structure and the characteristics of the blade body. Eighty typical points are selected as the measurement points on the blade body convex surface. This is because the Renishaw probe in this fixture cannot measure the blade body's concave surface, and the fixture in Figure 7 cannot be used for the blade tenon root CNC machining, and the measurement point's selection is no longer carried out according to Figure 6.
The blade deviation values between the measured points and the theoretical points are used to obtain the posture relationship of the blade. The measurement trajectory is shown in Figure 8    Adaptive CNC machining process of a specific blade measurement points are then used for the data calculation of the adaptive CNC machining process [3]. The blade tenon root and tip are first roughing machined to remove a large amount of the blade tenon root and tip based on the final blade pose state obtained with the above adaptive CNC machining process. A nose knife cutter and the waterbased cutting fluid are selected. The cutting tool path is shown in Figure 9, the main machining parameters for the blade roughing milling are shown in Table 1, and the experimental site is shown in Figure 10.

Adaptive CNC machining process and the blade tenon root machining contouring error
This experiment is mainly intended to explore the relationship between the adaptive CNC machining process and the contouring error of the blade tenon root pressure surface under the condition of the positioning error controlling the blade tenon root relative to the blade body.
First, the assumption is made that the clamping position of the blade in the fixture will change after the roughing process is conducted, which will cause the benchmark to lose its original accuracy position due to the large external cutting force load in the roughing machining process with the large material removed. In this case, the blade tenon root machining contouring error is ensured based on the adaptive CNC machining process. For this near-net-shaped blade, the contouring error of the tenon root pressure surface is required to be less than 0.007 mm, which is a very high requirement.
As shown in Figure 11, surface A is used as the measuring surface to measure the position deviation of the blade tenon root on the fixture, and the adaptive CNC machining process of the blade tenon root is carried out to ensure that the contour error of surface B is less than 0.007 mm. The position of the blade tenon root will accurately adjust based on the adaptive adjustment process (see Figure 11 (b)). This is because that only the blade body profile can be used as the benchmark in the roughing machining process, and the positioning error is This adaptive CNC machining process is used to ensure the contour error of the blade tenon root. In this case, the CNC machining process is mainly intended to improve the surface quality of the blade tenon root and tip. A ball-nosed cutter is selected to directly mill the R1 mm area for high-precision milling. Water-based cutting fluid is selected as the lubricating fluid to cool the cutting process. The specific process parameters are shown in Table 2, and the tool paths of each region are shown in Figure 12. The cutting process is at a high speed of 12000 rpm, and the milling depth is 0.02 mm. The experimental site is shown in Figure 13.

Results and discussion
4.1 Adaptive CNC machining process for the positioning benchmark error control Figure 14 shows the results of the adaptive CNC machining process for the positioning benchmark error control. It can be seen that the blade measurement model is far from the theoretical model in the initial installation state relative to the theoretical contour of the blade due to the existence of the initial geometric accuracy of 0.008-0.05 mm. The single-point deviation value of the blade convex profile is greater than 0 mm, while the single point deviation value of the blade concave profile is less than 0 mm (see Figure 14 (a)). In the first stage of the adaptive CNC machining process regulation, the single-point deviation values of the blade convex profile are all greater than 0 mm, some of the single-point deviation values of the blade concave profile are less than 0 mm, and some of the values are greater than 0 mm. The adaptive adjustment stage plays a very important regulation role in the blade concave profile (see Figure 14 Figure 14 have no obvious changes in the entire adaptive CNC machining process. Therefore, these two points can be understood as invalid measurement points and must be eliminated in the subsequent analysis.
In fact, the blade body surface has an initial geometric accuracy error of 0.008-0.05 mm compared with the theoretical model, which results in a benchmark error, and the benchmark error is a positive error, which shows that the blade is in the correct position only when the single-point deviation values of the blade convex profile and the concave profile are all positive values, and the measurement points are evenly distributed on both sides of the theoretical profile of the blade.
The single-point deviation values corresponding to Figure 14 are shown in Figure 15 (a). It can be seen that the initial geometric deviation of the blade is evenly distributed on the theoretical surface of the blade after multiple adaptive regulation, and the single-point deviation value of the blade concave profile and the blade convex profile effectively becomes positive value with the adaptive regulation. Figure 15 (b) shows that the single-point deviation change value of the blade convex profile is positive in the first stage of adaptive regulation, which shows that the single-point deviation value of the blade convex profile exhibits negative compensation, while the single-point deviation change value of the concave profile is negative, which indicates that the single-point deviation value of the concave profile is positively compensated for. In this way, the two-way compensation of the blade concave profile and the blade convex profile finally achieves single point deviation uniform distribution. In the final adaptive regulation stage, the single-point deviation change value of the blade concave profile and the blade convex profile becomes 0 mm, which indicates that the maximum adaptive regulation has been reached at this time. The blade adaptive regulation has obtained the best regulation performance. It can be considered that the position and posture of the blade have been adjusted to the maximum suitable position and attitude. Figure 16 shows the single-point deviation value during the precise adjustment at the end of the adaptive regulation stage. It can be seen that the single-point deviation value during the precise adjustment at the end of the adaptive regulation stage becomes very evenly distributed. However, due to the existence of the initial deviation of the blade geometry of 0.008-0.05 mm, this geometry deviation cannot be completely reduced. If these positive geometric deviations are evenly distributed on both sides of the theoretical model of the blade, the benchmark error at this time will be completely reduced, and the position of the blade tenon root can be guaranteed.  In summary, the single point deviations of the blade concave profile and the blade convex profile are all positive, and they tend to be uniform through adaptive adjustment due to the existence of the initial deviation of the blade geometry of 0.008-0.05 mm. At this time, it can be considered that the blade is in the correct position, and the position of the blade tenon root relative to the blade body has been obtained. Therefore, the adaptive CNC machining process reasonably reduces the inconsistent error of the initial geometric accuracy. However, the prerequisite is that these single-point deviation values must all be positive. This method is only feasible for the case of positive margins, and these positive margins can be evenly distributed on both sides of the blade theoretical model. Therefore, it must be noted that this method cannot be used when the blade margin is negative.
It can be concluded that the adaptive CNC machining process can reasonably reduce the machining error of the nearnet-shaped blade caused by the positioning benchmark when the positioning surface margin of the near-net-shaped blade is positive and the positive margin of the positioning benchmark surface is evenly distributed on both sides of the theoretical model by the adaptive CNC machining process.

Adaptive CNC machining process for machining position error control
The machining position error refers to the position deviation of the blade tenon root relative to the blade body. Relatively, the deviation value of the measurement point on the blade body can be used to indicate the position error of the blade tenon root when the blade tenon root is used as the measurement benchmark. Figure 17 shows the deviation between the blade on-machine measurement model and the blade theoretical model. Figure 17 (a) shows the deviation value obtained during the first measurement process after the blade is clamped, and the specific deviation distribution value is shown in Figure 18. It can be seen that deviation values of some measurement points of the blade body are negative values (less than 0 mm), while the deviation values of some measurement points are positive values (more than 0 mm), and the overall position of the blade is inclined in this case. The unstable blade clamping process and the initial deviation of the blade geometry causes the amount of inclination. Figure 17 (c) shows the blade body deviation value obtained through the adaptive CNC machining process adjustment. The specific deviation distribution value is shown in Figure 18. It can be seen that the overall deviation value of the blade tends to be stable. Figure 17 (b) shows the intermediate process of the intermediate adaptive CNC machining process. Figure 17 (c) shows the final acceptable deviation range after repeated iterations of the adaptive CNC machining process.
Specifically, the adjustment process from Figure 17 (a) to Figure 17 (c) is the overall position rotation and translation process of the blade. In the blade adaptive CNC machining adjustment, an initial translation matrix and a rotation matrix The contour error of B surface relative to A surface is less than 0.007mm Fig. 11 Blade on-machine measurement process for machining contour error control. (a) On-machine measurement model, (b) blade tenon root measurement point, and (c) on-machine measurement process Cutting fluid Water-based cutting fluids are firstly given, and then an iterative process is used to repeatedly solve the matching objective equation, which is shown in Formula 5. Finally, a rotation matrix and a translation matrix are determined in order to obtain the final adaptive adjustment position of the blade. The deviation variation amount of the blade position during the adjustment process is shown in Figure 19 (b). The blade body deviation is distributed regularly (see Figure 19 (b)) through the blade adaptive adjustment. Figure 19 (b) is the distribution of the deviation variation of the blade during the adaptive adjustment. The deviation variation of the blade is large during the intermediate state of the adaptive adjustment after the clamping is completed, which indicates that the adaptive adjustment plays a key role in the blade position, which has an obviously regulatory effect at this time. The deviation variation amount is basically 0 mm during the last adaptive adjustment, which is the acceptable final position of the blade at this time, and the upper limit of the position adjustment capability has been reached, which means that more iterations will be meaningless and the iteration should be stopped. Table 3 is the blade machining position error control capability of the adaptive CNC machining process. The corresponding normal distribution curve is shown in Figure 20. According to these 80 measurement points on the blade body, the lower deviation of the measurement model relative to the theoretical model is reduced from 0.0556 to 0.0482 mm after the blade position adjustment by the adaptive CNC machining process. The upper deviation of the blade is reduced from 0.0706 to 0.0532 mm, and the blade deviation band is reduced from 0.015 to 0.005 mm, which is reduced by 60%. The adaptive CNC machining process has good machining position error control ability.
The adaptive adjustment is very reasonable for optimizing the blade position after clamping. The blade body position obtained at this time is used as the benchmark surface to complete high precision the CNC machining process of the blade tenon root. A more accurate fit between the actual position of the blade body surface and the theoretical surface is obtained with the blade position adaptive adjustment according to the above-mentioned measurement results.
It can be concluded that the adaptive CNC machining process has good positioning errors control ability for the nearnet-shaped blade. The lower deviation of the measurement model relative to the theoretical model of the 80 measuring points of the blade body is reduced from 0.0556 to 0.0482 mm after the blade position and posture adjustment by the adaptive CNC machining process. The upper deviation of the blade is

Adaptive CNC machining process of contouring error control
In order to correlate the stiffness of the blade-fixture system and the adaptive CNC machining process, it is assumed that there is a large deviation in the roughing process for the large cutting force and the weak stiffness blade-fixture system. As shown in Figure 21, the machined blade position deviation from the theoretical model is in the area of − 1.0 to 0.7 mm. Obviously, such a large position deviation will inevitably cause a larger machining error copy phenomenon according to the machining error transmission model (Formula 3 and of Figure 3). The machining error copy phenomenon will cause the size of the final blade tenon root to be seriously uncontrollable, and the low machining accuracy blade tenon root will lead to an unqualified blade. The adaptive CNC machining process will cut off this machining error transfer mode according to the previously discussed theoretical models (see Figure 2, Figure 3, and Figure 5). Specifically, in the experimental results, Figure 21 (b) shows the position and status deviation after the first blade adaptive adjustment. The adjustment method is to move 0.87 mm along the x-axis direction, and the translation matrix is: The position of the blade tenon root is distributed around the blade theoretical model into a diagonal line (see Figure 22 (a)) after the blade translation. The position of the blade is tilted, and the blade needs to be rotated along the C-axis. There are not significant changes in the position deviation of the blade tenon root after the blade rotation. The position of the blade tenon root is still a diagonal line. However, the angle of the diagonal line will change. This means that the adjustment method of rotating by 0.05 deg along the C-axis direction has the opposite effect, which makes the blade deviate further from the blade theoretical position. Therefore, it is necessary to change the direction of rotation around the Caxis. Further, the direction of the rotation around the C-axis is − 0.15 deg, and the new rotation matrix is   However, the position error of the blade tenon root relative to the blade body may change after the adaptive adjustment due to the contouring error of the tenon root. Therefore, it is necessary to monitor the position deviation of the blade body to explore whether the blade body position has greatly changed during this adaptive adjustment, which can ensure the position error of the blade tenon root relative to blade body under the condition of ensuring the key position size of the blade tenon root, which refers to the contour error of the blade tenon root. Figure 23 is the distribution of the deviation value of the blade body surface measurement point, and the corresponding deviation data are shown in Figure 24. It can be seen that the blade body deviation is close to the theoretical data, and the blade body deviation is also controlled with the same rules in the process of rotation and translation. Figure 25 (b) shows the numerical distribution of the variation value of the blade body contour deviation. It can be seen that the deviation becomes smaller during the adaptive adjustment, which indicates that the adaptive adjustment adjusts the position and status of the blade body at this time.
The variation of the deviation amount during the last adjustment is small. The final position and the status of the blade can be adjusted by the blade tenon root side position deviation adaptive adjustment, which reaches the upper limit of the blade position and status adjustment ability. However, it can be seen in Figure 25 (b) that the single-point deviation variation value that reflects the adaptive adjustment capability and the upper limit of adaptive adjustment is not close to 0 mm at this time, which shows that there is still a big difference between the blade measurement model and the blade theoretical model. The position and the posture of the blade body at this time do not have the best pose compared with Figure 19 (b). The adaptive adjustment process with the goal of minimizing the contouring error of the tenon root pressure surface cannot  In other words, for the case where the blade has a large displacement deviation produced in the previous CNC machining procedure, the adaptive CNC machining process cannot accomplish the position error control and contouring error control of the blade at the same time. This experimental phenomenon is particularly important considering the precision requirements of the blade CNC machining process. Therefore, ensuring the stiffness of the blade-fixture system in the adaptive CNC machining process is a prerequisite for the adaptive CNC machining process. In summary, for the case where the blade has a large displacement deviation produced in the previous CNC machining procedure, the adaptive CNC machining process cannot accomplish the position error control and contouring error control of the blade at the same time. The sufficient stiffness of the blade-fixture system in the adaptive CNC machining process is a prerequisite for the adaptive CNC machining process.
Based on the above analysis, the adaptive CNC machining process of the near-net-shaped blade is obtained under the premise of sufficient process system stiffness of the blade-fixture system (see Figure 5). Figure 26 shows the contouring error test results of different parts of the blade tenon root and tip. The side contouring error of the blade tenon root and tip is below 0.003 mm (see Figure 26 (a)), which indicates that the adaptive CNC machining process meets the contouring requirement of the blade tenon root, which is less than 0.007 mm. Figure 27 shows the surface roughness test results of different parts of the blade tenon root and tip. The surface roughness of the flat part of the blade tenon root and tip is below Ra 0.2 μm. The surface roughness at the arc transition and some rounded corners is less than Ra 0.4 μm. This is because the actual cutting depth and the theoretical

Conclusions
In this work, the adaptive CNC machining process and the blade CNC machining errors are investigated with theoretical and experimental analyses. The results can be summarized as follows: 1. The adaptive CNC machining process can reasonably reduce the machining errors caused by the positioning benchmark when the positioning surface margin is positive, and the positive margin of the positioning surface is evenly distributed on both sides of the theoretical model based on the adaptive CNC machining process. 2. For a near-net-shaped blade, the adaptive CNC machining process has good positioning error control ability. The lower deviation of the measurement model relative to the theoretical model of the blade body is reduced from 0.0556 to 0.0482 mm after the blade position and the posture adjustment based on the adaptive CNC machining process, the upper deviation of the blade is reduced from 0.0706 to 0.0532 mm, and the blade deviation band is reduced from 0.015 to 0.005 mm, which is a reduction of 60%. 3. The adaptive CNC machining process has a good effect for optimizing the tenon root contouring error, while the position error of the blade tenon root is not in the best state when the stiffness of the blade-fixture system is insufficient. Furthermore, the position error and the contouring error of the blade tenon root will improve at the same time only when the adequate stiffness of the blade-fixture system is a prerequisite.
However, the adaptive CNC machining process adjustments for the blade machining position error and the contouring error increase the on-machine measurement time and the measurement data processing time, respectively, which will reduce the CNC machining efficiency. Determining how to reduce the measurement links, reduce the procedures, and improve the processing efficiency will be the research focus of CNC machining optimization. Data availability Not applicable.

Declarations
Ethics approval Not applicable.
Consent to participate Not applicable.
Consent for publication Not applicable.
Competing interests The authors declare no competing interests.