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Geometric modelling of thin-walled blade based on compensation method of machining error and design intent
Journal of Manufacturing Processes 44 (2019) 327–336
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Journal of Manufacturing Processes journal homepage: www.elsevier.com/locate/manpro
Geometric modelling of thin-walled blade based on compensation method of machining error and design intent
T
Yaohua Hou, Dinghua Zhang, Jiawei Mei, Ying Zhang, Ming Luo
⁎
Key Laboratory of Contemporary Design and Integrated Manufacturing Technology, Ministry of Education, Northwestern Polytechnical University, Xi’an, China
ARTICLE INFO
ABSTRACT
Keywords: Machining Aero-engine blade Secant compensation Geometric model Machining error
Due to the low stiffness of the thin-walled aero-engine blade milling process, the machining errors cannot match the tolerance requirements and thus limit the aerodynamic performance. Previous research mainly focused on developing the compensation method to decrease the errors in concave and convex surfaces milling processes through controlling the tool path. However, the solutions on solving the leading and trailing edges machining errors are lacking. This paper first utilizes the geometric modelling framework of the blade to solve the machining error compensation problem. Furthermore, the presented solution could be not only suitable for the concave and the convex surfaces milling processes, but also capable for decreasing the leading and the trailing edges machining errors. A novel geometric modelling of the blade is developed in this paper, including the reconstruction procedure of the concave and the convex curves based on the secant compensation method, and the construction strategy for the leading and trailing edges is derived through the design tangents of the leading and trailing edges points. The validation demonstrates that for the thin-walled part milling process, the proposed secant compensation method could decrease the machining errors significantly (the maximum and the average errors are reduced to 31.7% and 23.8%, respectively). And for the blade milling process, the average errors of concave and convex surfaces are reduced to 25.1% and 14.3%, respectively. The average errors of leading and trailing edges are reduced to 29.7% of uncompensated machining process.
1. Introduction
the predicted deflection error was compensated through a developed algorithm. However, the machining errors in the thin-walled parts milling process mainly come from the deformation of the workpiece, not the deflection of the cutter [6]. Ratchev et al. [3] and Chen et al. [7] predicted and compensated the machining errors based on the finite element (FE) analysis of the cutting forces and deformation of the thinwalled parts. However, they did not expand their approach to the blade machining process. Wang and Sun [8] applied the FE model into the compensation procedure of the blade machining process, however, their research also demonstrated that even the residual stress, roughness and deformation were considered, the FE simulation results were still different from the real machining process without considering the initial stress. For the second strategy, the measurement compensation method. The principle of this idea is derived from the compensation solution as well. Meanwhile, the application process is much easier because the detail reason of error generation is not considered. The application of measurement compensation method is to obtain the dimension of part using measuring skill and compare it with the design model to calculate the machining errors. According to these integrated errors, the process
The conventional computer numerical control (CNC) machining is still the main manufacturing solution for the aero-engine blade processing in industry [1]. The complex shape and low stiffness of the blade cause severe challenge for the machining process, resulting in the unacceptable machining errors and limiting the precision of the machined blade surfaces [2]. As one of the typical thin-walled parts, the blade faces the unexpected deformation under the variable cutting forces due to the low stiffness during the milling process. Therefore, the real cutting depth cannot match the ideal nominal cutting depth and thus causes the machining errors. These machining errors were normally compensated by the manufactures [3] and hence many compensation solutions have been developed to overcome this problem [4]. Generally, the error compensation methods for thin-walled parts could be classified into two strategies. The first one is the calculation compensation method based on the error simulation and the deformation prediction, and the second one is the measurement compensation method [5]. For the first strategy, Rao et al. [4] simplified the milling cutter as the cantilever beam to predict the deflection of the cutter, then
⁎
Corresponding author. E-mail address: [email protected] (M. Luo).
https://doi.org/10.1016/j.jmapro.2019.06.012 Received 13 June 2018; Received in revised form 21 May 2019; Accepted 18 June 2019 1526-6125/ © 2019 The Society of Manufacturing Engineers. Published by Elsevier Ltd. All rights reserved.
Journal of Manufacturing Processes 44 (2019) 327–336
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parameters or tool path is adjusted to machine the new part by using an off-line or in-line control methods [9]. After that, machining errors will be deceased iteratively [10,11]. In the implementation of this method, how to accurately obtain part errors is prerequisite. Cho et al. [12] proposed the error compensation system to compensate the tool path based on the on-machine measurement (OMM) and introduced the concepts of error zone and tolerance zone. Then the neural networks were utilized to process the machining errors obtained from the OMM and an approach which could update the cutting depth based on these resulted errors was developed [13]. Marinescu et al. [14] established a compensation processing system with OMM to predict, measure and compensate the turning process errors. These applications, however, only considered calculating errors directly through the measured data and ignored the relationship among the machining, systematic and random errors. Chen et al. [15] applied the regression equation to investigate the relationship among the machining, systematic and random errors, such relationship was then utilized to compensate the machining errors through the autocorrelation algorithm. Poniatowska [16] analyzed the spatial error of the measured data and established the error distribution model to develop the machine pattern model (MPM) for the machining errors compensation. However, only the mirror compensation method was taken as the basic algorithm to compensate the machining errors in the mentioned research. To develop a more effective compensation algorithm, Lim and Menq [17] proposed the single “zero-error” compensation strategy utilizing the straight lines to fit the relationship between the estimated machining error and the normal cutting depth. For the thin-walled milling process, Guiassa and Mayer [18] proposed the process-intermittent gauging to compensate the finish milling process utilizing the machining errors and the cutting depth of semi-finish milling process. Similarly, the “zero-error” strategy is proposed to calculate the command nominal cutting depth for finish milling process. Although it has been proved that the “zero-error” strategy is more effective than the mirror approach, the efficiency can be improved by using the higher order curves instead of straight lines or increasing the number of the iteration times that the strategy is been applied. The applications of the above compensation methods are only limited in the specific objects, few research of the blade compensation process was proposed. The aero-engine blades, as one of the representative thin-walled parts, are different from the normal parts [19]. The low stiffness, complex shape and quick change of curvature complicate the blade machining process, resulting in two problems to be solved. The first one is how to establish the detail mathematical model for blade machining, i.e. how to machine the different region of the blade. Huang et al. [20] applied OMM to measure and reconstruct the machined surface in 5-axis flank milling for thin-walled parts. Combined with the iterative compensation method, the tool path is modified to machine the surface again and decease the machine error, which was demonstrated in impeller machining. However, this research mainly focused on the flank milling tool path generation of the free-form surface which is only applied in the rough milling of blade machining process. Altintas et al. [21] introduced the virtual compensation process to construct the blade deflection for ball end milling operation. Nevertheless, only the mirror compensation method was considered and the geometric modelling of the blade was ignored, leading to the second problem. The second problem is how to design the proper geometric model of the blade in compensation algorithm to obtain the exact dimension of the machined blade. For blade machining process, the detail construction methods of blade modelling should be considered. Dong et al. [22] applied iterative mirror compensation method to obtain the casting blade considering the shrink effect. The mean line (ML) curve and the thickness of airfoil were used to adjust the shape of die for high precision. Considering the design intent in precision forging of blade, Zhang et al. [23] utilized the free-form deformation to modify the ML of the original blade and optimize the thickness distribution. The reconstructed geometric model of forging blade was obtained and
satisfied the measuring data. It could be commented that these blade modelling methods were based on the ML and thickness distribution using envelop method. However, the concave and convex surfaces are bending in different directions, leading to different error distributions. The geometric modelling method of ML and thickness distribution cannot satisfy the compensation method. Moreover, as the most important regions of blade, the leading edge (LE) and trailing edge (TE) curves influence the inlet and outlet angles. The construction method of these areas is significant. However, the research of machining leading and trailing edges precisely is still lacking. Therefore, how to reconstruct the LE and TE curves is necessary. Aiming to decreasing the machining errors caused by the deformation of the blade, the compensation mathematical model and geometric modelling for the blade machining are proposed in this paper. Based on the OMM data, the compensation model is applied to calculate the compensation data which will be transmitted to reconstruct the new geometric model after the first machining process. The new tool path will then be generated from this model for the next blade machining process. The process will be applied iteratively, and once the machining precisions satisfy the design tolerance, the current geometric model of process for this blade could be considered as the standard model for the current machining condition. In detail geometric modelling, the concave and convex surfaces are reconstructed by the secant compensation method. The constraint of design tangent is proposed to reconstruct the LE and TE for compensated blade model. At last, the steel block and titanium blades are implemented to verify the compensation mathematical model and compensation algorithm for blade process. 2. Geometric modelling of compensated blade based on the design intent This section is focusing on constructing a certain error compensation model to redesign the blade for generating the new tool path which will be utilized to control the machining error. It has been agreed that the adjustment of NC code, as a direct modification method, cannot satisfy the complex shape change of the blade since the normal of cutter contact point, cutter axis and cutter location will make the compensation process more complex. Based on the design intent, the redesign of blade model is more convenient and has been utilized widely [21,22]. According to the expected contents, the framework is shown in Fig. 1. The compensation process is applied to reconstruct the CAD model of the blade based on the machining results to generate the CAM process iteratively until the machining results satisfying the design tolerance. Currently, the blade geometry design comes from the aero-engine designer. However, with the development of the design and manufacture integration process, the design process of the blade should also be taken into account for the manufacture. For the aero-engine designer, the primary request of the blade design is to satisfy the aerodynamics performance. Secondly, stiffness and strength of the blade should also be considered. To design the blade of the aero-engine (see an example shown in Fig. 2), the design step could be classified as preliminary design, throughflow design, 2D blade design and 3D blade design. The preliminary and throughflow design are utilized to confirm the average geometric performance parameters, twist and matching of the blade. 2D blade design is applied to calculate the 2D shape of the airfoil (see Fig. 3), which will be stacked to construct the 3D blade (see Fig. 2) by the bended stacking line. Therefore, the 2D blade design method is the most important step in the blade design. Geometric parameter method and spline parameterization method are the most popular methods for the 2D blade design. Geometric parameter method stacks the thickness distribution of the blade to the ML curve to construct the concave (pressure) and convex (suction) curves referring to [23]. Spline parameterization method divides the blade section curve into LE, TE, concave and convex curves with smooth connection. The LE and TE curves are expressed by arcs while 328
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Fig. 1. The compensation framework of aero-engine blade.
compensation models of thin-walled parts and fit the compensated points of concave and convex curves. 3.1. Compensation modelling of the machining errors for thin-walled parts Machining is a complex elastoplastic deformation process with the generation and transformation of cutting force [24] and heat [25]. In the meantime, the existences of different ways of clamping, variation of stiffness and instability of machine tool result in unavoidable cutting vibrations and deformation. Therefore, the machining process could be expressed as a complex function with inputs and outputs. The inputs include machine tool, cutter, fixture, workpiece, cutting parameters (cutting depth, spindle speed and feed rate). The outputs are the status of workpiece after cutting. When machining the thin-walled parts, the deformation of parts always influences the machining precision. Especially for the blade parts, the unique fixture and complex shape of the blade lead to the low stiffness at tip and edges. The corresponding deformation errors are much larger than the design tolerance. Since the shape of blade and stiffness cannot be changed, adjusting the parameters of process has become a popular way to control the machining error. Therefore, the compensation model could be utilized to control the machining error through optimizing the cutting depth. Fig. 2. Constitutions of a 3D blade.
3.1.1. Machining error model for thin-walled parts The nature of cutting process is the contest between force and stiffness of various parts of the processing system. The thin-walled parts exhibit the low rigidity and cause the errors, based on the Hooke’s law, the parts deformation caused by the cutting force could be express as
=
F K
(1)
where ε is the deformation value, F is the normal cutting force, K is the system equivalent stiffness. For a certain milling process (see Fig. 4a), the material between the design and the allowance surfaces should be removed to achieve dimensional accuracy. For the existence of machining deformation, the non-linearity between machining error and cutting depth could be calculated as
Fig. 3. Constitutions of a 2D blade.
the concave and convex curves are expressed by splines. It has been commented in Section 1 that the geometric parameter method cannot satisfy the geometric modelling of compensated blade. Therefore, the spline parameterization method is applied to reconstruct the 2D airfoil curves for compensated geometric modelling of blade. Following the description of this method, the compensated construction methods should be divided into the compensated fit processes of concave, convex curves and LE, TE arcs.
e (x ) = H
x + (x ) + s (x ) + …
(2)
where H is design allowance, x is nominal cutting depth, e is machining error, s is inelastic error and so on, as shown in Fig. 4(a). In thin-walled parts machining, since the deformation value ε is the key component of the comprehensive machining error, it has been considered to represent the machining error [18] in the following calculation. Fig. 4 shows the relationship among parameters of general, 0th and nth compensation processes. The compensation modelling and solving are shown in the following sections.
3. Geometric compensation modelling of concave, convex curves When machining the concave and convex surfaces, for the special fixture and geometrical feature, the machining deformation satisfies the elastic deformation. Therefore, the geometric compensation modelling of concave and convex curves could be employed to construct the 329
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Fig. 4. Parameters of (a) general, (b) 0th and (c) nth compensation processes.
3.1.2. General error compensation model for thin-walled parts For thin-walled parts processing, the design allowance H is set to constant before the machining. Focusing on the cutting depth, the relation between nominal cutting depth x and real cutting depth y could be expressed as Eq. (3). (3)
y = f (x ) According to Fig. 4(a), the machining error can be expressed as
e (x ) = H
y=H
(4)
f (x )
Therefore, the compensation method of machining error is selecting a more suitable nominal cutting depth x to make the real cutting depth y equal to the design allowance H and achieve the purpose of zero error. Iteration method can be introduced to solve the nonlinear equation as a successive approximation approach. The iterative error compensation equation, which is the general error compensation model, is established to calculate the nominal cutting depth of the next cutting process as:
x0 = H xk + 1 = xk +
k+1
ek (k = 0,1,
)
Fig. 5. Convergence process of mirror method.
(5)
where ρ is the compensation coefficient, k is the times of compensation process. It should be noted that the compensation process is required to be processed among multiple parts until the compensation value is stable. The case that k = 0 corresponding to the condition that the part is machined without compensation.
red curve represents the cutting process f. The black dash lines are marked as nominal and real cutting depth. The allowance value of real cutting depth is described as green line to show the final cutting position. The dash dot line defines the ideal cutting depth D. The calculating direction of the nominal cutting depth is presented as blue line. For the inherent limitation of the mirror method, it could be observed that the convergence speed is slow. The compensated amount is too small to counteract the error. The limitation can be proved by the ideal cutting depth D and ideal error E. It can be observed from Eq. (6) that when k = 0, the process has not been compensated, the nominal cutting depth is x 0 = H . If the current cutting process is rigid, then the deformation will not happen, i.e. D0 = H y = x . The ideal cutting depth D and error E are . However, the E0 = 0 variations of cutting force and stiffness will change the real cutting y = f (x 0 ) < D0 state. The result is described as 0 . e0 = H y0 > E0 During the (n+1)th compensated process, the current nominal n cutting depth could be calculated as x n+ 1 = H + i = 0 ei by using the iteration. If the current deformation relationship between the real and nominal cutting depth is the same as the previous one, then the ideal cutting depth D could be expressed as:
3.1.3. Solutions of general error compensation model Based on the error model of the nonlinear cutting process for thinwalled parts, a general error compensation model could be established. However, such general model cannot be applied in a real milling process without the detail calculation of cutting depth. The purpose of error compensation is to reduce the machining error to fit the accuracy range. For the cutting error model (Eq. (4)), error compensation is developed to calculate the x when e = 0, which is the zero-point problem of e(x). Two solutions with different convergence speeds are provided to solve the general model in the following. 3.1.3.1. Mirror method. Mirror method has been widely employed in calculating the compensation value for both error prediction and measurement compensation processes. Such method adds the error caused by previous machining process to current nominal depth of cut to improve the manufacturing accuracy, as described in the following formula:
x0 = H xk + 1 = xk + ek (k = 0,1,
)
Dn+ 1 = xn + 1
(6)
By updating the cutting depth through the mirror method, some machining errors could be reduced. Especially when sufficient iterative calculating and processing have been conducted, the machining error can be controlled in the desired area. However, the lower rate of convergence of the mirror method should be noticed. The convergence process of mirror method is provided in Fig. 5. The
yn xn
(7)
Eq. (7) could also be rewritten as
(
Dn+ 1 = H + Therefore, 330
n i=0
the
ei
)
H H+
ideal
en n 1 e i=0 i cutting
(8) depth
D
and
error
E
are
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Y. Hou, et al. en
Dn + 1 = H
H+ n i = 0 ei n 1 i = 0 ei
en
En + 1 =
H+
n i = 0 ei n 1 i = 0 ei
. Thus, even for the ideal situation that the
stiffness is not weakened in the current cutting process, the ideal cutting error calculated by the mirror method is not 0. Furthermore, during the real process the stiffness would be decreased, and hence the real cutting depth is smaller than the ideal cutting depth and the real machining errors are larger than the ideal error, which is described as
yn + 1 < Dn + 1 . en + 1 > En + 1 Therefore, based on the discussion above, it could be commented that the small compensation amount leads to the slow rate of convergence and low cutting efficiency due to the inherent of the mirror method. Thus, a more efficiency compensation method is required. Fig. 6. Convergence process of secant method.
3.1.3.2. Secant method. Based on the mirror method with the smaller real cutting depth, the secant method is established to obtain a larger cutting depth. The nominal cutting depth of the next process is expressed as the summation between the nominal cutting depth of the previous process and the compensation amount:
x0 = H xk + 1 = xk + z k + 1 (k = 0,1,
n
k+1
x0 = H xk + 1 = xk +
zi
(12)
i=1
It can be observed from Eq. (9) that the uncompensated process can be achieved when k = 0 (see Fig. 4b), where the nominal cutting depth is x0 and machining error is e0, thus the springback coefficient could be calculated as: 0
=
e0 x0
zi
(16)
xk ek (k = 0,1, yk
)
(13)
Considering that the current stiffness is fixed and utilizing the springback coefficient, the nominal cutting depth of the next compensated process, x1, could be calculated under the control of the ideal error, zero. The relationship between the nominal cutting depth and increment of next compensated process is expressed as
(x 0 + z1 )
0
(14)
= z1
(17)
With the above analyses, the secant compensation model is applied to reconstruct the concave and convex curves due to its high efficiency and precision. However, the secant compensation model is to solve the relationship between the cutting depth and error while the geometric modelling is to fit the point sets of contours and construct a series of smooth curves. The adjustment of compensation algorithm is applied to solve the calculation of the point sets. Based on the general error compensation model, the allowance curve, design curve, the (k-1)th machining error curve and the kth compensated curve are presented as shown in Fig. 7. Then the
k
= ek +
i=1
3.2. Geometric compensation calculation of concave and convex curves
In Eq. (11), ε is the deformation value presented in Eq. (1), and could be expressed as: k
n
The convergence process of the secant method is demonstrated in Fig. 6. The error iteration sequence of this method is linearly convergent and hence the rate of convergence of the secant method is faster than the mirror method.
(11)
x
= zn+1 +
Thus, the relations that z n + 1 = H e en , and x n+ 1 = xn + zn + 1 are n obtained. Combined with the general compensation model Eq. (5), the above equation is written as
Demonstrating that the compensation amount is varied by adjusting ρ. Thus, the quotient of springback and nominal cutting depth, the springback coefficient, is introduced to this topic and presented in the following:
=
n
xn
(10)
ek
(15)
xn
(x n + z n + 1 )
where zk+1 represents the increment between the nominal cutting depths of the (k+1)th and kth compensated processes. Unlike the mirror method, the compensation amount z of the secant method is adjusted dynamically according to the real machining error. According to the general compensation model Eq. (5), it can be written as
z k+1 =
n z i=1 i
en +
Considering a constant stiffness in the machining process. Setting the ideal error En+1 = 0. The increment zn+1 of the (n+1)th compensated process is expressed as
(9)
)
=
Therefore, the formulas that z1 = H e e0 , and x1 = x 0 + z1 could be 0 obtained. During the nth compensated process (see Fig. 4c), the nominal cutting depth is xn and the machining error is en, hence the springback coefficient could be calculated as: x0
Fig. 7. Compensation process of compensated points. 331
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parameters of Eq. (17) could be calculated from this figure. Considering that Pd is the point of the design curve, Pc,k is the corresponding compensated point of the kth compensated process and nd is the material removal direction, the calculation of the error compensation points is obtained combined with the iteration equation of secant method Eq. (17). The ith compensated points of the concave or convex curves could be expressed as
Pc , i, k = Pd, i + nd, i (xk, i
Hi ), (i = 0,1,
, m)
(18)
where m is the total number of the measurement points in concave or convex curves. After obtaining the compensated points, the fitting process of these points is followed to generate the relative curve. However, due to the vibration, residual height, measuring plan and other reasons, generally the measured points are not in a smooth curve, and thus resulting in the unsmooth compensated points. Therefore, the interpolation of unsmooth points could lead to an unsmooth curve which is not appropriate in manufacture, and thus the approximation of the fitted curve is required. To this end, the least square is employed to conduct the approximation of the fitted curve. For the approximation of points, the NURBS fitting method is introduced. With the certain knot vector and curve degree, the fitted curve is reconstructed easily. Following the above process of calculating and fitting compensated points, the compensated concave and convex curves could then be reconstructed.
Fig. 8. LE model of design type.
Ld (s ) = LEPd + s tn
(19)
Step 3. Calculating the compensated LE curve (CAd,k) for the compensated concave and convex curves.
4. Geometric compensation modelling of LE and TE The shapes of LE and TE within the blade can be designed as circular, elliptical or curved referring to [26]. For processing convenience, the shapes of LE and TE are designed as circle. Such simplification sacrifices the aerodynamic performance to increase the precision of manufacture since the elliptical and curved edges complicate the manufacturing process. However, the circular edges are still challenging for machining and the large error is existed in the machined blade. To decrease the machining error of the edges, the compensation method is introduced. Unlike the compensation method of concave and convex curves, the detail deformations principles of edges are complicated. The secant compensation model cannot be applied directly. Therefore, from the view of geometry modelling, the new LE and TE are reconstructed to connect the compensated concave and convex curves smoothly, which is the reflection of the design intent of the blade. For the same shapes of LE and TE, the following steps take LE as an example to describe the detail.
4.2. Arc fitting method based on ML For the special geometry constitution, the smooth connections of LE and concave, convex curves are required. Therefore, the compensated curves should be connected smoothly. Moreover, the tangent Ld of LE is the other constraint of calculating the LE. Thus, the constraints of three curves (tangent, concave and convex curves) are the keys in calculation of the compensated LE. Generally, the research of reconstructing the arc in the constraints of the three curves is to calculate an inscribed circle, which is tangent to L, CV and CC as shown in Fig. 9. To this end, the inscribed circle is expressed as follow
4.1. LE modelling based on tangent constraint The shape of LE influences the region of attack angle and flow separation, based on the designed LE, the new LE is developed to construct the compensation model with constant chord length of the blade. In the following description, the new LE is treated as the compensated LE. In the compensation model of the blade, it is assumed that the concave and convex curves satisfy the elastic deformation principle for the thin-walled feature, the nearby area of the compensated LE is defined to be rigid. Therefore, when constructing the compensation model, the position of the compensated leading edge point (LEP) is the same as the position of designed LEP (see Fig. 8). As the result, the chord length remains the same. Based on these features, the tangent constraint of LEP is introduced to constraint the LE arc which will be reconstructed in the following steps. Step 1. Calculating the designed LEP (LEPd). Based on the designed ML (MLd) of airfoil, the intersection point, LEPd, is calculated by the extended ML and designed LE curves. Step 2. Calculating the tangent Ld of LEP. The direction of tangent tn is found to calculate the designed tangent Ld of LE, which is
Fig. 9. Inscribed arc of concave, convex curves and tangent line. 332
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fCV (PCV ) (PCV
O) = 0
fCC (PCC ) (PCC f L (PL ) (PL |PCV |PCC
O) = 0 O) = 0
O| = |PCC O| = |PL
O| O|
(20)
where fCV , fCC and f L are the first derivatives of CV, CC and L, respectively, PCV , PCC and PL are the corresponding points in CV, CC and L, respectively, and O is the coordinate of the circle center. By solving the above equations, the inscribed circle with center O and radius R = |PL O| is obtained. However, the function of the freeform curves is complicated to applied in the equations solving problem. Therefore, the ML (ML) of CV and CC is introduced to simplify the problem referring to [27]. Considering that the points of ML are the centres of inscribed circles of CV and CC, the radius function R(s) and the centre of the circle O(s) could be constructed with all points of ML, where s is the arc parameter of ML. The Eq. (20) is simplified as
f L (PL ) (PL |PL
O (s )) = 0
O (s )| = R (s )
(21)
Fig. 11. On-machine measurement setup.
Moreover, the distance function d(s) which is the distances between points and tangent line L with s shows the relation among PL , O (s ) and L. Then the following expression could be obtained to replace the Eq. (21).
t (s ) = R (s )
d (s )
compare different algorithms, three cutting depths which are separately calculated by the original model, mirror compensation and secant compensation methods, were applied in the finish milling. The machining errors of different cutting depths calculated by three compensation strategies are shown in Fig. 12. It could be observed that the compensation results based on both mirror and secant methods are better than the original model. The observations could be concluded as follows:
(22)
When the t(s) is zero, the relative circle CA with parameter s is tangent to L, CV and CC, which satisfies the requirement of reconstructing the compensated LE. The complicated equations solving problem of Eq. (20) with five variables is simplified as a single-variable equation which is convenient to be solved. Back to Step 3 in Section 4.1, the algorithm of inscribed circle mentioned in Eq. (22) is utilized to calculate the compensated LE (CAd,k), which is tangent to the concave, convex curves and tangent lines.
1) comparing to the uncompensated method, the maximum and the average errors of mirror method were separately reduced to 57.7% and 61.4%. 2) comparing to the uncompensated method, the maximum and the average errors of secant method were separately reduced to 31.7% and 23.8%. 3) comparing to the mirror method, the maximum and average errors of secant method were reduced to 54.9% and 38.8%, respectively.
5. Verification and discussion To validate the proposed method, the mathematical and geometric model for error compensation of blade machining were established. Moreover, the mathematical model was applied in geometry modelling. Two trials are presented in the following as the example to conduct the verification.
5.2. Verification of the compensated geometric model for blade To verify the geometric model of error compensation for aero-engine blade milling process, the uncompensated and compensated experiments were conducted to demonstrate the error diminution of different regions of blade. Two blades were machined by a 5-aixs machine
5.1. Verification of the compensation model for thin-walled part To verify the secant compensation model proposed in Section 3.1, the rectangular block in steel with cantilevered fixture was designed (see Fig. 10). A YHVT850Z 4-axis machine tool was employed as shown in Fig. 11. The cutting condition is: spindle speed = 3000 rpm, feed rate = 300 mm/min, axis depth = 11 mm, radial depth = 0.25 mm, tool radius = 6 mm, tool type = end mill, cutting type = flank mill. To
Fig. 10. Dimensions (mm) and sampled points of machining face.
Fig. 12. Error distributions of different cutting depths. 333
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Moreover, the machining errors of LE and TE regions in the uncompensated and compensated models are shown in Fig. 16. The average errors of edge regions fitted by tangent constraint of compensated machining process are reduced to 29.7% of uncompensated machining process. It has been proved that the machining precision is improved by applying the compensation algorithm in the above experiments. However, due to the machining error measurement and the compensation calculation, the total time in the compensated process is longer than the uncompensated process. Nevertheless, in the industry field, the compensation framework proposed in this paper can be applied to the volume production, where only the first few parts are taken as the specimens and the error measurements are needed. After the compensation value is stable, the modified tool path can be utilised in the remaining parts without any further compensation and the machining time could be saved from the compensated path. Thus, the calculation time of the compensation can be ignored for the volume production. Therefore, three conclusions could be concluded from these experiments: 1) The experiment of thin-walled parts validates that the secant compensation method is more effective than the current mirror compensation method. The average errors of secant compensation method are reduced to 38.8% of the mirror compensation method. 2) The application of secant compensation method to the construction procedures of the concave and convex curves has been proved to improve the overcut and undercut conditions. The average errors of concave and convex surfaces are reduced to 25.1% and 14.3%, respectively. 3) The fitting method of the LE and TE considering the constraint of design tangent was verified as the proper way to construct the most suitable arc shape comparing to the uncompensated model. The average errors of edge regions of design type are reduced to 29.7% of uncompensated machining process.
Fig. 13. Measuring process of machined blade.
tool, and the dimensions were measured by a CMM. The blade model is shown in Fig. 2 and the measuring process is shown in Fig. 13.The secant compensation method was applied to fit the new curves of concave and convex regions while the new leading and trailing edges were reconstructed based on the tangent constraint for compensation milling process. The actual machining errors of the uncompensated and the compensated concave and convex surfaces are shown in Fig. 14 as the cloud figures. Without the compensation, the machined concave surface is overcut while the convex surface is undercut (see Fig. 14a and c, respectively). Applying the compensated milling process, the overcut of concave surface and undercut of convex surface are corrected shown in Fig. 14(b) and (d). The average errors of concave surface are reduced to 25.1% while the average errors of convex surface are reduced to 14.3%. The measured error distributions and the fitted curves in LE and TE regions are shown in Fig. 15. The red measured points represent the actual machining error distributions of the uncompensated blade which is deviated from the designed section curves (green points). After the compensation process, the blue measured points show that the compensated curves are much closer to the design curves, indicating that the overcut and undercut are compensated.
6. Conclusion Due to the elastic deformation caused by the low stiffness of the aero-engine blade, the machining error of the blade milling process has become one of the most severe factors which limit the aerodynamic performance. In this paper, a novel geometric modelling of the blade has been firstly applied into the compensation procedure to decrease the machining errors of the blade milling process. The key contributions could be concluded as follow:
Fig. 14. The error distributions of concave surfaces with (a) uncompensated and (b) compensated methods, and the error distributions of convex surfaces with (c) uncompensated and (d) compensated methods. 334
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Fig. 15. Measured points of specific areas after compensating.
this framework, the undercut and overcut of concave and convex surface could be compensated, and the leading and trailing edges were optimized to satisfy the design shape and decrease the machining errors. Opening the avenue to improve the processing quality and the efficiency of the blade, and greatly reduce the rejection rate. Acknowledgements This study is co-supported by the Major National Science and Technology Projects (No. 2015ZX04001202), the Fundamental Research Funds for the Central Universities (No. 3102018jcc004) and the 111 project of China (No. B13044). References [1] Luo M, Han C, Hafeez HM. Four-axis trochoidal toolpath planning for rough milling of aero-engine blisks. Chin J Aeronaut 2018. https://doi.org/10.1016/j.cja.2018. 09.001. [2] Yan Q, Luo M, Tang K. Multi-axis variable depth-of-cut machining of thin-walled workpieces based on the workpiece deflection constraint. Comput Des 2018;100:14–29. [3] Ratchev S, Liu S, Huang W, et al. An advanced FEA based force induced error compensation strategy in milling. Int J Mach Tools Manuf 2006;46(5):542–51. [4] Rao VS, Rao PVM. Tool deflection compensation in peripheral milling of curved geometries. Int J Mach Tools Manuf 2006;46(15):2036–43. [5] Aguado S, Santolaria J, Samper D, et al. Improving a real milling machine accuracy through an indirect measurement of its geometric errors. J Manuf Syst 2016;40:26–36. [6] Ratchev S, Liu S, Huang W, Becker AA. A flexible force model for end milling of lowrigidity parts. J Mater Process Technol 2004;153–154:134–8. [7] Chen W, Xue J, Tang D, et al. Deformation prediction and error compensation in multilayer milling processes for thin-walled parts. Int J Mach Tools Manuf 2009;49(11):859–64. [8] Wang MH, Sun Y. Error prediction and compensation based on interference-free tool paths in blade milling. Int J Adv Manuf Technol 2014;71(5-8):1309–18. [9] Xie K, Camelio JA, Izquierdo LE. Part-by-part dimensional error compensation in compliant sheet metal assembly processes. J Manuf Syst 2012;31(2):152–61. [10] Fiorentino A, Feriti GC, Giardini C, et al. Part precision improvement in incremental sheet forming of not axisymmetric parts using an artificial cognitive system. J Manuf Syst 2015;35:215–22. [11] Ratchev S, Liu S, Huang W, Becker AA. Milling error prediction and compensation in machining of low-rigidity parts. Int J Mach Tools Manuf 2004;44:1629–41. [12] Cho MW, Seo TI, Kwon HD. Integrated error compensation method using OMM system for profile milling operation. J Mater Process Technol 2003;136(136):88–99. [13] Myeong-Woo CHO, Gun-Hee KIM, Tae-Il SEO, et al. Integrated machining error compensation method using OMM data and modified PNN algorithm. Int J Mach
Fig. 16. Average errors of 6 regions using design type.
1) The concave and convex surfaces of the blade have been constructed by the fitted points from the secant compensation method calculation. This is the first time that the secant method has been utilized into the compensation procedure of the blade body region. Such method has been validated through the thin-walled part and the blade milling processes, the secant compensation method proposed in this paper was proved to be a better solution to decrease the machining error as the average errors were reduced to 38.8% of mirror method. 2) The compensation procedures of the leading and the trailing edges for the blade milling process have been first conducted in this paper. The construction methods have been applied to fit the new shape of the edge regions. The validation demonstrated that the machined edge regions under the constraint of design tangent matches the design tolerance. The average errors of edge regions were reduced to 29.7% of uncompensated machining process. The proposed geometric modelling framework of the blade firstly solves the machining error problem for the whole blade shape. Through 335
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Y. Hou, et al. Tools Manuf 2006;46(12-13):1417–27. [14] Marinescu V, Constantin I, Apostu C, et al. Adaptive dimensional control based on in-cycle geometry monitoring and programming for CNC turning center. Int J Adv Manuf Technol 2011;55(9):1079–97. [15] Chen Y, Gao J, Deng H, et al. Spatial statistical analysis and compensation of machining errors for complex surfaces. Precis Eng 2013;37(1):203–12. [16] Poniatowska M. Free-form surface machining error compensation applying 3D CAD machining pattern model. Comput Des 2015;62:227–35. [17] Lim EM, Menq CH. Error compensation for sculptured surface productions by the application of control- surface strategy using predicted machining errors, transactions of ASME. J Manuf Sci Eng 1997;119:402–9. [18] Guiassa R, Mayer JRR. Predictive compliance based model for compensation in multi-pass milling by on-machine probing. CIRP Ann Manuf Technol 2011;60:391–4. [19] Luo M, Yan D, Wu B, et al. Barrel cutter design and toolpath planning for highefficiency machining of freeform surface. Int J Adv Manuf Technol 2016;85(912):2495–503. [20] Huang N, Bi Q, Wang Y, et al. 5-axis adaptive flank milling of flexible thin-walled
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parts based on the on-machine measurement. Int J Mach Tools Manuf 2014;84(6):1–8. Altintas Y, Tuysuz O, Habibi M, et al. Virtual compensation of deflection errors in ball end milling of flexible blades. CIRP Ann Manuf Technol 2018;67(1):365–8. Dong Y, Zhang D, Bu K, et al. Geometric parameter-based optimization of the die profile for the investment casting of aerofoil-shaped turbine blades. Int J Adv Manuf Technol 2011;57(9-12):1245–58. Zhang Y, Chen ZT, Ning T. Reverse modeling strategy of aero-engine blade based on design intent. Int J Adv Manuf Technol 2015;81(9):1–16. Luo M, Luo H, Axinte D, et al. A wireless instrumented milling cutter system with embedded PVDF sensors. Mech Syst Signal Process 2018;110:556–68. Cui D, Zhang D, Wu B, et al. An investigation of tool temperature in end milling considering the flank wear effect. Int J Mech Sci 2017;131:613–24. Hamakhan IA, Korakianitis T. Aerodynamic performance effects of leading-edge geometry in gas-turbine blades. Appl Energy 2010;87(5):1591–601. Hou Y, Zhang Y, Zhang D. Geometric error analysis of compressor blade based on reconstructing leading and trailing edges smoothly. Procedia CIRP 2016;56:272–8.
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Journal of Manufacturing Processes journal homepage: www.elsevier.com/locate/manpro
Geometric modelling of thin-walled blade based on compensation method of machining error and design intent
T
Yaohua Hou, Dinghua Zhang, Jiawei Mei, Ying Zhang, Ming Luo
⁎
Key Laboratory of Contemporary Design and Integrated Manufacturing Technology, Ministry of Education, Northwestern Polytechnical University, Xi’an, China
ARTICLE INFO
ABSTRACT
Keywords: Machining Aero-engine blade Secant compensation Geometric model Machining error
Due to the low stiffness of the thin-walled aero-engine blade milling process, the machining errors cannot match the tolerance requirements and thus limit the aerodynamic performance. Previous research mainly focused on developing the compensation method to decrease the errors in concave and convex surfaces milling processes through controlling the tool path. However, the solutions on solving the leading and trailing edges machining errors are lacking. This paper first utilizes the geometric modelling framework of the blade to solve the machining error compensation problem. Furthermore, the presented solution could be not only suitable for the concave and the convex surfaces milling processes, but also capable for decreasing the leading and the trailing edges machining errors. A novel geometric modelling of the blade is developed in this paper, including the reconstruction procedure of the concave and the convex curves based on the secant compensation method, and the construction strategy for the leading and trailing edges is derived through the design tangents of the leading and trailing edges points. The validation demonstrates that for the thin-walled part milling process, the proposed secant compensation method could decrease the machining errors significantly (the maximum and the average errors are reduced to 31.7% and 23.8%, respectively). And for the blade milling process, the average errors of concave and convex surfaces are reduced to 25.1% and 14.3%, respectively. The average errors of leading and trailing edges are reduced to 29.7% of uncompensated machining process.
1. Introduction
the predicted deflection error was compensated through a developed algorithm. However, the machining errors in the thin-walled parts milling process mainly come from the deformation of the workpiece, not the deflection of the cutter [6]. Ratchev et al. [3] and Chen et al. [7] predicted and compensated the machining errors based on the finite element (FE) analysis of the cutting forces and deformation of the thinwalled parts. However, they did not expand their approach to the blade machining process. Wang and Sun [8] applied the FE model into the compensation procedure of the blade machining process, however, their research also demonstrated that even the residual stress, roughness and deformation were considered, the FE simulation results were still different from the real machining process without considering the initial stress. For the second strategy, the measurement compensation method. The principle of this idea is derived from the compensation solution as well. Meanwhile, the application process is much easier because the detail reason of error generation is not considered. The application of measurement compensation method is to obtain the dimension of part using measuring skill and compare it with the design model to calculate the machining errors. According to these integrated errors, the process
The conventional computer numerical control (CNC) machining is still the main manufacturing solution for the aero-engine blade processing in industry [1]. The complex shape and low stiffness of the blade cause severe challenge for the machining process, resulting in the unacceptable machining errors and limiting the precision of the machined blade surfaces [2]. As one of the typical thin-walled parts, the blade faces the unexpected deformation under the variable cutting forces due to the low stiffness during the milling process. Therefore, the real cutting depth cannot match the ideal nominal cutting depth and thus causes the machining errors. These machining errors were normally compensated by the manufactures [3] and hence many compensation solutions have been developed to overcome this problem [4]. Generally, the error compensation methods for thin-walled parts could be classified into two strategies. The first one is the calculation compensation method based on the error simulation and the deformation prediction, and the second one is the measurement compensation method [5]. For the first strategy, Rao et al. [4] simplified the milling cutter as the cantilever beam to predict the deflection of the cutter, then
⁎
Corresponding author. E-mail address: [email protected] (M. Luo).
https://doi.org/10.1016/j.jmapro.2019.06.012 Received 13 June 2018; Received in revised form 21 May 2019; Accepted 18 June 2019 1526-6125/ © 2019 The Society of Manufacturing Engineers. Published by Elsevier Ltd. All rights reserved.
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parameters or tool path is adjusted to machine the new part by using an off-line or in-line control methods [9]. After that, machining errors will be deceased iteratively [10,11]. In the implementation of this method, how to accurately obtain part errors is prerequisite. Cho et al. [12] proposed the error compensation system to compensate the tool path based on the on-machine measurement (OMM) and introduced the concepts of error zone and tolerance zone. Then the neural networks were utilized to process the machining errors obtained from the OMM and an approach which could update the cutting depth based on these resulted errors was developed [13]. Marinescu et al. [14] established a compensation processing system with OMM to predict, measure and compensate the turning process errors. These applications, however, only considered calculating errors directly through the measured data and ignored the relationship among the machining, systematic and random errors. Chen et al. [15] applied the regression equation to investigate the relationship among the machining, systematic and random errors, such relationship was then utilized to compensate the machining errors through the autocorrelation algorithm. Poniatowska [16] analyzed the spatial error of the measured data and established the error distribution model to develop the machine pattern model (MPM) for the machining errors compensation. However, only the mirror compensation method was taken as the basic algorithm to compensate the machining errors in the mentioned research. To develop a more effective compensation algorithm, Lim and Menq [17] proposed the single “zero-error” compensation strategy utilizing the straight lines to fit the relationship between the estimated machining error and the normal cutting depth. For the thin-walled milling process, Guiassa and Mayer [18] proposed the process-intermittent gauging to compensate the finish milling process utilizing the machining errors and the cutting depth of semi-finish milling process. Similarly, the “zero-error” strategy is proposed to calculate the command nominal cutting depth for finish milling process. Although it has been proved that the “zero-error” strategy is more effective than the mirror approach, the efficiency can be improved by using the higher order curves instead of straight lines or increasing the number of the iteration times that the strategy is been applied. The applications of the above compensation methods are only limited in the specific objects, few research of the blade compensation process was proposed. The aero-engine blades, as one of the representative thin-walled parts, are different from the normal parts [19]. The low stiffness, complex shape and quick change of curvature complicate the blade machining process, resulting in two problems to be solved. The first one is how to establish the detail mathematical model for blade machining, i.e. how to machine the different region of the blade. Huang et al. [20] applied OMM to measure and reconstruct the machined surface in 5-axis flank milling for thin-walled parts. Combined with the iterative compensation method, the tool path is modified to machine the surface again and decease the machine error, which was demonstrated in impeller machining. However, this research mainly focused on the flank milling tool path generation of the free-form surface which is only applied in the rough milling of blade machining process. Altintas et al. [21] introduced the virtual compensation process to construct the blade deflection for ball end milling operation. Nevertheless, only the mirror compensation method was considered and the geometric modelling of the blade was ignored, leading to the second problem. The second problem is how to design the proper geometric model of the blade in compensation algorithm to obtain the exact dimension of the machined blade. For blade machining process, the detail construction methods of blade modelling should be considered. Dong et al. [22] applied iterative mirror compensation method to obtain the casting blade considering the shrink effect. The mean line (ML) curve and the thickness of airfoil were used to adjust the shape of die for high precision. Considering the design intent in precision forging of blade, Zhang et al. [23] utilized the free-form deformation to modify the ML of the original blade and optimize the thickness distribution. The reconstructed geometric model of forging blade was obtained and
satisfied the measuring data. It could be commented that these blade modelling methods were based on the ML and thickness distribution using envelop method. However, the concave and convex surfaces are bending in different directions, leading to different error distributions. The geometric modelling method of ML and thickness distribution cannot satisfy the compensation method. Moreover, as the most important regions of blade, the leading edge (LE) and trailing edge (TE) curves influence the inlet and outlet angles. The construction method of these areas is significant. However, the research of machining leading and trailing edges precisely is still lacking. Therefore, how to reconstruct the LE and TE curves is necessary. Aiming to decreasing the machining errors caused by the deformation of the blade, the compensation mathematical model and geometric modelling for the blade machining are proposed in this paper. Based on the OMM data, the compensation model is applied to calculate the compensation data which will be transmitted to reconstruct the new geometric model after the first machining process. The new tool path will then be generated from this model for the next blade machining process. The process will be applied iteratively, and once the machining precisions satisfy the design tolerance, the current geometric model of process for this blade could be considered as the standard model for the current machining condition. In detail geometric modelling, the concave and convex surfaces are reconstructed by the secant compensation method. The constraint of design tangent is proposed to reconstruct the LE and TE for compensated blade model. At last, the steel block and titanium blades are implemented to verify the compensation mathematical model and compensation algorithm for blade process. 2. Geometric modelling of compensated blade based on the design intent This section is focusing on constructing a certain error compensation model to redesign the blade for generating the new tool path which will be utilized to control the machining error. It has been agreed that the adjustment of NC code, as a direct modification method, cannot satisfy the complex shape change of the blade since the normal of cutter contact point, cutter axis and cutter location will make the compensation process more complex. Based on the design intent, the redesign of blade model is more convenient and has been utilized widely [21,22]. According to the expected contents, the framework is shown in Fig. 1. The compensation process is applied to reconstruct the CAD model of the blade based on the machining results to generate the CAM process iteratively until the machining results satisfying the design tolerance. Currently, the blade geometry design comes from the aero-engine designer. However, with the development of the design and manufacture integration process, the design process of the blade should also be taken into account for the manufacture. For the aero-engine designer, the primary request of the blade design is to satisfy the aerodynamics performance. Secondly, stiffness and strength of the blade should also be considered. To design the blade of the aero-engine (see an example shown in Fig. 2), the design step could be classified as preliminary design, throughflow design, 2D blade design and 3D blade design. The preliminary and throughflow design are utilized to confirm the average geometric performance parameters, twist and matching of the blade. 2D blade design is applied to calculate the 2D shape of the airfoil (see Fig. 3), which will be stacked to construct the 3D blade (see Fig. 2) by the bended stacking line. Therefore, the 2D blade design method is the most important step in the blade design. Geometric parameter method and spline parameterization method are the most popular methods for the 2D blade design. Geometric parameter method stacks the thickness distribution of the blade to the ML curve to construct the concave (pressure) and convex (suction) curves referring to [23]. Spline parameterization method divides the blade section curve into LE, TE, concave and convex curves with smooth connection. The LE and TE curves are expressed by arcs while 328
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Fig. 1. The compensation framework of aero-engine blade.
compensation models of thin-walled parts and fit the compensated points of concave and convex curves. 3.1. Compensation modelling of the machining errors for thin-walled parts Machining is a complex elastoplastic deformation process with the generation and transformation of cutting force [24] and heat [25]. In the meantime, the existences of different ways of clamping, variation of stiffness and instability of machine tool result in unavoidable cutting vibrations and deformation. Therefore, the machining process could be expressed as a complex function with inputs and outputs. The inputs include machine tool, cutter, fixture, workpiece, cutting parameters (cutting depth, spindle speed and feed rate). The outputs are the status of workpiece after cutting. When machining the thin-walled parts, the deformation of parts always influences the machining precision. Especially for the blade parts, the unique fixture and complex shape of the blade lead to the low stiffness at tip and edges. The corresponding deformation errors are much larger than the design tolerance. Since the shape of blade and stiffness cannot be changed, adjusting the parameters of process has become a popular way to control the machining error. Therefore, the compensation model could be utilized to control the machining error through optimizing the cutting depth. Fig. 2. Constitutions of a 3D blade.
3.1.1. Machining error model for thin-walled parts The nature of cutting process is the contest between force and stiffness of various parts of the processing system. The thin-walled parts exhibit the low rigidity and cause the errors, based on the Hooke’s law, the parts deformation caused by the cutting force could be express as
=
F K
(1)
where ε is the deformation value, F is the normal cutting force, K is the system equivalent stiffness. For a certain milling process (see Fig. 4a), the material between the design and the allowance surfaces should be removed to achieve dimensional accuracy. For the existence of machining deformation, the non-linearity between machining error and cutting depth could be calculated as
Fig. 3. Constitutions of a 2D blade.
the concave and convex curves are expressed by splines. It has been commented in Section 1 that the geometric parameter method cannot satisfy the geometric modelling of compensated blade. Therefore, the spline parameterization method is applied to reconstruct the 2D airfoil curves for compensated geometric modelling of blade. Following the description of this method, the compensated construction methods should be divided into the compensated fit processes of concave, convex curves and LE, TE arcs.
e (x ) = H
x + (x ) + s (x ) + …
(2)
where H is design allowance, x is nominal cutting depth, e is machining error, s is inelastic error and so on, as shown in Fig. 4(a). In thin-walled parts machining, since the deformation value ε is the key component of the comprehensive machining error, it has been considered to represent the machining error [18] in the following calculation. Fig. 4 shows the relationship among parameters of general, 0th and nth compensation processes. The compensation modelling and solving are shown in the following sections.
3. Geometric compensation modelling of concave, convex curves When machining the concave and convex surfaces, for the special fixture and geometrical feature, the machining deformation satisfies the elastic deformation. Therefore, the geometric compensation modelling of concave and convex curves could be employed to construct the 329
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Fig. 4. Parameters of (a) general, (b) 0th and (c) nth compensation processes.
3.1.2. General error compensation model for thin-walled parts For thin-walled parts processing, the design allowance H is set to constant before the machining. Focusing on the cutting depth, the relation between nominal cutting depth x and real cutting depth y could be expressed as Eq. (3). (3)
y = f (x ) According to Fig. 4(a), the machining error can be expressed as
e (x ) = H
y=H
(4)
f (x )
Therefore, the compensation method of machining error is selecting a more suitable nominal cutting depth x to make the real cutting depth y equal to the design allowance H and achieve the purpose of zero error. Iteration method can be introduced to solve the nonlinear equation as a successive approximation approach. The iterative error compensation equation, which is the general error compensation model, is established to calculate the nominal cutting depth of the next cutting process as:
x0 = H xk + 1 = xk +
k+1
ek (k = 0,1,
)
Fig. 5. Convergence process of mirror method.
(5)
where ρ is the compensation coefficient, k is the times of compensation process. It should be noted that the compensation process is required to be processed among multiple parts until the compensation value is stable. The case that k = 0 corresponding to the condition that the part is machined without compensation.
red curve represents the cutting process f. The black dash lines are marked as nominal and real cutting depth. The allowance value of real cutting depth is described as green line to show the final cutting position. The dash dot line defines the ideal cutting depth D. The calculating direction of the nominal cutting depth is presented as blue line. For the inherent limitation of the mirror method, it could be observed that the convergence speed is slow. The compensated amount is too small to counteract the error. The limitation can be proved by the ideal cutting depth D and ideal error E. It can be observed from Eq. (6) that when k = 0, the process has not been compensated, the nominal cutting depth is x 0 = H . If the current cutting process is rigid, then the deformation will not happen, i.e. D0 = H y = x . The ideal cutting depth D and error E are . However, the E0 = 0 variations of cutting force and stiffness will change the real cutting y = f (x 0 ) < D0 state. The result is described as 0 . e0 = H y0 > E0 During the (n+1)th compensated process, the current nominal n cutting depth could be calculated as x n+ 1 = H + i = 0 ei by using the iteration. If the current deformation relationship between the real and nominal cutting depth is the same as the previous one, then the ideal cutting depth D could be expressed as:
3.1.3. Solutions of general error compensation model Based on the error model of the nonlinear cutting process for thinwalled parts, a general error compensation model could be established. However, such general model cannot be applied in a real milling process without the detail calculation of cutting depth. The purpose of error compensation is to reduce the machining error to fit the accuracy range. For the cutting error model (Eq. (4)), error compensation is developed to calculate the x when e = 0, which is the zero-point problem of e(x). Two solutions with different convergence speeds are provided to solve the general model in the following. 3.1.3.1. Mirror method. Mirror method has been widely employed in calculating the compensation value for both error prediction and measurement compensation processes. Such method adds the error caused by previous machining process to current nominal depth of cut to improve the manufacturing accuracy, as described in the following formula:
x0 = H xk + 1 = xk + ek (k = 0,1,
)
Dn+ 1 = xn + 1
(6)
By updating the cutting depth through the mirror method, some machining errors could be reduced. Especially when sufficient iterative calculating and processing have been conducted, the machining error can be controlled in the desired area. However, the lower rate of convergence of the mirror method should be noticed. The convergence process of mirror method is provided in Fig. 5. The
yn xn
(7)
Eq. (7) could also be rewritten as
(
Dn+ 1 = H + Therefore, 330
n i=0
the
ei
)
H H+
ideal
en n 1 e i=0 i cutting
(8) depth
D
and
error
E
are
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Y. Hou, et al. en
Dn + 1 = H
H+ n i = 0 ei n 1 i = 0 ei
en
En + 1 =
H+
n i = 0 ei n 1 i = 0 ei
. Thus, even for the ideal situation that the
stiffness is not weakened in the current cutting process, the ideal cutting error calculated by the mirror method is not 0. Furthermore, during the real process the stiffness would be decreased, and hence the real cutting depth is smaller than the ideal cutting depth and the real machining errors are larger than the ideal error, which is described as
yn + 1 < Dn + 1 . en + 1 > En + 1 Therefore, based on the discussion above, it could be commented that the small compensation amount leads to the slow rate of convergence and low cutting efficiency due to the inherent of the mirror method. Thus, a more efficiency compensation method is required. Fig. 6. Convergence process of secant method.
3.1.3.2. Secant method. Based on the mirror method with the smaller real cutting depth, the secant method is established to obtain a larger cutting depth. The nominal cutting depth of the next process is expressed as the summation between the nominal cutting depth of the previous process and the compensation amount:
x0 = H xk + 1 = xk + z k + 1 (k = 0,1,
n
k+1
x0 = H xk + 1 = xk +
zi
(12)
i=1
It can be observed from Eq. (9) that the uncompensated process can be achieved when k = 0 (see Fig. 4b), where the nominal cutting depth is x0 and machining error is e0, thus the springback coefficient could be calculated as: 0
=
e0 x0
zi
(16)
xk ek (k = 0,1, yk
)
(13)
Considering that the current stiffness is fixed and utilizing the springback coefficient, the nominal cutting depth of the next compensated process, x1, could be calculated under the control of the ideal error, zero. The relationship between the nominal cutting depth and increment of next compensated process is expressed as
(x 0 + z1 )
0
(14)
= z1
(17)
With the above analyses, the secant compensation model is applied to reconstruct the concave and convex curves due to its high efficiency and precision. However, the secant compensation model is to solve the relationship between the cutting depth and error while the geometric modelling is to fit the point sets of contours and construct a series of smooth curves. The adjustment of compensation algorithm is applied to solve the calculation of the point sets. Based on the general error compensation model, the allowance curve, design curve, the (k-1)th machining error curve and the kth compensated curve are presented as shown in Fig. 7. Then the
k
= ek +
i=1
3.2. Geometric compensation calculation of concave and convex curves
In Eq. (11), ε is the deformation value presented in Eq. (1), and could be expressed as: k
n
The convergence process of the secant method is demonstrated in Fig. 6. The error iteration sequence of this method is linearly convergent and hence the rate of convergence of the secant method is faster than the mirror method.
(11)
x
= zn+1 +
Thus, the relations that z n + 1 = H e en , and x n+ 1 = xn + zn + 1 are n obtained. Combined with the general compensation model Eq. (5), the above equation is written as
Demonstrating that the compensation amount is varied by adjusting ρ. Thus, the quotient of springback and nominal cutting depth, the springback coefficient, is introduced to this topic and presented in the following:
=
n
xn
(10)
ek
(15)
xn
(x n + z n + 1 )
where zk+1 represents the increment between the nominal cutting depths of the (k+1)th and kth compensated processes. Unlike the mirror method, the compensation amount z of the secant method is adjusted dynamically according to the real machining error. According to the general compensation model Eq. (5), it can be written as
z k+1 =
n z i=1 i
en +
Considering a constant stiffness in the machining process. Setting the ideal error En+1 = 0. The increment zn+1 of the (n+1)th compensated process is expressed as
(9)
)
=
Therefore, the formulas that z1 = H e e0 , and x1 = x 0 + z1 could be 0 obtained. During the nth compensated process (see Fig. 4c), the nominal cutting depth is xn and the machining error is en, hence the springback coefficient could be calculated as: x0
Fig. 7. Compensation process of compensated points. 331
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parameters of Eq. (17) could be calculated from this figure. Considering that Pd is the point of the design curve, Pc,k is the corresponding compensated point of the kth compensated process and nd is the material removal direction, the calculation of the error compensation points is obtained combined with the iteration equation of secant method Eq. (17). The ith compensated points of the concave or convex curves could be expressed as
Pc , i, k = Pd, i + nd, i (xk, i
Hi ), (i = 0,1,
, m)
(18)
where m is the total number of the measurement points in concave or convex curves. After obtaining the compensated points, the fitting process of these points is followed to generate the relative curve. However, due to the vibration, residual height, measuring plan and other reasons, generally the measured points are not in a smooth curve, and thus resulting in the unsmooth compensated points. Therefore, the interpolation of unsmooth points could lead to an unsmooth curve which is not appropriate in manufacture, and thus the approximation of the fitted curve is required. To this end, the least square is employed to conduct the approximation of the fitted curve. For the approximation of points, the NURBS fitting method is introduced. With the certain knot vector and curve degree, the fitted curve is reconstructed easily. Following the above process of calculating and fitting compensated points, the compensated concave and convex curves could then be reconstructed.
Fig. 8. LE model of design type.
Ld (s ) = LEPd + s tn
(19)
Step 3. Calculating the compensated LE curve (CAd,k) for the compensated concave and convex curves.
4. Geometric compensation modelling of LE and TE The shapes of LE and TE within the blade can be designed as circular, elliptical or curved referring to [26]. For processing convenience, the shapes of LE and TE are designed as circle. Such simplification sacrifices the aerodynamic performance to increase the precision of manufacture since the elliptical and curved edges complicate the manufacturing process. However, the circular edges are still challenging for machining and the large error is existed in the machined blade. To decrease the machining error of the edges, the compensation method is introduced. Unlike the compensation method of concave and convex curves, the detail deformations principles of edges are complicated. The secant compensation model cannot be applied directly. Therefore, from the view of geometry modelling, the new LE and TE are reconstructed to connect the compensated concave and convex curves smoothly, which is the reflection of the design intent of the blade. For the same shapes of LE and TE, the following steps take LE as an example to describe the detail.
4.2. Arc fitting method based on ML For the special geometry constitution, the smooth connections of LE and concave, convex curves are required. Therefore, the compensated curves should be connected smoothly. Moreover, the tangent Ld of LE is the other constraint of calculating the LE. Thus, the constraints of three curves (tangent, concave and convex curves) are the keys in calculation of the compensated LE. Generally, the research of reconstructing the arc in the constraints of the three curves is to calculate an inscribed circle, which is tangent to L, CV and CC as shown in Fig. 9. To this end, the inscribed circle is expressed as follow
4.1. LE modelling based on tangent constraint The shape of LE influences the region of attack angle and flow separation, based on the designed LE, the new LE is developed to construct the compensation model with constant chord length of the blade. In the following description, the new LE is treated as the compensated LE. In the compensation model of the blade, it is assumed that the concave and convex curves satisfy the elastic deformation principle for the thin-walled feature, the nearby area of the compensated LE is defined to be rigid. Therefore, when constructing the compensation model, the position of the compensated leading edge point (LEP) is the same as the position of designed LEP (see Fig. 8). As the result, the chord length remains the same. Based on these features, the tangent constraint of LEP is introduced to constraint the LE arc which will be reconstructed in the following steps. Step 1. Calculating the designed LEP (LEPd). Based on the designed ML (MLd) of airfoil, the intersection point, LEPd, is calculated by the extended ML and designed LE curves. Step 2. Calculating the tangent Ld of LEP. The direction of tangent tn is found to calculate the designed tangent Ld of LE, which is
Fig. 9. Inscribed arc of concave, convex curves and tangent line. 332
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fCV (PCV ) (PCV
O) = 0
fCC (PCC ) (PCC f L (PL ) (PL |PCV |PCC
O) = 0 O) = 0
O| = |PCC O| = |PL
O| O|
(20)
where fCV , fCC and f L are the first derivatives of CV, CC and L, respectively, PCV , PCC and PL are the corresponding points in CV, CC and L, respectively, and O is the coordinate of the circle center. By solving the above equations, the inscribed circle with center O and radius R = |PL O| is obtained. However, the function of the freeform curves is complicated to applied in the equations solving problem. Therefore, the ML (ML) of CV and CC is introduced to simplify the problem referring to [27]. Considering that the points of ML are the centres of inscribed circles of CV and CC, the radius function R(s) and the centre of the circle O(s) could be constructed with all points of ML, where s is the arc parameter of ML. The Eq. (20) is simplified as
f L (PL ) (PL |PL
O (s )) = 0
O (s )| = R (s )
(21)
Fig. 11. On-machine measurement setup.
Moreover, the distance function d(s) which is the distances between points and tangent line L with s shows the relation among PL , O (s ) and L. Then the following expression could be obtained to replace the Eq. (21).
t (s ) = R (s )
d (s )
compare different algorithms, three cutting depths which are separately calculated by the original model, mirror compensation and secant compensation methods, were applied in the finish milling. The machining errors of different cutting depths calculated by three compensation strategies are shown in Fig. 12. It could be observed that the compensation results based on both mirror and secant methods are better than the original model. The observations could be concluded as follows:
(22)
When the t(s) is zero, the relative circle CA with parameter s is tangent to L, CV and CC, which satisfies the requirement of reconstructing the compensated LE. The complicated equations solving problem of Eq. (20) with five variables is simplified as a single-variable equation which is convenient to be solved. Back to Step 3 in Section 4.1, the algorithm of inscribed circle mentioned in Eq. (22) is utilized to calculate the compensated LE (CAd,k), which is tangent to the concave, convex curves and tangent lines.
1) comparing to the uncompensated method, the maximum and the average errors of mirror method were separately reduced to 57.7% and 61.4%. 2) comparing to the uncompensated method, the maximum and the average errors of secant method were separately reduced to 31.7% and 23.8%. 3) comparing to the mirror method, the maximum and average errors of secant method were reduced to 54.9% and 38.8%, respectively.
5. Verification and discussion To validate the proposed method, the mathematical and geometric model for error compensation of blade machining were established. Moreover, the mathematical model was applied in geometry modelling. Two trials are presented in the following as the example to conduct the verification.
5.2. Verification of the compensated geometric model for blade To verify the geometric model of error compensation for aero-engine blade milling process, the uncompensated and compensated experiments were conducted to demonstrate the error diminution of different regions of blade. Two blades were machined by a 5-aixs machine
5.1. Verification of the compensation model for thin-walled part To verify the secant compensation model proposed in Section 3.1, the rectangular block in steel with cantilevered fixture was designed (see Fig. 10). A YHVT850Z 4-axis machine tool was employed as shown in Fig. 11. The cutting condition is: spindle speed = 3000 rpm, feed rate = 300 mm/min, axis depth = 11 mm, radial depth = 0.25 mm, tool radius = 6 mm, tool type = end mill, cutting type = flank mill. To
Fig. 10. Dimensions (mm) and sampled points of machining face.
Fig. 12. Error distributions of different cutting depths. 333
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Moreover, the machining errors of LE and TE regions in the uncompensated and compensated models are shown in Fig. 16. The average errors of edge regions fitted by tangent constraint of compensated machining process are reduced to 29.7% of uncompensated machining process. It has been proved that the machining precision is improved by applying the compensation algorithm in the above experiments. However, due to the machining error measurement and the compensation calculation, the total time in the compensated process is longer than the uncompensated process. Nevertheless, in the industry field, the compensation framework proposed in this paper can be applied to the volume production, where only the first few parts are taken as the specimens and the error measurements are needed. After the compensation value is stable, the modified tool path can be utilised in the remaining parts without any further compensation and the machining time could be saved from the compensated path. Thus, the calculation time of the compensation can be ignored for the volume production. Therefore, three conclusions could be concluded from these experiments: 1) The experiment of thin-walled parts validates that the secant compensation method is more effective than the current mirror compensation method. The average errors of secant compensation method are reduced to 38.8% of the mirror compensation method. 2) The application of secant compensation method to the construction procedures of the concave and convex curves has been proved to improve the overcut and undercut conditions. The average errors of concave and convex surfaces are reduced to 25.1% and 14.3%, respectively. 3) The fitting method of the LE and TE considering the constraint of design tangent was verified as the proper way to construct the most suitable arc shape comparing to the uncompensated model. The average errors of edge regions of design type are reduced to 29.7% of uncompensated machining process.
Fig. 13. Measuring process of machined blade.
tool, and the dimensions were measured by a CMM. The blade model is shown in Fig. 2 and the measuring process is shown in Fig. 13.The secant compensation method was applied to fit the new curves of concave and convex regions while the new leading and trailing edges were reconstructed based on the tangent constraint for compensation milling process. The actual machining errors of the uncompensated and the compensated concave and convex surfaces are shown in Fig. 14 as the cloud figures. Without the compensation, the machined concave surface is overcut while the convex surface is undercut (see Fig. 14a and c, respectively). Applying the compensated milling process, the overcut of concave surface and undercut of convex surface are corrected shown in Fig. 14(b) and (d). The average errors of concave surface are reduced to 25.1% while the average errors of convex surface are reduced to 14.3%. The measured error distributions and the fitted curves in LE and TE regions are shown in Fig. 15. The red measured points represent the actual machining error distributions of the uncompensated blade which is deviated from the designed section curves (green points). After the compensation process, the blue measured points show that the compensated curves are much closer to the design curves, indicating that the overcut and undercut are compensated.
6. Conclusion Due to the elastic deformation caused by the low stiffness of the aero-engine blade, the machining error of the blade milling process has become one of the most severe factors which limit the aerodynamic performance. In this paper, a novel geometric modelling of the blade has been firstly applied into the compensation procedure to decrease the machining errors of the blade milling process. The key contributions could be concluded as follow:
Fig. 14. The error distributions of concave surfaces with (a) uncompensated and (b) compensated methods, and the error distributions of convex surfaces with (c) uncompensated and (d) compensated methods. 334
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Fig. 15. Measured points of specific areas after compensating.
this framework, the undercut and overcut of concave and convex surface could be compensated, and the leading and trailing edges were optimized to satisfy the design shape and decrease the machining errors. Opening the avenue to improve the processing quality and the efficiency of the blade, and greatly reduce the rejection rate. Acknowledgements This study is co-supported by the Major National Science and Technology Projects (No. 2015ZX04001202), the Fundamental Research Funds for the Central Universities (No. 3102018jcc004) and the 111 project of China (No. B13044). References [1] Luo M, Han C, Hafeez HM. Four-axis trochoidal toolpath planning for rough milling of aero-engine blisks. Chin J Aeronaut 2018. https://doi.org/10.1016/j.cja.2018. 09.001. [2] Yan Q, Luo M, Tang K. Multi-axis variable depth-of-cut machining of thin-walled workpieces based on the workpiece deflection constraint. Comput Des 2018;100:14–29. [3] Ratchev S, Liu S, Huang W, et al. An advanced FEA based force induced error compensation strategy in milling. Int J Mach Tools Manuf 2006;46(5):542–51. [4] Rao VS, Rao PVM. Tool deflection compensation in peripheral milling of curved geometries. Int J Mach Tools Manuf 2006;46(15):2036–43. [5] Aguado S, Santolaria J, Samper D, et al. Improving a real milling machine accuracy through an indirect measurement of its geometric errors. J Manuf Syst 2016;40:26–36. [6] Ratchev S, Liu S, Huang W, Becker AA. A flexible force model for end milling of lowrigidity parts. J Mater Process Technol 2004;153–154:134–8. [7] Chen W, Xue J, Tang D, et al. Deformation prediction and error compensation in multilayer milling processes for thin-walled parts. Int J Mach Tools Manuf 2009;49(11):859–64. [8] Wang MH, Sun Y. Error prediction and compensation based on interference-free tool paths in blade milling. Int J Adv Manuf Technol 2014;71(5-8):1309–18. [9] Xie K, Camelio JA, Izquierdo LE. Part-by-part dimensional error compensation in compliant sheet metal assembly processes. J Manuf Syst 2012;31(2):152–61. [10] Fiorentino A, Feriti GC, Giardini C, et al. Part precision improvement in incremental sheet forming of not axisymmetric parts using an artificial cognitive system. J Manuf Syst 2015;35:215–22. [11] Ratchev S, Liu S, Huang W, Becker AA. Milling error prediction and compensation in machining of low-rigidity parts. Int J Mach Tools Manuf 2004;44:1629–41. [12] Cho MW, Seo TI, Kwon HD. Integrated error compensation method using OMM system for profile milling operation. J Mater Process Technol 2003;136(136):88–99. [13] Myeong-Woo CHO, Gun-Hee KIM, Tae-Il SEO, et al. Integrated machining error compensation method using OMM data and modified PNN algorithm. Int J Mach
Fig. 16. Average errors of 6 regions using design type.
1) The concave and convex surfaces of the blade have been constructed by the fitted points from the secant compensation method calculation. This is the first time that the secant method has been utilized into the compensation procedure of the blade body region. Such method has been validated through the thin-walled part and the blade milling processes, the secant compensation method proposed in this paper was proved to be a better solution to decrease the machining error as the average errors were reduced to 38.8% of mirror method. 2) The compensation procedures of the leading and the trailing edges for the blade milling process have been first conducted in this paper. The construction methods have been applied to fit the new shape of the edge regions. The validation demonstrated that the machined edge regions under the constraint of design tangent matches the design tolerance. The average errors of edge regions were reduced to 29.7% of uncompensated machining process. The proposed geometric modelling framework of the blade firstly solves the machining error problem for the whole blade shape. Through 335
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