Prediction of contact characteristics of abrasive belt compliant grinding for aircraft blades

Due to the characteristics of thin-walled curved surface, wall thickness variations and processing cantilever �xtures, the mechanical state of the different contact positions of aircraft engine blades varies signi�cantly during the grinding process. The different contact interactions between contact wheel and blade result in changes of material removal e�ciency and surface quality. To achieve contact state control during blade grinding process, a novel �exible abrasive belt grinding device was designed and developed considering the compliance of rubber contact wheel. The signi�cant effect of compliance parameters on grinding contact state was veri�ed through simulation. The grinding contact pressure distribution and normal contact force at different positions in the blade width and length directions were studied, and a prediction model for the maximum contact pressure and normal contact force was established based on BP neural networks. The results showed that with the increase in contact wheel compliance, the effective contact range increased, the pressure distribution gradually became uniform, and showed a double-elliptical distribution. The maximum contact pressure was signi�cantly reduced, with a reduction of up to 46.00%. As the grinding contact position moved towards the weak rigidity area of the blade, the contact pressure distribution became more uniform. And the normal contact force was signi�cantly reduced, with a maximum reduction of 68.49%. The mean average percentage error (MAPE) of the prediction model was small, verifying the effectiveness of the model. The research results of this manuscript laid a foundation for achieving consistent control of blade grinding material removal rate through contact wheel compliance adjustment.

of the blade, the contact pressure distribution became more uniform.And the normal contact force was signi cantly reduced, with a maximum reduction of 68.49%.The mean average percentage error (MAPE) of the prediction model was small, verifying the effectiveness of the model.The research results of this manuscript laid a foundation for achieving consistent control of blade grinding material removal rate through contact wheel compliance adjustment.

Speci c remarks 1) What is your main contribution to the eld?
The main contribution of the manuscript is that a novel belt compliant device is proposed and used in blade grinding process.The in uence of grinding wheel compliance and different grinding positions on grinding contact characteristics is studied in detail.And the research will help to develop suitable belt grinding process control methods for blades, and realize the prediction of grinding contact pressure and normal contact force.
2) What is novel?In theory, in experimental techniques, or a combination of both?
The novel lies in a combination of both theory and experimental techniques.The novel grinding device is designed considering the effect of hyperelastic rubber wheel deformation on grinding contact mechanism.And the different contact characteristics at different grinding positions of thin-walled blades are analyzed in order to establish the contact pressure and force model based on the simulation and experiment results.
3) Does your paper have industrial applications?If yes, who are the likely user?
Yes, this study has an extensive industrial application.The paper establish an important theoretical foundation for e cient, precise, and intelligent control of blade grinding processes.The enterprises involving the eld of aircraft engine manufacturing, free-form surface workpiece manufacturing, especially the blade manufacturing, are the potential users.

Introduction
Blades are one of the major critical components of aircraft engines, whose pro le accuracy and surface consistency have a signi cant impact on the aerodynamic e ciency and fatigue life of the engine [1,2].Due to their thin-walled and weakly rigid structure, as well as the complex curvature and di cult-tomachine materials, the high-e ciency and precision machining of aircraft blades is facing great challenges.
Currently, precision milling is one of the main technologies for obtaining blade pro les.However, milling processes cannot guarantee the actual pro le accuracy and surface quality due to the characteristics of blade thickness, curvature variation, etc.Therefore, precision grinding is often required to remove residual machining allowances of milling process, ensure the pro le dimensional accuracy and improve the surface integrity.
Compared with rigid grinding process, belt grinding is an elastic contact grinding process that uses abrasive grains uniformly planted in an elastic substrate to form a grinding tool.Under the guidance of a hyperelastic rubber contact wheel, abrasive belt contacts the blade surface and makes relative movement to remove material from workpiece.It is an important means for the precision grinding process of weakly rigid curved blades [3][4][5].
However, due to the weak rigidity of blade and the hyperelastic effect of rubber contact wheel, different regions of blades exhibit complex and variable contact states during abrasive belt grinding.And this makes it di cult to control grinding pressure effectively and affects material removal e ciency and processing quality [6].Therefore, the current blade manufacturing still mostly adopts the ine cient "processing-inspection" method to ensure processing quality [7].
To address these challenges, scholars have focused on the research of robotic abrasive belt grinding force/position control [8][9][10][11][12], parameter optimization [13][14][15], compensation control [16,17], etc. Xu et al. [18,19] proposed a robotic belt grinding force control method that combines force/position compound control with PI/PD control, and analyzed the precision, stability and reliability of the system.Zhang [20] et al. explored a constant force control algorithm based on the pressure release model and model-based reinforcement learning, which is used in the different stages like impact and processing stages.And the optimal parameters were obtained to improve the machining quality.Combining neural network and genetic algorithm, Mohammad et al. [21] proposed a grinding and polishing optimization method for material removal and surface quality improvement.Chen et al. [22] proposed a polishing end effector for intelligent robots and developed a gravity compensation force controller, with which compensation control of polishing force was achieved to attain preferable surface quality.
In the actual manufacturing process, the in uence of hyperelastic effect of rubber contact wheel on contact characteristics has not been fully explained.Additionally, the differences of processing state at different grinding points produce a great impact on the grinding contact and machining quality.
Therefore, based on the proposed novel grinding device, the in uence of compliant contact wheel on grinding pressure distribution and normal contact force was indicated in this manuscript.And the contact mechanism affected by different grinding characteristics of blade at different grinding positions was revealed.Finally, a prediction model of contact pressure was established.The research will help to develop suitable belt grinding process control methods for blades, and realize the prediction of grinding contact pressure and force.Simultaneously, it establish an important theoretical foundation for e cient, precise, and intelligent control of blade grinding processes.

Belt grinding characteristics of blade
Aircraft engine blades consist of parts such as concave part, convex part, leading and trailing edges.And the leading and trailing edges are usually designed as high-order theoretical free-form surfaces with very small transition sizes to the concave and convex parts.The minimum thickness of some compressor blades is even less than 0.1 mm.Therefore, different ranges of processing deformation are easily accompanied during blade grinding process.The rubber material contact wheel and the blade workpiece are in exible contact, which is different from the planar contact problem in rigid processing.Under the grinding trajectory control with different technological parameters, a large elastic contact deformation of contact wheel is produced.Meanwhile, the surface structure characteristics (e.g.curvature radius, normal wall thickness) vary at different machining positions, so that a complex and changeable machining state is presented during grinding process.And differences in material removal effect and processing surface quality are caused ultimately.Additionally, blade grinding is usually carried out by cantilever clamping method, which aggravates the complex changes in belt grinding process.

grinding device with compliance adjustable contact wheel
In order to adapt to the geometric and rigid state changes, a exible grinding device with compliance adjustable contact wheel was designed as shown in Fig. 1(a) on the basis of basic grinding theory.The compliance of contact wheel is adjusted depending on the support state of hyperelastic rubber material wheel [23,24].Through the adjustment of compliance of contact wheel, the contact state between contact wheel and blade can be adaptive controlled.
The designed grinding device consists of rod, tool handle, bearing seat, ange, ribs, guide part, contact wheel, abrasive belt and so on.The core of this grinding device lies in the special structure inside the rubber contact wheel, as shown in Fig. 1(b).The contact wheel is uniformly arranged with arc-shaped rib hole channels inside.The position change of ribs in the channel plays a different supporting role on the compliance of contact wheel.The control of contact state during the grinding process is realized additionally.The movement of the ribs in the channels of contact wheel is driven by the servo-driven bearing seat under the constraints of guide part.The servo-driven linear displacement variable λ (mm) is de ned as 0 mm when all internal ribs are fully inserted into channels, which is used for characterizing the contact wheel compliance.

Constitutive model of rubber contact wheel
The rubber contact wheel is one of the key components for abrasive belt grinding.The belt is guided to realize multi-axis machining of curved components.Different from the mechanical properties of metal materials, rubber material is a typical hyperelastic material which has the properties of large elastic deformation, incompressibility and viscoelasticity.Common hyperelastic constitutive models include Mooney-Rivlin, Neo-Hookean, Ogden, Yeoh, etc., which have different application ranges as shown in Table 1.Since the grinding process described in this article belongs to precision machining, with small contact wheel shear deformation of less than 75%, the commonly used Mooney-Rivlin [25] is selected as the constitutive model of contact wheel.

Mooney-Rivlin
suitable for rubber with small and medium deformation, but cannot accurately simulate rubber with carbon black Yeoh suitable for the large deformation behavior of rubber with carbon black, but can't accurately describe the situation of small deformation Ogden Suitable for large deformation environment, still applicable when the strain reaches 700%

Neo-Hookean
The parameters required by the model are simple, and suitable for tensile environment Mooney Rivlin model is: where W stands for the strain energy density of rubber material; I 1 and I 2 represent the invariants of deformation tensors; C 1 and C 2 denote the material elastic coe cients related to rubber hardness.Since the elastic effect of rubber material is much greater than the elastic property of abrasive belt substrate, the in uence of abrasive belt microscopic state on the contact state is not considered in the subsequent research.

Contact simulation analysis
To reveal the effectiveness of contact wheel compliance in adjusting the machining process, a simulation contact model between the wheel and the blade is established as shown in Fig. 2. The origin of coordinate system O 1 -XYZ is located at the center of the convex surface of blade xed end, where X represents the direction along the width of the blade, Y represents the direction along the length, and Z represents the direction perpendicular to the blade surface.O 2 is the central point of contact wheel.In the simulation model, the aluminum alloy blade is chosen as the object to be machined.The rib is made of 45 steel material, and the contact wheel is made of rubber material with a hardness of 75Hs/A.Other physical property parameters are detailed in Table 2. Due to the large deformation of the rubber material wheel during the grinding contact process with aluminum alloy blade workpiece, large de ection is initiated during the simulation.The blade surface in contact pair is set as the target surface, while the outer circular surface of contact wheel is set as the contact surface.High-precision hexahedral elements are adopted for the aluminum alloy blade, and the meshes between contact wheel and rebars are segmented.Tetrahedral SOLID92 elements are used to re ne the mesh near the contact surface.
In the contact process simulation, the linear length L of the blade is 80mm, the linear width B 1 is 40 mm, the curvature radius R 1 is 40 mm, the contact wheel radius R 2 is 40mm, and the contact wheel width B 2 is 15 mm.A cantilever single-end xed constraint is applied to blade.The contact wheel speed r is 2000 r/min.The theoretical cutting depth a p is 0.1 mm.The projection point of the axial center of the contact wheel in the XOY coordinate system is de ned as the contact point (x, y).During grinding penetration, the radial direction of contact wheel is always perpendicular to the blade surface.Three points O 1 , A and O 2 are collinear.

Simulation results
The contact state changes between the contact wheel and the workpiece surface during penetration at the contact point (x = 0 mm, y = 7.5 mm) are simulated and analyzed, respectively, under λ = 0, 1, 2 and 5 mm.And the stress distribution of the contact area is derived as shown in Fig. 3.
The preliminary simulation results demonstrate that the wheel compliance λ has a signi cant impact on the contact pressure distribution between contact wheel and blade workpiece.And the pressure distribution is approximately elliptical.With the increase of λ, the cavity volume inside the contact wheel channel increases, and the compliance of rubber material is enhanced, which makes the contact area larger.However, the pressure distribution of the contact area tends to be more uniform, and the maximum contact pressure decreases signi cantly.Accordingly, the pressure distribution gradually changes to a double elliptical peak distribution.When λ = 0 and 1 mm, the contact pressure distribution is unimodal, and the maximum contact pressures are 0.191 MPa and 0.125 MPa, respectively.When λ = 2 and 5 mm, the maximum contact pressures are reduced to 0.110 MPa and 0.103 MPa, respectively.Compared to the case of λ = 0 mm, the maximum contact pressure drops by about 0.080 MPa when λ = 5 mm, with a decrease of 46%.By equivalent conversion of contact pressure distribution, the normal contact force is obtained as shown in Fig. 4. With the increase of wheel compliance λ, the normal contact force decreases from 4.611 N to 3.191 N. It can be clearly seen that wheel compliance parameter has a great in uence on maximum contact pressure, pressure distribution and distribution area, which can provide a positive effect for grinding process control compensation.

Contact characteristic at different positions of blade
Due to the in uence of different factors such as wall thickness and curvature at different positions, the contact state consistency of the thin-walled blade clamped by cantilever is poor during the grinding process.And it is di cult to attain stable grinding quality.As the grinding position changes with the X direction, the contact state is affected by the twisting deformation of blade.As the contact position changes with the Y direction, the contact state is affected by the bending deformation of blade.
To clarify the variation trends of grinding contact state at different blade positions, the contact pressure distributions at blade contact points A 1 (0 mm, 7.5 mm), A 2 (10 mm, 7.5 mm) and A 3 (20 mm, 7.5 mm) are initially investigated under λ = 0 mm as shown in Fig. 5.Meanwhile, the equivalent normal contact force under the cross in uence of compliance parameter λ and different X-direction positions is obtained as shown in Fig. 6.
The simulation results show that as the contact position deviates in X direction to approach the leading and trailing edge of blade, the wall thickness gradually decreases and the machining rigidity weakens.The contact pressure distribution gradually changes from a single peak value to a multi-peak value distribution state, and the maximum contact pressure gradually decreases from 0.191 MPa to 0.123 MPa, with a decrease of 35.6%.Nonetheless, the pressure distribution changes more uniform.As is clear from the equivalent normal contact force results in Fig. 6, when the contact position changes in X direction, the normal contact force decreases rapidly.When λ = 0 mm, the normal contact force decreases from 4.611 N to 4.113 N and nally to 1.594 N, with two decreases 10.8% and 61.2%, respectively.When λ = 5 mm, the normal contact force decreases from 3.191 N to 2.999 N and nally to 1.295 N, with two decreases of 6.02% and 56.8%, respectively.At the same position in the X direction and under different compliance parameter, there are also signi cant differences in normal contact force.
Additionally, the different contact positions in Y direction have a signi cant impact on the grinding contact state of the cantilever-clamped thin-walled workpiece.According to the above mentioned simulation method, the contact simulation of different grinding contact positions within the 7.5-72.5mmregion in Y direction is carried out to obtain the contact pressure distribution state as shown in Fig. 7 and the equivalent normal contact force at different positions as shown in Fig. 8.
As the contact position deviates in Y direction and moves away from the cantilevered end of the workpiece, the support rigidity of blade gradually weakens, and the contact pressure distribution gradually becomes atter and more uniform.Overall, the contact pressure presents a single peak elliptic area distribution, and the maximum contact pressure gradually decreases.At a contact position y = 72.5 mm away from the cantilever end, the maximum contact pressure decreases to 0.084 MPa.Compared with the maximum contact pressure of 0.191 MPa at point y = 7.5mm, the overall decrease is 56.0%.
Figure 8 shows that when λ = 0 mm, the equivalent normal contact force decreases signi cantly as the grinding contact point is away from the constrained end.At the contact position x = 0 mm, the normal contact force decreases from 4.611N to 1.453N.
In summary, the geometrical and spatial characteristics of blade vary in different grinding contact positions in X and Y directions, which affect the machining rigidity at different grinding positions and cause signi cant differences in grinding contact state.Meanwhile, the wheel compliance parameter λ plays an important role in the grinding state.Hence, to compensate for the differences in grinding process and quality caused by different grinding positions, the contact wheel compliance parameter can be adjusted, which effectively adapts to the changes of blade machining state and provide a method for obtaining consistent machining accuracy and surface quality.

Grinding experiments
To verify the reliability of grinding contact simulation, a grinding contact experiment was carried out using a JD50 3-axis CNC machining center.The workpiece used in the experiment was an aluminum alloy blade with the same size and processing parameters as those in the simulation.The normal contact force during the grinding contact process was acquired in real time using an LH-SZ-02 3-axis force sensor.The grinding experimental setup is shown in Fig. 9.
Initially, the grinding contact experiments were carried out under different contact wheel compliance λ.
The grinding contact position (0 mm, 20.5 mm) of the blade was selected, and the normal contact force was detected in real time under the conditions of under λ = 0, 1, 2 and 5 mm.The collected normal contact force data is shown in Fig. 10.And the comparison with the simulation results under identical conditions is shown in Fig. 11.
The experimental measurements were assumed to be true values.The experimental results show that at the grinding contact position (0 mm, 20.5 mm), the simulation data has high reliability under different compliance parameter conditions.The maximum error at λ = 0 mm is -0.25 N, and the maximum and minimum relative errors of simulations are 5.53% and 4.12%, respectively.The mean absolute percentage error is 4.82%, and the root mean square error is 0.177 N. The contact state during compliant abrasive belt grinding is described accurately through simulation.
Secondly, to verify the in uence of blade grinding position on the contact state, six positions A ~ F on blade pro le were selected, as shown in Fig. 12.The coordinate values were A (20 mm, 20.5 mm), B (20 mm, 46.5 mm), C (20 mm, 72.5 mm), D (0 mm, 20.5mm), E (0 mm, 46.5 mm) and F (0 mm, 72.5 mm), respectively.The grinding contact experiments were carried out under the condition of λ = 0 mm.
The experimental results are compared with the simulation results as shown in Fig. 13.It is clear that under the conditions of λ = 0 mm, x = 0 mm (contact points D, E, F) and λ = 0 mm, x = 20 mm (contact points A, B, C), the normal contact forces decrease with the increase of the distance from the constraint end.The simulation values are in good agreement with the experimental values.The maximum absolute error is -0.38 N, the maximum relative error is 14.29%, the mean absolute percentage error is 10.16%, and the root mean square error is 0.198 N.This indicates that the simulation data has high reliability and also indicates that different contact positions of the blade have a signi cant effect on its grinding contact state.

BP neural network-based prediction of blade contact characteristics
To grasp the variation trends of contact state during blade grinding and realize the prediction of grinding characteristics for targeted control of machining process, a simulation research was carried out by comprehensively considering the in uence of wheel compliance parameter λ and contact positions (x, y).Other parameters involved in the simulation process are the same as those in Section 2.2.The variation of the maximum contact pressure and normal contact force is shown in Fig. 14.
The change of contact position actually affects the normal wall thickness, cantilever distance and curvature direction, etc. of blade in the machining process, resulting in the difference of machining state.With the increase of the contact position in Y direction, away from the blade constraint end, the clamping and support effect of the blade is weakened, and the maximum contact pressure of the grinding contact position is signi cantly reduced.The normal contact force also decreases to varying degrees.When λ = 0 mm and x = 0 mm, with the increase of y, the maximum contact pressure decrease by 55.96% from 0.190 MPa to 0.084 MPa, and the normal contact force decreases by 68.49% from 4.611 N to 1.453 N.
With the increase of the contact position in the X direction, the grinding point tends to approach the thinner leading and trailing edge of blade, and the processing rigidity of the blade is weak and torsional deformation is enhanced.The maximum contact pressure and normal contact force change signi cantly.Among them, the change range of maximum contact pressure is relatively small.The highest decrease in maximum contact pressure when λ = 0 mm can reach 40.55%.In case λ = 1, 2 and 5 mm, the maximum contact pressure decreases by less than 10%.Under different compliance parameters, with the change of contact position in X direction, the normal contact force decreases signi cantly by about 60%.
In summary, with the movement of the grinding contact position in Y direction, the blade bending deformation intensi es and the contact state changes to a large extent.With the movement of the grinding contact position in X direction, the torsional deformation intensi es and the contact state also changes to varying degrees.For traditional grinding methods with xed process parameters, the difference in grinding contact state leads to great differences in the material removal effect, ultimately affecting the consistency and integrity of machined surface.Meanwhile, the compliance parameter variation of contact wheel affects the effective contact area between grinding tool and workpiece during machining process, as well as the material removal effect.It can adapt to the changes of characteristics in different blade grinding positions and dynamically control the grinding process.
Using the grinding contact results obtained in the above simulation as the dataset, a total of 180 groups of data were established as shown in Table 3.The compliance parameter λ and different contact positions (x, y) in X and Y directions were used as inputs, and the maximum contact pressure and normal contact force were used as outputs.Among them, 126 groups of sample data were randomly selected as training data sets (accounting for 70%), while 27 groups of data (accounting for 15%) were selected as validation and test data sets, respectively.BP neural network was used to predict the contact characteristic.The neural network predictions were comparatively analyzed and compared with the nite element simulation results as shown in Fig. 17.Taking the simulation data as the true value, the relative error of predicted maximum contact pressure is 14.65% at highest and 7.80% at lowest, while the relative error of predicted normal contact force is 15.56% at highest and 6.48% at lowest.The relative errors are all small, proving high reliability of the neural network-based prediction model, which can accurately predict the grinding contact state of blade at different machining positions.The ndings provide guidance for regulating the blade surface precision and quality consistency, and improving the uncertain effect of blade rigidity difference on machining quality.

Conclusions
1.In response to the problems of easy deformation and poor consistency of machining thin-walled aircraft blades, a novel abrasive belt grinding technique with adjustable compliance is proposed based on the blade structure characteristics and xture characteristics.The simulation results verify that the proposed adjustment method has a signi cant impact on the processing state.Under different compliance parameters, the normal contact force changes signi cantly, with a maximum change rate of 30.80%, which can affect the blade grinding contact state and contribute to obtain consistent processing quality.

Contact state variations in
Figures

Figure 2 Contact
Figure 2

Figure 3 Effect
Figure 3

Figure 4 Effect of λ on equivalent normal contact force Figure 5
Figure 4

Figure 7 Pressure
Figure 7

Figure 9 The
Figure 9

Figure 10 Normal
Figure 10

Figure 13 Normal
Figure 13

Figure 15 Comparison
Figure 15

Table 3
As mentioned above, the number of input layer nodes was 3, and the number of output layer nodes was 2. To improve the training e ciency of the network, the hidden layer was set as a single layer, and the initial value of its node number p was selected according to the number of input layer and output layer nodes after comparing the model accuracy and validity.The structure of the BP neural network-based prediction model for blade grinding contact state was preliminarily determined to be 3-p-2.The commonly used training functions include Levenberg-Marquardt (trainlm), Bayesian Regularization (traingbr) and Scaled Conjugate Gradient (trainscg), etc. whose training results are shown in Fig.15.It can be seen that the trainscg function failed to reach the target accuracy of 1e-04, while the trainbr function achieved the target accuracy 1e-04 after 534 training sessions.As for the trainlm function, it reached 5.646e-05 only after 82 training sessions, which met the accuracy requirements and had a fast convergence speed.Hence, the trainlm was selected as the training function for the BP neural network-based prediction model.By setting the hidden layer node number of the trainlm-based BP neural network separately at 5, 7, 9 and 10, the root-mean-square error was obtained by calculating the network model several times, and the results are detailed in Table4.The node number of the network hidden layer was determined to be 10.The topological structure of the constructed BP neural network was 3-10-2, and its learning rate was lr = 0.01.The maximum training epochs were set as 1,000, while the allowable error was set as 0.001.The BP neural network model was constructed in MATLAB, and sample data were used to train the network.The contact characteristic prediction model was established nally.As shown in Fig.16 (a), the training regression R is very close to 1.And as shown in Fig. 16 (b), the network training error is less than 0.001 at 82 steps, which achieving the training goal.The Generalization ability of neural network refers to the ability of BP neural network to predict uncertain data.Despite a powerful generalization function of BP neural network for information within the range of training samples, whether it can attain higher estimation accuracy for information outside the training sample range is the core of evaluating the training model[27].Therefore, the generalization ability of the above-mentioned neural network model was tested, and 6 groups of original data sets that do not exist in the training sample were added for veri cation of the neural network prediction model as shown in Table5.