Removal of Belt Polishing for Blade Complex Surface

To improve the polishing precision of blade complex surface, the quantitative material removal method for belt polishing of blade complex is researched in this paper. The removal rate model for belt polishing of blade complex surface based on Preston equation is firstly established. Then, by analyzing the contact process between blade surface and contact wheel, the contact force model, relative velocity model as well as contact time model are established. The material removal depth is calculated by integrating the material removal rate within the contact time between the blade and the contact wheel. Relationships of material removal depth relative to contact force, feed rate, belt speed as well as workpiece curvature are acquired by simulation analysis. Finally, the material removal depth model is testified by experiments, and measurement error is analyzed. Theoretical analysis and experimental results show that the established model in this paper is correct, and we can design reasonably machining parameters and realize quantitative material removal of complex surface by applying it.


Introduction
As one of the important finishing processes of complex free-form surfaces, polishing has a crucial impact on surface quality and has been investigated for decades.In polishing of complex surfaces, such as turbine blade, there are strict dimensional accuracy requirements on the workpiece.Traditionally, the complex blade surfaces are polished manually.However, manual polishing is not only time and labor consuming but highly depends on labor's experience and technology.Recently, computer-controlled belt polishing has been introduced into precision manufacturing for its characteristics of flexibility, high-efficiency and labor liberation that make it very suitable for manufacturing workpiece with complex free-form surfaces [1][2][3].However, the dimensional accuracy is prone hard to control as large amounts of factors contribute to the removal effect simultaneously, such as belt speed, in feed rate, workpiece geometry, the belt material, the elasticity of contact wheel, grit size, grain distance and so on.
Over the years, many academics made a systematically theoretical study on computer -controlled belt polishing process.The calculation of contact force between the elastic contact wheel and the rigid complex surfaces is one of the key works to analysis this process.The signorini method, which is focuses on the contact problem between an elastic body and a rigid body, has been used to solve the deformation of the elastic contact wheel.Blum and Suttmeier worked out a FEM model to solve this signorini contact problem and used an optimized mesh discretization strategy to enhance the efficiency and accuracy of the FEM model [4,5].X. Ren proposed energy minimization principle to further help solving the signorini contact problem [6].However, the modeling process is too computationally expensive since a small mesh size is required to ensure the calculating precision.Zhang developed a new model using support vector regression method as the learning machine to speed the calculation [7], but it is also time-consuming in the training phase and not suitable in real-time applications.S.H. Wu developed a super position force model based on Hertz method to approximately calculate pressure distribution in the workpiecewheel contact area [8], it was proven not only time saving but also of high accuracy, for which the model will be used in this paper.Since the local removals are not homogeneously distributed in the whole contact area due to the complex geometry of workpiece [8], any changes of workpiece curvature would influence the material removal accuracy, so the influence of workpiece curvature on material removal is also studied in this paper.
In this paper, a quantitative material removal depth model is implemented.This model can be used to achieve the quantitative material removal with little calculation work.The rest of this paper is organized as follows.In Section II, the material removal rate model and the material removal deep model are established.In Section III, simulation and analysis to model are done.The experiments and error analysis are described in Section IV, followed by conclusions in Section V.

Material Removal Rate Model
Figure 1 is the schematic diagram of belt polishing process.For surfaces with nonuniform curvature, the contact state between the contact wheel and the blade surfaces varies according to the changes of surface curvature.The distribution of contact force and the relative velocity in different contact area are different.So it is meaningful to select appropriate process parameters based on the shape characteristics of blade surface to achieve quantitative removal.The traditional model of material removal rate is based on Preston equation [9,10]. .
In the type: K P is the comprehensive influence coefficient decided by experiments, P c is the contact force, V s is the relative velocity, and d h is the material removal depth during contact time d t .
Many factors influence the material removal rate simultaneously in belt polishing process.These factors include polishing parameters such as belt speed, feed rate and the contact force, and tool features such as the tension of the belt, grit size, grit density, belt wear rate and contact wheel hardness.In (1), the impact of all the factors, except the contact force and the relative velocity, attribute to an integrated constant K P , which can be decided by experiments.Two most important factors on material removal rate, contact force and relative velocity, which are controllable, are modeled in next session based on the characteristics of blade surface and features of belt polishing tools.

Contact Force Model of Polishing Zone
According to (1), the force distribution in contact area is one of the most important factors for calculation of material removal rate.It can be calculated by FEA (finite element analysis) method preciously, but the method is time-consuming.Compared to FEA method, the Hertz method has been proven computationally easy and can effectively estimate the contact area and calculate the pressure distribution in contact area [8].For the purpose of analyzing force distribution problem in the belt-polishing process of blade surfaces with Hertz method, the blade-wheel contact can be approximated by the contact between two cylinders based on the following facts [11]: 1.The polishing path is well planned so that when the workpiece-wheel contact occurs during the polishing process, the workpiece surface has minimum principal curvature along the axial direction of the contact wheel.
2. The deformation that occurs during the workpiece-wheel contact is much less than the radius of the contact wheel and the local radius of the workpiece.The contact status between contact wheel and blade is shown in Figure 2. Assuming the blade is a rigid body that has no deformation while the contact wheel is of elasticity and deforms when the contact happens.For complex blade surfaces, it is also assumed that the contact area between the contact wheel and the blade surface is a line contact [12].The contact force distribution between the contact wheel and workpiece can be calculated from (2) by applying Hertz method.

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In the type: P is the contact pressure which can be further formulated as F/W, (F is the normal force; W is the thickness of contact wheel).R 1 and R 2 are the radiuses of the contact wheel and the local radius of workpiece respectively, R * is the equivalent radius.ν 1 and ν 2 are the Poisson's ratios of the contact wheel and the workpiece respectively.E 1 and E 2 are the Young's modulus of the contact wheel and the workpiece respectively, E * is the equivalent Young's modulus.a is half of the width of the contact area.P Hertz (x) is the pressure at the point with an x coordinate of x.
We can get the force distribution of the contact zone by using Hertz model when . The H ertz force distribution of polishing contact zone is shown in Figure 3.
Figure 3a shows that when the radius of workpiece is constant, the larger the normal force of contact wheel is, the greater the force distribution of contact zone is. Figure 3b shows that when the normal force of contact zone is constant, the greater the radius of workpiece is, the less the force distribution of contact zone in the middle is, while the greater the force distribution of contact zone in the edge is.

Relative Velocity Model of Polishing Zone
The relative velocity which is along the tangential direction of contact point, is another important factor that affects the material removal rate greatly.As shown in Figure 4, the contact wheel moves along the polishing trajectory.The velocity of the contact wheel for its self-rotation on a point with coordinate of x is Vm, the feed rate of the contact wheel along the polishing trajectory is Va, the rotational angular velocity of the contact wheel is w.So the relative velocity Vs along tangential direction between the contact wheel and the workpiece can be calculated with (4).
In Figure 4, δ is the maximum deformation of the contact wheel [8], it can be calculated by (5).
According to (4) and Figure 4, we can get the (6).

Material Removal Depth Model
The material removal rate model is firstly built by integrating ( 2) and ( 6) into (1).To obtain the contact time between the blade and the contact wheel, the polishing process should be analyzed.As is shown in Figure 6, in the process of polishing the blade surface, the contact between the contact wheel and blade is approximately linear.When the contact occurs in normal direction of the point i-m between the contact wheel and blade, the point i on the blade begin to contact with the contact wheel and its polishing begins too.The polishing of point i is over as the contact occurs in normal direction of the point i+m.Therefore, the material removal depth of point i is the integration of its material removal rate from the time of i-m (t1) to i+m (t2), as shown in (8).  .In addition, in the polishing process, the relationship between the coordinate of the point i and time can be presented with (9).

Simulation Analysis
The material removal depth model is simulated to forecast the quantitative relationship between the polishing parameters and the material removal depth.The workpiece material is Q235 steel with elastic modulus E 2 of 210Gpa and poisson ratio ν 2 of 0.3 while the contact wheel is rubber wheel with elastic modulus E 1 of 2.18N/ mm 2 and poisson ratio ν 1 of 0.48.Kp is 0.09 that is introduced from X. Ren [6].The true value of Kp can be defined by (10) to apply to the belt polishing of blade surface.The radiuses of contact wheel and workpiece are R 1 and R 2 respectively.
There are three unknown polishing parameters in the material removal model that can be controlled in actual polishing process, include the contact force, feed rate and belt speed.To clearly show their influence on material removal depth, simulations were developed to study the relationships between them.Distribution of the contact force in contact area is a major factor that influences the removal depth, but it is hardly measured in real-time process.However, it can be expressed by the normal force F and their relationship has been discussed in section II.In Figure 7a, we chose R 1 is 50mm and it is unchangeable, R 2 changed from 10mm to 1000mm.It can be seen from Figure 7a that the material removal increase sharply with the increasing of normal force F, but it increases slightly with the increasing of R 2 .
For the required removal depth, it is able to select proper F based on the curvature of blade to achieve quantitative removal.In Figure 7b, the blade curvature R 2 is stable while R 1 changes from 10mm to 100mm.It can be seen that the removal depth increase largely with the increasing of R 1 .With this simulation, the appropriate contact wheel can be chosen according to the actual processing requirements.
The relative velocity Vs is another major factor which impacts the removal depth.It is composed of two parts: feed rate Va and belt speed Vm.  8 show that the material removal depth increases with the increasing of normal force F and belt speed Vm.This is because with the increasing of normal force F, the material removal rate increases.Similarly, with the increasing of belt speed Vm, the grinding grain number per unit time in polishing, so the removal rate increases.Figure 9 shows that the feed rate Va has little impact on the material removal depth.This is because in abrasive belt polishing, the workpiece feed rate is far less than the belt speed.
The relationship between feed rate and the material removal depth.

Experiments of Material Removal Depth
The material removal depth model of belt polishing is testified by experiments.The material removal depth is obtained by adjusting machining parameters which are the normal force F, the belt relative velocity Vs and the feed rate Va.When a parameter is adjusted, other two parameters remain constant.When theoretical value is calculated by using (8), the comprehensive influence coefficient Kp is set as 1.Then Kp is revised by contrast and analysis between the measurement value and theoretical value.The revision equation is shown in (11).
Error of revised theoretical value and measurement can be calculated as shown in (12).
Where n is measurement times in each group.Here we set n as 30.
The machine tool used in the experiment is an adaptive grinding machine which is developed by the research group, as shown in Figure 10.We use disk force sensor as force measuring device.The force sensor is arranged on the lower part of the work piece, and is connected with the computer.When the tool head is pressed down, as the downward displacement increases, the force of the work piece increases, the force of the sensor increases, and the force of the sensor is displayed on the computer software.This force is the normal force of the workpiece during the machining process.The depth measuring device is Kean super depth VHK-900 optical microscope.The experiment selected medium-granular belt, which is P120 alumina.We use closed grinding patterns and lateral row cutting path.

Effect of Normal Force on Material Removal Depth
In the experiment, the workpiece is polished by increasing gradually the normal force F of the head of tool.Parameters are set as: Vs=0.25m/s,Va=0.1mm/s.Theoretical value and experimental value of material removal depth with different normal force is shown in Figure 11a.
According to (11), KP can be revised as 1.5290.Effect of the normal force F on the material removal depth which is calculated according to revised KP is shown in Figure 11b.
The error between the measured value and the theoretical value is 4.172%, which is got by (2).

Effect of Relative Velocity On Material Removal Depth
In the experiment, the relative velocity Vs is increased by increasing gradually the rotate speed of driving motor.Parameters are set as: F=8N, Va =0.1mm/s.Theoretical value and experimental value of material removal depth H with different relative velocity Vs is shown in Figure 12a.
According to (11), KP can be revised as 1.3572.Effect of the relative velocity Vs on the material removal depth which is calculated according to revised is shown in Figure 12b.

Effect of Feed Rate on Material Removal Depth
In the experiment, the workpiece is machined by increasing gradually the feed rate of the contact wheel along the polishing trajectory Va.Parameters are set as: F=8N, Vs=0.25m/s.Theoretical value and experimental value of material removal depth with different normal force is shown in Figure 13a.
According to (11), KP can be revised as 1.3728.Effect of the feed rate Va on the material removal depth which is calculated according to revised is shown in Figure 13  (b).
According to (12), we can get: e=2.32%.By comparing Figure 11, 12 and 13 to Figure 7, 8 and 9, we can see that the influence trend of main parameters to material removal depth H in experiments and simulation are accordance, which shows that established theoretical material removal depth model is correct.And we find that KP is approximated in three experiments, so we can calculate average value of KP as comprehensive influence coefficient.

1) Accuracy error of machine tool
There are some errors in the machine tool used in the experiment, which include the error of the machine tool slide guide and the transmission error of the machine tool.The flatness error of the slide guide has certain influence on the machining precision.The transmission errors that exist in the slide guide feed motion process lead to the change of machining parameters, thus the machining error is produced.

2) Belt wear
In the experiment, abrasive grains increasing with time gradually fall off, which lead to the decline of grinding ability, reduce the removal amount with same conditions.It produces a certain error to the removal result.

3) Machine tool fixture error
In the direction of workpiece longitudinal stress, workpiece fixture is located by bolts which are on the both sides of fixture, which lead to greater friction to fixture.When the tangential force increases, workpiece fixture will have tiny movement, which offsets parameters change trend, and has a certain impact on the final machining precision.

4) Force sensor error
The experiment depends on the force sensor installed at the bottom of the workpiece to realize the control of the normal force of the workpiece.Due to the operation of the staff and the vibration of the machine tool, the feedback value of the force sensor has a certain range of fluctuations, which has a certain impact on the final processing results.

5) Measurement error
After the workpiece is processed, the depth of the material is measured by optical microscope VHK-900.Measurement results are greatly affected by human factors.
When the processing department 3D molding is made, operators will adjust the stage, the adjustment of the speed will result in a certain impact on the measurement results.

Conclusion
A quantitative material removal depth model that focuses on belt polishing of complex blade surfaces is established in this paper, and the relationship between the polishing parameters and material removal depth is established.
Through simulation to this model, it is testified that material removal with different curvature of blade complex surface is changeable when other machining parameters are invariable, and the contact force and belt speed have a great impact on removal while the feed rate of the wheel is of little influence, so that they can be combined properly to achieve precious polishing.It can also be deduced from the simulation that the smaller contact wheel can achieve a stable polishing, which is beneficial to realize precision manufacturing.Through experiments, it is testified that the model established is correct, and we can design reasonably machining parameters and realize quantitative material removal of complex surface by applying it.

Figure 1 .
Figure 1.The diagram of belt polishing process.

Figure 2 .
Figure 2. Contact status between contact wheel and workpiece.

Figure 3 .
Figure 3.The diagram of Hertz force distribution of polishing contact zone.

Figure 4 .
Figure 4.The distribution of relative velocity.

Figure 5 .
Figure 5.The diagram of the distribution of relative velocity.

Figure 6 .
Figure 6.Polishing process of point i.
6/22/19 4:11 PMIt can be calculated that the arc from i-m to i+m is

Figure 7
Figure7and Figure8show that the material removal depth increases with the increasing of normal force F and belt speed Vm.This is because with the increasing of normal force F, the material removal rate increases.Similarly, with the increasing of belt speed Vm, the grinding grain number per unit time in polishing, so the removal rate increases.

Figure 7 .Figure 8 .
Figure 7.The relationship between normal force and the material removal depth.

Figure 10 .
Figure 10.Adaptive grinding machine for complex surface.

Figure 11 .
Figure 11.Theoretical value and experimental value of material removal depth with different normal force.

Figure 12 .
Figure 12.Theoretical value and experimental value of material removal depth with different relative velocity.

Figure 13 .
Figure 13.Theoretical value and experimental value of material removal depth with different feed rate.