The ultrasonic vibration–assisted polishing material removal model of tilted cylindrical polishing tool based on BK7 glass

BK7 glass is widely used in various optical instruments as a common ceramic material. However, there are still several difficulties in the polishing process due to its material properties. Ultrasonic vibration–assisted polishing (UVAP) is used widely for its excellent material removal properties. However, there are few researches on the model of UVAP material removal for tilted. Therefore, this paper investigates this problem. A model of UVAP material removal was established for titled column polishing; The pressure distribution and velocity distribution in the contact zone were modeled by means of experiments and geometric analysis. A series of orthogonal experiments were conducted, and the experimental results showed that the coefficient of determination was above 0.9 after fitting the actual profile to the predicted profile. The results of the orthogonal experiments were also analyzed by ANOVA. The predict error of maximum removal depth was less than 13.12%. The error of material removal rate (MRR) was less than 9.24%. By power spectral density (PSD) analysis, it was found that the introduction of ultrasonic amplitude has a good effect on suppressing the medium and high-frequency errors on the machined surface. The novel model can be used to optimize machining parameters and provide theoretical support for complex parts polishing.


Introduction
As science and technology improve by leaps and bounds, more and more precision instruments are put into daily use. Hard and brittle materials [1][2][3][4] are widely used in these instruments for their wear resistance, corrosion resistance, and excellent mechanical properties. The processing methods of hard and brittle materials have also been studied by scholars [5,6]. BK7 is widely used as a general optical glass in optical systems [7][8][9]. However, due to the influence of material properties, BK7 glass encounters many problems during processing [10]. As an ultra-precision machining method, polishing is an important method to achieve nano-scale machining. And the polishing process is usually preceded by ultra-precise grinding and lapping to give the brittle material a satisfactory surface shape [11][12][13][14]. Ultrasonic-assisted grinding of BK7 glass has been studied by many scholars internationally [15,16], and process optimization has been performed in grinding force, grinding thickness, and grinding temperature [17][18][19][20]. Although, ultrasonic vibration-assisted machining has many advantages in improving processing efficiency and processing surface quality. However, the UVAP mechanism of BK7 glass has been less studied, and the method has not been widely applied in practical processing. Compared with the traditional polishing method, UVAP can significantly improve the polishing efficiency, so it is necessary to conduct an indepth study on the polishing mechanism of UVAP. Many researchers have investigated material removal models and process parameter optimization for UVAP. Zhang et al. [21] introduced the micromorphology of polishing tools during the study the polishing forces of UVAP. A model of the microscopic contact state during UVAP was developed. The accuracy of the predicted model was verified experimentally, which is a guide for UVAP processing of brittle materials. Liang et al. [22] introduced an ultrasonic atomization device in UVAP. Taking BK7 optical glass as an example, the material removal mechanism of UVAP was analyzed and a material removal model was developed. Qu et al. [23] developed a material removal model for tilted-axis UVAP with a spherical polishing tool. The correctness of the predicted model was verified by comparing the material removal depth, MRP, and MRR. Moreover, the experimental results show that larger ultrasonic amplitude and tilt angle can effectively improve the material removal.
In addition to the process parameters, the shape of the polishing tool also has a great influence on the polishing effect. Xiao et al. [24] investigated the polishing force for small inclination machining of circular polishing discs. The relationship between polishing force and contact depth, inclination angle, polishing disc, and workpiece radius of curvature was established. The model was validated by finite element simulations and experiments. This study provides a theoretical basis for achieving uniform material removal. Wang et al. [25] analyzed the material removal model after machining tilting polished discs. A material removal map was generated based on this removal model, and its correctness was verified by experiments. This study is essential for the selection and optimization of processing parameters. Zhang et al. [26] proposed a surface profile model for UVAP by using a cylindrical polishing tool. The study introduced the concept of the Preston correction factor. It was found that the prediction results were consistent with the experimental results, which is an important guideline for the adjustment and selection of polishing parameters. Feng et al. [27] developed a corresponding material removal model for a tilted elastic polishing disc. The step size between polishing paths was investigated based on the model. The correctness of the model was verified by experiments.
In summary, in previous studies, there have been sufficient theoretical studies on material removal models for tilted column polishing tools, but few studied the effect of ultrasonic on it. Predictions of the surface profile of the UVAP formed by the straight feed of the column polishing head have been presented, but little research has been conducted on the material removal characteristics of UVAP of the tilted cylindrical polishing tool. Therefore, this article attempts to fill the gap in this study through theoretical and experimental analysis.

The model of material removal
The polishing tool can deform considerably due to the polishing force when the spherical polishing tool was performed. And this deformation can have an impact on the accuracy of the material removal model. However, cylindrical polishing tools have less deformation and greater contact zone for the same processing parameters. In this paper, the model of UVAP material removal for tilted column polishing tools was investigated.
The contact diagram between the workpiece and the polishing tool is shown in Fig. 1. The polishing tool with radius R is mounted on the spindle of the polishing; point O is the critical point between the deformed and undeformed areas of the polishing tool. The feed direction is identical to the positive direction of the x-axis. α is the angle between the column polishing tool and the workpiece. The dashed part is the compression area of the polishing tool, whose depth h is defined as the displacement of the polishing tool along the normal of the workpiece surface. The spindle applies ultrasonic vibration of frequency f to the polishing tool.
In the process of polishing, the function of material removal is usually described by the Preston equation, which is given as follows: where the material removal depth is denoted by dh(x,y); the relative speed between the workpiece and the polishing tool is denoted by v(x,y); and the polishing pressure is denoted by p(x,y); and k is the Preston coefficient; the value of k needs to be determined experimentally.

Polishing pressure distribution model
As shown in Fig. 2, during the plane polishing process, when the compression of the polishing tool is E, the graphics enclosed by the line CB and the arc BGC can be represented as the polished contact zone. Where CB denotes the contact zone width 2 W and OG denotes the contact zone length L, the geometric relationship gives the following: When ultrasonic vibration is introduced, E′ can be expressed as follows: where A and f are the ultrasonic amplitude and ultrasonic frequency; φ is the initial phase of ultrasonic vibration; in this study, = 0 . Substituting Eq. (4) into Eq. (2): (1) dh(x, y) dt = kv(x, y)p(x, y) Due to the specificity of the contact zone in this paper, the pressure distribution in the contact zone is analyzed by Feng's method [28]. Assume that the contact pressure at any position within the contact zone is directly proportional to the deformation variable, the specific relationship is as follows: K is a constant that needs to be calculated by subsequent experiments. Considering the characteristics of ultrasonic vibrations, the pressure at any point in the contact zone can be expressed as follows: The relationship between polishing force and compression amount is established through experiments. The polishing force is measured by a dynamometer (Kistler 9257B). The experiment chose 8 mm diameter column polishing tool, the inclination angle was set to 5°, 10°, 15°, and 20°, respectively, and the uniform compression amount was taken each time. The compression amount and polishing force curves are shown in Fig. 3. When the inclination angle is 5° and the ultrasonic amplitude is 0, the force-compression expression is as follows: According to the polishing force equation: The value of the coefficient K can be calculated from different angles. where y′ is the longitudinal coordinate of point C in Fig. 2 represents the arc BCG, and f (y) = √ R 2 − y 2 − R + L. As shown in Fig. 4, P is a point on the workpiece; position 1 is the polished contact zone just touching point P, which is the critical position for point P to produce material removal. When the polishing tool moves to position 2, point P is completely within the polishing area. Position 3 indicates that the polished contact zone is completely leaving point P. As shown in Fig. 5, to analyze the distribution of polishing pressure, the coordinate system x-o-y is established. The coordinate of point P1 is (L, y), and the coor- , when the polishing tool feed speed is v f : When point P moves between P1 and P2, it is not subjected to pressure. The pressure distribution in the stage . When the polishing tool moves from P1 to P3, the polishing pressure can be indicated as follows:

Polishing area linear velocity distribution model
When the feed speed of polishing tool is 0, the velocity at any point of the polished area can be denoted: The angular velocity of the spindle can be expressed as . The velocity distribution function in the polished contact zone can be represented by the linear velocity v s of the polishing tool and the feed velocity v f along the trajectory direction: where is the angle between v s and v f , and 0 < θ < π. Figure 6 shows a schematic diagram of the velocity along a straight line polished when the contact point is located in the positive direction of the y-axis. When the contact point is in the negative direction of the y-axis: Combined with Eqs. (13)- (16), the linear velocity distribution function in the contact zone can be represented:

Material removal modeling
According to Eq. (7), the effect of ultrasonic vibration working time t was considered in this model. Combined with Eq.
(1), the MRP at any point in the polished area can be written as a function of the ultrasonic vibration working time t: Assume that the distance of polishing tool movement in dt time is dl, then: Fig. 7 Five-axis polishing machine tool According to Fig. 4, for any point in the polishing area, the relationship between the working time dt of ultrasonic vibration and the feed rate v f can be expressed as follows: Substituting Eqs. (19) and (20) into Eq (18): The material removal model can be expressed as follows:

Experimental workpiece and experimental equipment
The experimental workpiece is a 50-cm × 50-cm × 5-cm BK7 optical glass. Before experimenting, the workpiece was placed in an ultrasonic cleaner with alcohol for 10 min to ensure the cleanliness of the workpiece. The polishing experiment was performed on the five-axis CNC polishing machine tool. The specific parameters of the polishing machine tool can be found in the previous research [28]. The specific machine structure is shown in Fig. 7.

Determination of experimental parameters
The Preston coefficients in this paper were determined by pre-experiment prior to each group of experiments. By substituting Eq. (7) and Eq. (13) into Eq. (1): Assuming that the total polishing time is T d , the material removal function can be given as follows: In a further derivation of Eq. (24), taking the contact zone as the integration zone, the following formula can be obtained:

Fig. 9 Comparison between the predicted and experimental results
In Eq. (25), volume V can be used to represent ∫ ∫ H(x, y)dxdy ; then, the Preston coefficient k can be denoted as follows: The Preston coefficients in this paper are calculated by measuring the experimental removal volume by 3D laser microscopy. To verify the correctness of the material removal model established in this study, a series of polishing experiments were performed. The experiments variables and parameters were selected as shown in Table 1.

Analysis of experiments results
The experiments workpieces were examined with a 3D laser electron microscope. The measurement results will be compared with the prediction model in four aspects: MRP, MRR, maximum removal depth, and surface roughness (SR). The specific measurement results are shown in   Fig. 8

MRP analysis
Figure 9a-p show the results of experiments measured MRP compared with the predicted profiles. To evaluate the prediction effect more intuitively, the Fréchet distance of the simulated and experimental profile curves was calculated, and the results are shown in Table 3. However, certain evaluation criteria are missing when using Fréchet distance as the evaluation index, so the determination coefficients of the two curves were calculated in this paper using linear interpolation and linear fitting methods. According to the calculated results, the determination coefficient of both simulated and experimental removal profile curves can reach above 0.9, which proves that the two curves are well-fitted. Combined with the above analysis, the prediction model can accurately predict the MRP and the polishing profile for tilted UVAP of columnar polishing tools. The error between the theoretical model and the actual result is mainly caused by the deformation of the polishing tool and the processing error of the equipment. In the experiments, it was found that the feeding direction and rotation direction of the polishing tool would have a significant impact on the utilization rate of the polishing solution. Reasonable choice of feed direction and rotation direction can effectively prevent polishing solution splash. During the polishing process, the impact of the abrasive grains in the polishing solution on the surface of the workpiece can have a significant impact on the material removal. The splashing of the polishing solution reduces the number of effective abrasive grains, which is one of the main reasons for the difference between the theoretical model and the actual situation.

MRR analysis
According to the experimental data in Table 2 and Fig. 10, it can be found that the MRR errors of simulations and experiments range from 0.76 to 13.12%. The maximum error is for the 11th experimental group; the minimum error is for the 3rd experimental group. The prediction model is in better agreement with the actual processing. The experimental MRR was used as response data for ANOVA of the experimental results. The mean response table and main effects plot are shown in Table 4 and Fig. 11, respectively. The results of the analysis showed that the four variables of the experiment affected the MRR in the order of the rotational speed, compression amount, ultrasonic amplitude, and inclination angle. To improve the processing efficiency, the polishing process should maximize the MRR, so the optimal level is reached when the speed is 6000 rpm, the compression is 0.2 mm, the amplitude is 0 μm, and the tilt angle is 10°.

Maximum removal depth analysis
According to the experimental data in Table 2 and Fig. 12, it can be found that the errors between the predicted and experimental values range from 0.38 to 9.24%. The largest error was found in group 14; the smallest error was found in group 10. The prediction model is in better agreement with the actual situation. The experimental results were analyzed by ANOVA using the experimental maximum depth of removal as response data. The mean response table and main effects plot are shown in Table 5 and Fig. 13, respectively. The results of the analysis showed that the four variables of the experiment influenced the maximum removal depth in the order of inclination angle, rotational speed, compression amount, and ultrasonic amplitude. The optimum level for maximum removal depth is when the inclination angle is 15°, the speed is 6000 rpm, the compression is 0.2 mm, and the amplitude is 9 μm.

SR analysis
According to the experimental data in Table 2, it can be found that the Sa value of the surface roughness of the polished workpiece is 53-62 nm. ANOVA was performed on the experimental results using the SR of the machined  workpiece as response data. The mean response table is shown in Table 6, and the SR main effects plot is shown in Fig. 14. The analysis results show that the four variables in the experiment affect the SR of the machined workpiece in the order of ultrasonic amplitude, rotational speed, inclination angle, and compression. Considering the need to reduce the SR of the workpiece, the optimum level is reached when the amplitude is 6 μm, the inclination angle is 10°, the speed is 10000 rpm, and the compression is 0.4 mm. Figure 15 is a schematic diagram of part of the measurement area. When the rotational speed is low, the removal ability of abrasive grains is weak. As the rotational speed increases, the relative speed between the polishing tool and the abrasive grains increases, which improves the material removal ability of the abrasive grains and makes the surface quality improved. When the speed continues to increase, too fast a speed increases the centrifugal force, resulting in some of the abrasive grains being thrown out and an uneven distribution of the effective abrasive grains involved in polishing. Therefore, too fast rotational speed will have some effect on the surface quality.
Through Fig. 9, it can be seen that with the increase of the compression amount, the polishing force gradually becomes larger. When the compression amount is small, the polishing force is weak, the depth of the abrasive grain being pressed into the workpiece is shallow, and the removal effect is poor. With the increase of the compression amount, the polishing force gradually becomes larger, and the contact force between the abrasive grain and the workpiece increases, which improves the removal ability of the abrasive grain and the surface quality is improved. However, when the compression amount is too large, the polishing fluid between the polishing tool and the workpiece is significantly reduced and cannot play a sufficient role in heat dissipation and lubrication. And less polishing fluid cannot bring out the debris produced during polishing, which reduces the surface quality of the workpiece.
When the ultrasonic amplitude is 0, the material removal in the polishing process is mainly achieved by the sliding of the abrasive grains, and the surface quality of the workpiece is poor. With the increase of amplitude, the abrasive grains gain a certain kinetic energy due to ultrasonic vibration, which increases the removal capacity and improves the surface quality of the processed workpiece. However, excessive ultrasonic amplitude will cause the abrasive grains to press into the surface of the workpiece too deep and have greater kinetic energy, which will scratch the surface of the workpiece to a certain extent and have a certain impact on the surface quality.

PSD analysis
In order to further investigate the surface condition after polishing, the polished surface was analyzed with the original surface for power spectral density (PSD) analysis, and the results of the analysis are shown in Fig. 16. No. 0 is the original surface and no. 1-no. 16 are the polished surface. It can be found that ultrasonic vibration-assisted polishing can have an effective reduction of medium and high-frequency errors and an extremely significant effect on the reduction of high frequency errors.
To further investigate the surface evolution mechanism during UVAP, BK7 glass was polished using different ultrasonic amplitudes, respectively, and the results of the PSD analysis are shown in Fig. 17. It can be found that when the frequency is low (frequency is less than the position of X point marked in this figure), ultrasonic vibration has less effect on the spatial frequency error; with the increase of spatial frequency, ultrasonic vibration plays a certain effect on reducing the error. With the increase of ultrasonic amplitude, the spatial frequency error has a tendency to gradually decrease. Therefore, ultrasonic vibration-assisted polishing can reduce the high-frequency error in the processed workpiece surface and help improve the optical properties of Fig. 15 Image of part of the measurement area processed BK7 glass, and the processing effect will become better with the increase of amplitude.

Conclusion
1. An effective UVAP model of material removal for a tilted column polishing tool was developed. The distribution model of polishing pressure within the contact zone was determined by normal loading tests on BK7 glass. The distribution of linear velocity within the contact zone was also investigated using geometric analysis. Compared with previous material removal models, this model takes into account the effect of ultrasonic vibrations. 2. A series of experiments were conducted to verify the reliability of the prediction model from several aspects.
The experimental results showed that the coefficient of determination between the predicted curve and the actual removal profile was above 0.9. The MRR error values ranged from 0.76 to 13.12%, and most of the error values were below 10%. The maximum removal depth error values are in the range of 0.37-9.24%. The surface roughness of the machined workpiece is 38-85 nm. 3. The ANOVA was performed based on the results of an orthogonal test. It shows that the four variables chosen in the experiment influence the MRR in the order of rotational speed, compression amount, ultrasonic amplitude, and inclination angle. The degree of influence on the maximum removal depth was in the order of inclination angle, rotational speed, compression amount, and amplitude. The degree of influence on the SR of the machined workpiece was determined in the order of amplitude, inclination angle, rotational speed, and compression amount. And the optimal level of each factor was determined. 4. PSD analysis of the experimental results was conducted.
The analysis results show that UVAP can effectively reduce the medium frequency and high frequency errors on the processed workpiece surface. And the processing effect gradually improves with the increase of ultrasonic amplitude.
Based on the above, this paper establishes a UVAP model of material removal for tilted column polishing tools, which can effectively predict the surface profile after UVAP processing. It is a good guide to achieving material deterministic removal by UVAP. In future work, curved surfaces will be studied as objects for more in-depth material removal models.