Processing of high-precision ceramic balls with a spiral V-groove plate

As the demand for high-performance bearings gradually increases, ceramic balls with excellent properties, such as high accuracy, high reliability, and high chemical durability used, are extensively used for highperformance bearings. In this study, a spiral V-groove plate method is employed in processing high-precision ceramic balls. After the kinematic analysis of the ball-spin angle and enveloped lapping trajectories, an experimental rig is constructed and experiments are conducted to confirm the feasibility of this method. Kinematic analysis results indicate that the method not only allows for the control of the ball-spin angle but also uniformly distributes the enveloped lapping trajectories over the entire ball surface. Experimental results demonstrate that the novel spiral Vgroove plate method performs better than the conventional concentric V-groove plate method in terms of roundness, surface roughness, diameter difference, and diameter decrease rate. Ceramic balls with a G3-level accuracy are achieved, and their typical roundness, minimum surface roughness, and diameter difference are 0.05, 0.0045, and 0.105 μm, respectively. These findings confirm that the proposed method can be applied to high-accuracy and high-consistency ceramic ball processing.


Introduction
The demand for high-accuracy, high-efficiency, and high-automation machinery is increasing concomitant with the rapid development of the industrial technology. As indispensable components of machinery, bearings determine the accuracy, performance, life span, and reliability of the host machine; therefore, the development of the bearing industry is necessary [1][2][3]. However, traditional metal bearings can hardly meet the requirements when the working conditions demand high precision and high stability as well as involve high overload, high temperature, and acidic environment. Under these conditions, wherein defects in rolling bodies and raceway are likely to occur, metal bearings are recommended to be replaced by ceramic ones, whose excellent properties include low density, high hardness, high wear resistance, and high corrosion resistance [4][5][6]. In recent years, ceramic bearings have been extensively applied in aerospace, precision machinery, petrochemical industry, national defense, and other fields. Consequently, the demand for high-precision ceramic balls increases along with the extensive application of ceramic bearings. Ceramic balls are widely used as the rolling bodies for bearings; thus, their precision must reach at least the G5 level. The details of the precision ball grades are listed in Table 1. Furthermore, as an important factor in bearing performance, high-precision ceramic balls present high commercial profit because their commercial demand increases by 15% each year [6]. However, ensuring the quality of highprecision ceramic balls is challenging due to the different precision levels of semi-finished products and the difficulty of manufacturing ceramic balls. This issue provides a significant opportunity to conduct research on the processing of high-precision ceramic balls.
In general, the following two key points must be satisfied in high-precision ceramic ball processing: 1) The different points on the ball surface must have the same probability of being processed; 2) the processing of balls that are of relatively large in size should be given priority. Generally, different processing methods vary in their capability to satisfy these two points: The stronger the capability of the method is, the higher the ball quality becomes. In other words, the processing method significantly influences the ball quality. Currently, the traditional metal ball processing methods are mainly utilized to produce high-accuracy ceramic balls, including concentric/ eccentric V-groove plate method, controllable ball-spin angle method, and magnetic abrasive slurry processing method. In the controllable ball-spin angle method, the Vgrooves are separate, and the sole concentric V-groove is formed by the inner and outer plates. The ball-spin angle is used to describe the self-motion behavior of the ball, and a dynamic ball-spin angle can result in an enveloped lapping trajectories over the ball surface [7,8].
In the concentric V-groove plate method [9,10], the ballspin angle is kept constant during processing, and a stirring device is utilized to realize a randomly changeable ballspin angle. The eccentric V-groove method achieves a dynamic ball-spin angle by setting an eccentric distance between the upper and lower plates [11]. Although this technique improves ball roundness, it increases the diameter deviation and requires high installation precision of the employed machine. Controllable ball-spin angle by the separated V-groove method was proposed to control the ball-spin angle accurately. In this method, the velocity difference between the inner and outer plates are generated; however, the quantity of balls that can be simultaneously machined in a batch is limited by having only one V-groove [12]. In the magnetic abrasive slurry processing method, a slurry produced by blending abrasives into a magnetic fluid is used to process the balls. This method considerably improves the material removal rate during processing in the given magnetic field [13,14].
The processing of high-precision ceramic balls should simultaneously meet the following requirements: Largescale processing and high accuracy. In terms of processing cost, the concentric V-groove plate method is the best method. Aside from low processing cost, it is also widely adopted because of it presents a simple mechanical structure, accommodates large-scale processing. However, its drawback is its requirement for a stirring device to change the ball-spin angle randomly for the achievement of an enveloped lapping trajectory [15]. Zhao et al. [16] proposed a new method that can achieve high accuracy, high consistency, circular processing, and variable ballspin angle in one cycle by substituting the concentric Vgroove plate with variable-radius V-groove plate and using a recycler for balls. This technique has been successfully applied in steel ball processing. However, only few studies on ceramic ball processing have been conducted using this new method. In this study, we draw on the method of Zhao et al. [16] to develop a novel precision processing technique for ceramic balls. In our proposed method, a spiral V-groove plate is used to replace the concentric Vgroove plate. We conduct a kinematic analysis of the ballspin angles in two directions as well as ceramic ball processing experiments to confirm the validity of the proposed processing technique.
2 Ball processing with a spiral V-groove plate Figure 1 shows the schematic diagram of the system for the proposed ball processing technique. As shown, an Archimedes spiral V-groove plate is used. The upper plate with a spiral V-groove is placed on top of the flat lower plate. In addition, a loading mechanism and a recycler are attached to the upper plate. At a certain distance to the center of the upper plate, an inclined through-hole is machined in the upper plate and used as the inlet for the balls. The outlet is located at the edge of the lower plate. A recycler, which links the inlet to the outlet, supplies the balls to be processed through the inlet and collects the balls processed from the outlet. If necessary, the recycler is used to supply and collect the balls in the next cycle of processing. A cycle is completed when a ceramic ball passes through the inlet to the processing zone and exits the outlet. A motor supplies the power that drives the lower plate to rotate. During processing, the materials of the workpieces are removed to improve the ball roundness and surface roughness by the combined effects of loading, rotation, and slurry. given by R = aφ (φ 0 < φ < φ e ). In this equation, a is a constant, and φ 0 and φ e represent the polar angles determining the initial and end points of the spiral, respectively ( Fig. 2(a)). When φ is equal to 0, the ball enters the processing zone from the center of the upper plate. As φ increases, the ball moves forward along the spiral. As shown in Figs. 2(a) and 2(b), the upper and lower plates are installed concentrically, that is, Point O 0 and the center Point O of the working surface of the lower plate are on the same vertical line. A is the contact point of the ball with the lower plate, whereas B and C are contact points of the ball both sides of the V-groove with the angles of α and β on the upper plate. In Fig. 2(a), ρ A , ρ B , and ρ C are the curve radii of the spiral tracks at contact Points A, B, and C, respectively; P A , P B , and P C are the centers of the curve radii ρ A , ρ B , and ρ C , respectively. A rectangular coordinate system XYZ is fixed on the ball with a radius of r b , and the origin of this coordinate system is located at the geometric center O b of the ball, as shown in Fig. 2(c). In the same figure, Ω s denotes the ball-spin angular speed vector, which is divided into two vectors, namely, Ω b and Ω g in XY-plane and Z-axis, respectively. The angles between Ω b and the X-axis and between Ω g and  s are denoted by q and g, which are regarded as the ballspin angles. Moreover, the angular rotation speeds of the ball around X-, Y-, and Z-axes are given by Ω b cosq, Ω b sinq, and Ω g , respectively. The angular rotation speeds of the upper and lower plates are denoted by n 0 and n 1 , respectively. The angular rotation speed of the geometric center of the ball around the axis that is vertical to the lower plate and intersects the lower plate at Point P A is denoted by n. The distance between Points A and O is R A , and the distances of Points B and C to Point O 0 are R B and R C , respectively. The velocities of Points A, B, and C and those of P A , P B, and P C can be divided into two components in the Xand Z-axes, respectively, as shown in Fig. 2(d). The following assumptions of pure rolling motion are considered in the analysis: 1) The pushing and impact forces between balls are ignored; 2) The slide motions between the balls and plates are disregarded; 3) All the components are regarded as rigid bodies. Subsequently, the velocity balance equations are obtained as follows: The absolute velocity of the ball center point O b can be expressed by V O b O ¼ R A n 1 ; thus, the projection velocities along the Xand Z-axes respectively. The absolute velocity expression of Point P A is V P A O ¼ R P A n 1 ; thus, its projection velocities along the Xand Z-axes are V t By the way, n 0 , n 1 represent the angular speed of upper and lower plate and n is the revolution speed of the ball. Subsequently, the same definitions are applied to the rest of the contact points. Finally, the equations are deduced on the basis of the velocity balance equations and given by During this process, the upper plate is fixed; thus, n 0 = 0, and α is equal to β. The relationships between the parameters in Eq. (1) are defined by the following equations: 2) Angles Substituting Eqs. (2)-(5) into Eq. (1) yields the following kinematic equations: Subsequently, the angle g is obtained and represented by According to Eq. (6) the curve radius ρ A and the distance of P A to O are required to obtain the ball-spin angles. Both of these parameters can be obtained by the following equations [17]: Finally, the numerical solution of the spin angles can be performed using the given values of some parameters. The simulation results are presented and discussed in Section 2.3.

Enveloped lapping trajectories
The ball-spin angles have been obtained in the previous section. In this section, the method by which to calculate the trajectories of contact Points A, B, and C during processing is explained. The initial coordinates of Points A, B, and C are (0, -r b , 0), (r b cosα, r b sinα, 0), and ( -r b cosβ, r b sinβ, 0), in the rectangular coordinate system XYZ, respectively. They are transformed into the new coordinates N A , N B , and N C , which are expressed as follows: In addition, the rotation angle expressions are as follows: where i is the serial number of sampling time, x ði þ 1Þ is the next value of the rotation angle x ðiÞ, and x ðiÞ is the increment of the rotation angle at the current moment. The same explanation applies to the rest. The value of the increment is obtained as follows: where Dt is the step interval calculated by The rotational coordinate transformation matrix around each axis is expressed as Finally, the coordinates of the contact points at each step interval can be calculated by The coordinates of the contact points are obtained and transformed from rectangular coordinates to spherical coordinates by using the above calculation. The trajectories of the contact points are subsequently deduced.

Analysis results and discussion
In practice, r b and α are constant; thus, the q value based on the q expression in Eq. (6) only varies with the curve radius ρ A , whose values differ across different positions along the track of the spiral V-groove. In other words, q can be controlled by the curve radius. By contrast, g (Eq. (7)) depends on several variable parameters, and its control is relatively complex compared with q.
where the R A is the radius of a circle along the tip of the concentric V-groove. Figures 3(a) and 3(b) show that the obtained ball-spin angle and enveloped lapping trajectories in the conventional concentric V-groove plate method, respectively. In Fig. 3(b), the enveloped lapping trajectories of contact Points A, B, C are color coded in red, green, and blue, respectively. The ball-spin angle remains the same during processing because the V-groove radius is constant at R A (Eq. (16)) and the trajectory of each contact point is round. These results demonstrate that the ball-spin angle behaves invariably at 2.4°, and the enveloped lapping trajectories are only three circles over the ball surface, confirming that a stirring device is essential for ball processing in the conventional concentric V-groove plate method for the uniform distribution of the enveloped lapping trajectories over the entire ball surface. Figures 4(a) and 4(b) show the theoretically obtained two ball-spin angles and enveloped lapping trajectories in the spiral V-groove plate method, respectively. As shown in Fig. 4(a), q initially rapidly decreases from 90°to 10°w ithin a short processing time and subsequently slightly changes. The ball-spin angle g varies between -90°and 90°at the beginning and then almost remained constant at around 82°. For the enveloped lapping trajectories of contact Points A, B, and C, Fig. 4(b) shows that the lapping trajectories can uniformly envelope the entire ball surface in only one cycle. Thus, this new method generally has the capacity to control the ball-spin angle and uniformly envelope the entire ball surface with the lapping trajectories in only one cycle.

Experimental rig, conditions, and procedure
An experimental rig was constructed by replacing the concentric V-groove plate with a spiral V-groove plate and attaching a cycler onto the existing concentric V-groove plate-type ball-processing machine, as shown in Fig. 5(a). This experimental rig is used to confirm the kinematic analysis in the previous section and validate the performance of the new method in the precision processing of ceramic balls. In this study, two rounds of experiments were designed and conducted on the constructed rig. The first test was conducted to validate the new method, in which the spiral V-groove with the following track expression was used: Figure 5(b) presents a picture of the spiral V-groove plate, which was made from cast iron. It was 300 mm in diameter and 30 mm in thickness. The V-groove had an 8 mm width, 8 mm depth, and 45°α and β angles. For comparison, experiments were also performed using a concentric V-groove plate (Fig. 5(c)) with V-grooves that are same in structure and dimensions as those on the spiral V-groove plate. However, unlike in the spiral V-groove plate method, in which the V-groove plate was used as the upper plate, the V-groove plate was used as the lower plate, and a hole was created in the upper plate for the artificial stirring of the ball-spin angle in the concentric V-groove plate method.
The second round of experiments were conducted to elucidate the potential of the new method for the highaccuracy processing of ceramic balls. In this round, the spiral V-groove with the following track equation was utilized: The V-groove had a 6 mm width, 6 mm depth, and 45°α and β angles. As previously mentioned, a lower φ initial value can attain greater magnitude of ball-spin angle variation. However, currently, the initial value of φ cannot be set to 0 due to the limitation of mechanical structure. Thus, in the second round, the initial value of φ was set to 5p, which is smaller than that in the first round, to achieve higher-accuracy balls.
In the first round, silicon nitride ceramic balls were used as the workpieces. The original accuracy of the used balls met the G16-G20 standards. Figures 6(a)-6(c) respectively present the initial roundness, surface roughness, and diameter of the 10 balls randomly collected from 300 raw silicon nitride ceramic balls utilized in the second round of experiments. The mean values of the roundness, surface roughness, and diameter were 0.38 μm, Ra 0.035 μm, and 4.7636 mm, respectively. Table 3 lists the experimental conditions. In the second round of experiments, three lapping steps, namely, coarse, middle, and fine, were performed in succession. Coarse lapping was conducted to achieve high material removal for improved roundness.
Middle lapping was performed to ensure roundness and improve surface roughness. Fine lapping was conducted to eliminate surface defects and achieve high surface roughness. After each step, the 10 balls were randomly collected from the processed ones and then washed by ultrasonic cleaning, and their roundness, surface roughness, and diameter were subsequently measured.

Experimental results and discussion
Figures 7(a)-7(c) present the measured roundness values, surface roughness values, and diameters of the 10 balls, respectively, in the first round of experiments. The balls processed using the spiral V-groove plate method exhibited higher accuracy than those processed by the concentric Vgroove plate method. The final mean values of the roundness and surface roughness achieved by the spiral V-groove plate method were lower by approximately 20% and 22%, respectively, than those by the concentric Vgroove plate method. Furthermore, the diameter deviation achieved by the spiral V-groove plate method was 0:000073 ¼ j6:731100 -6:731173j, which was lower by  approximately 38% than that by the concentric V-groove plate method 0:00012 ¼ j6:73116 -6:73128j. This result may be attributed to the fact that the ball moves for a considerably longer distance along the spiral V-groove in one cycle of the spiral V-groove plate method. Thus, material removal was relatively higher in this method compared with that in the concentric V-groove plate    Fig. 8(a), the average value of roundness after coarse lapping was approximately 0.18 µm, reaching the G10 level. After the middle and fine lapping, the average values of roundness improved to 0.1 and 0.07 µm, respectively. The roundness of the vast majority of balls reached the G3 level. In the first two lapping steps, the process parameters were set for the primary goal of removing the material of the workpieces. Consequently, the defects on the ball surface, such as scratches and microcracks, were generated, resulting in higher roughness values. Through fine lapping, most defects were eliminated to a certain extent, and the roundness was subsequently improved. Figure 8(b) shows that the mean roughness after coarse lapping was approximately 0.025 µm; thus, the G10 level requirement was met. The lapping load on a single ball was 2 N in this step, the abrasives completely contacted with the ball to increase the cutting depth. As a result of the high rotation speed of the lower plate, the increased cutting frequency of the abrasives increased the material removal rate. However, balls are prone to slip and collide in this situation. Thus, surface defects occurred, and surface roughness consequently increased. In middle lapping, the average value of surface roughness decreased to 0.014 μm as the abrasive size and concentration as well as the load and lower plate speed are altered. After fine lapping, the average value of the surface roughness was further improved to less than 0.0075 μm, and the minimum value was 0.0045 μm.
Figures 8(c) and 8(d) present the measured ball diameters and diameter decrease rates of the 10 randomly collected balls, respectively. The diameter decrease rate is defined as the decrease in the ball diameter per hour. In addition, Fig. 8(d) also shows the diameter difference, which is the difference between the diameters of the maximum-sized ball and minimum-sized ball. Each ball experiences nearly the same situation during coarse lapping; thus, the diameter difference reached 0.12 μm, Fig. 8 (a) Roundness, (b) surface roughness, (c) diameters, (d) diameter differences and diameter decrease rates of the balls after the second round of processing reaching the G3 level; however, the diameter decrease rate was high. This outcome suggested that the proposed processing method can rapidly improve the ball diameter consistency. The diameter difference further decreased to 0.105 μm after fine lapping. However, little material was removed during fine lapping; thus, the ball diameters remained unchanged.
Eventually, the final roundness was maintained below 0.09 mm, indicating that the roundness values of a majority of the balls met the accuracy requirement of the G3 level (0.08 μm). The surface roughness and diameter difference, which reflect ball lot consistency, also reached the G3 level. Figures 9(a)-9(c) show the ball appearance after processing, the typical ball roundness measurement result (0.05 μm), and the typical ball surface roughness measurement result (0.0045 μm), respectively. These results demonstrated that the novel method can be applied to high-accuracy and high-consistency ceramic ball processing.

Conclusions
A spiral V-groove plate method was utilized for processing high-precision ceramic balls. After the kinematic analysis of the ball-spin angle and enveloped lapping trajectories, an experimental rig was constructed, and two rounds of experiments were conducted. The experimental results confirmed the feasibility of the novel method for processing ceramic balls. The obtained results can be summarized as follows: 1) The kinematic analysis results showed that, in the conventional concentric V-groove plate method, the ballspin angle was constant and the lapping trajectories were three concentric rings over the ball surface. By contrast, in the novel spiral V-groove plate method, the ball-spin angles were varied and controllable, and the lapping trajectories uniformly envelope the entire ball surface.
2) The first round of experiments demonstrated that the novel spiral V-groove plate method performed better than the conventional concentric V-groove plate method in terms of roundness, surface roughness, diameter difference, and diameter decrease rate.
3) Under the given processing parameters, ceramic balls with G3-level accuracy were achieved in the second round of experiments. The typical roundness, minimum surface roughness, and diameter difference of the balls were 0.05, 0.0045, and 0.105 μm, respectively. These results suggested that this novel method could be applied to highaccuracy and high-consistency ceramic ball processing.