Mechanistic study of continuous polishing Download: 654次
1. Introduction
Continuous polishing, which is widely used in optical processing, is very important to obtain high-precision and supersmooth surfaces. This process is also applied in high-power laser facilities[1, 2] such as the National Ignition Facility in America and the Ligne d’Integration Laser in France, which require numerous large-aperture slabs with minimal defects and low roughness. To date, the surface shape has been difficult to control accurately in continuous polishing. Polishing with polyurethane laps is also widely used, in which the interactions between the workpiece and the lap are similar to those in continuous polishing. This type of polishing method is limited by the different radial positions of the lap, which have different friction distances from the workpiece. This characteristic causes the lap to deteriorate regardless of the surface shape of the workpiece, even if it is an ideal plane. Scientists and engineers from the Lawrence Livermore National Laboratory have proposed a new polishing method called convergent polishing[3, 4]. In this new method, septa with specific shapes are applied to compensate for the uneven wear to the polishing lap so the surface that has been conditioned can be maintained for a long period. This study won the 2014 R&D100 Award. However, the new method has a drawback; that is, the lack of autorotation of the septum such that the polishing lap and the septum fit perfectly only if the lap is rotationally symmetric, which may cause the surfaces to become aspheric. Continuous polishing differs from polyurethane polishing because the former uses a conditioning disk. According to the study results, the shape of the pitch lap surface in continuous polishing can also be maintained for long periods with a particular position of the conditioning disk, indicating an equilibrium state. The ideal mathematical model of continuous polishing is built first in this paper. The system characteristics are discussed, and the theoretical foundations for high-efficiency and high-precision polishing are established. Continuous polishing is also established as an important step for a deterministic process.
2. Ideal continuous polishing model
The continuous polishing system consists of a massive rotating table with a polishing annulus, a conditioning disk and workpieces. Preston’s equation[5], which is widely used to calculate the material removal rate for the polishing process, can be described as follows:
In the continuous polishing process, the material removal equation is applied to the workpiece and the conditioning disk with different constants . To a widely used pitch lap, this feature is similar to a series of independent springs because of the grooves on the surface. The lap is simplified to a one-dimensional Maxwell fluid based on the thickness, so the Preston constant considers lap wear, and viscous flow can be set for the pitch lap.
Winkler’s hypothesis is applied to calculate the contact pressure for simplification and can be written as follows:
The following assumptions should also be considered prior to modeling:
Given the force balance of the workpiece and the conditioning disk, the contact pressure distribution can be determined on the basis of Winkler’s hypothesis and assumption (4). The changes in the pitch lap surface, workpiece and conditioning disk can then be established. To simplify the calculation, the contact pressure is assumed to remain unchanged in a cycle, and the surfaces at the end of the cycle are modified. A computer is used to control the surface according to the program flow chart in Figure
The initialization includes every size of the lap, conditioning disk and workpiece, the position and density of the conditioning disk and workpiece, Young’s modulus of the lap, the Preston constants and rotation number, calculation errors, increments for and , and some parameters for integral operation and output file path.
Setting the initial surface figure refers to setting the profiles along the radial directions of the workpiece, lap and conditioning disk. The profiles can be described as functions or height values at given radial positions in this program.
Table 1. Parameters for calculation and experiments.
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The workpiece and conditioning disk have different values of and , but we consider that the computing method for both of them is identical, so and refer to both the workpiece and conditioning disk. On the basis of the scanning and variable step method, values of and that satisfy Equation (
A large number of numerical tests have been performed to verify the accuracy and consistency of the model. For example, it has been found that increasing the Preston constants causes faster changes in the surfaces, and changes in the surface figure from different initial shapes, such as concave, convex, ‘M’-shape and ‘W’-shape, are found to be consistent with experiment.
3. Study of the continuous polishing characteristics
3.1. Basic rules of surface figure adjustment
According to the model, two basic rules are found:
(1) Without moving the conditioning disk, the surface peak–valleys (PVs) of the pitch lap, conditioning disk and workpiece will change at constant speeds after their surfaces completely match.
This rule indicates that an inherent feature of the system is the rate of surface change rather than the specific surface. The rate of change in the workpiece surface PV based on the parameters in Table
Some differences are found between the calculated and experimental results because the Preston equation is not completely satisfied, as a result of the nonlinear characteristic of the pitch lap. Errors exist in the experiments and calculation because of interference and inaccurate setting of the Preston constants. However, the basic rules shown by the model are accurate because the shapes of the two curves are similar.
(2) An equilibrium state exists in the system, indicating that the surface shape could remain unchanged, and the shape could be flat or spherical with a small curvature.
The conditioning disk position corresponding to the equilibrium state is the equilibrium position marked in Figure
3.2. Workpiece polishing in the equilibrium state
Different radial positions of the pitch lap have different friction distances from the workpiece. Thus, the lap deteriorates regardless of the shape of the workpiece surface, even though it may be an ideal plane. Figure
If a conditioning disk is applied, the partial damage can be repaired naturally. Thus, the main task is to repair the general damage, which is most important in determining the workpiece surface figure. The workpiece surface can remain high-precision under equilibrium in flat cases, according to the above analysis. Further calculations show that the equilibrium state is insensitive to the surface error of the workpiece. If equilibrium with flat surfaces is obtained, exchanging the workpiece with another workpiece with surface error can also allow polishing to very high accuracy without moving the conditioning disk. This phenomenon indicates that in the process the curvature change is very small between the case of a mismatching workpiece and pitch lap and the perfectly matched case in the equilibrium state. These results can lead to a theoretical basis for high-efficiency and high-precision polishing. Figure
Fig. 6. Change in the workpiece surface caused by motion in the radial direction of the lap.
The equilibrium state can be found by measuring the rates of change of the workpiece surface under different conditioner positions (Figure
Given the delayed response of the pitch lap, measurement results beyond the equilibrium state cannot represent the true surface shape, which makes the lap surface difficult to control accurately. The model shows a delayed response in which the surface does not stop changing immediately after the conditioning disk is pushed to the equilibrium position. To adjust the lap surface, the shape of the workpiece surface should be measured in the equilibrium state. Then, from the surface error and the change rates measured, the distance of movement for the conditioner and the conditioning time is determined. At the end of this calculated time, the conditioner is pushed back to the equilibrium position, and the surface figure is again checked when it is stable. If an error still exists, it may have been caused by an environmental disturbance or calculation errors. The disturbance and errors are eliminated, and the lap is reconditioned.
The key to stable machining is maintenance of the pitch lap surface. From the analyses above, a lap in the equilibrium state is insensitive to the surface error of the workpiece. However, when the position or size of the workpiece changes without moving the conditioning disk, the equilibrium state disappears, and the lap surface will deteriorate. Thus, the conditioner position should be adjusted according to the workpiece to prolong the ideal state.
Figure
4. Conclusion and discussion
An ideal mathematical model for a continuous polishing system is built. According to the model, the mechanism of surface changes in the system is examined, and the equilibrium state, which indicates that the surfaces remain unchanged, is found in the polishing system. The model shows that although the surface of the workpiece is an ideal plane, it will also damage the lap. Therefore, the damage caused by workpieces with surface errors may not be that serious – and this phenomenon has been proved theoretically. The change in curvature is very small between the case of workpiece–lap mismatch to that of a perfect match in the equilibrium state. This characteristic lays the theoretical foundations for high-efficiency and high-precision polishing. The procedures for obtaining an ideal equilibrium state are performed and verified experimentally.
Calculations show that the polishing process is actually deterministic, and the shapes of the workpiece surfaces can be precisely controlled. An unstable environment (such as fluctuations in temperature and humidity, and irregular airflows) and imperfect factors in polishing (such as the nonuniform temperature distribution of the lap and the workpieces, uneven polishing powders on the surface of the lap, unstable rotating speeds of the workpieces, and so on) will seriously impact the machining precision and make the polishing a nondeterministic process. However, with technological developments, such as a higher control accuracy of the environment temperature and humidity, enhanced performance of polishing machines, studies on the polishing materials and material removal mechanisms, as well as the application of computer simulations in conventional polishing, conventional polishing will become a deterministic process.
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Article Outline
Xiang Jiao, Jianqiang Zhu, Quantang Fan, Yangshuai Li. Mechanistic study of continuous polishing[J]. High Power Laser Science and Engineering, 2015, 3(2): 02000001.