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Abstract
Magnetorheological finishing (MRF) technology is characterized by its high convergence rate and minimal subsurface damage as advantages. However, the non-Gaussian type tool influence function (TIF) it generates may cause mid-frequency errors and oriented surface texture issues. Magnetorheological precession finishing (MRPF) technology capable of generating Gaussian-like removal functions, lacks a clearly defined removal function model. This study acquired polishing spots in tilted polishing, discrete precession, and continuous precession modes via fixed-point polishing experiments. Using Multiphysics simulation software, stress and velocity distribution in the contact area were simulated. A TIF model, incorporating the synergistic effects of pressure and shear force and multiple influence coefficients, was proposed based on velocity characteristics across the three modes. To accurately predict the TIF, surface topographies with varying coefficients were constructed using this model, analyzing the coefficients’ impact on the TIF profile. Optimal coefficients were identified using a least fit error algorithm. Further analysis of the TIF's internal textures revealed that the precession mode of MRPF yields superior surface quality, thereby elucidating the material removal mechanism of MRPF and laying a theoretical groundwork for advancing processing technologies.
© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The development of fields such as astronomical observation [1], laser fusion [2], and extreme ultraviolet lithography [3–5] has driven the demand for complex curved optical components, thereby setting higher requirements for surface treatment technologies. Numerous sub-aperture polishing technologies, including computer controlled optical surfacing (CCOS), bonnet polishing, and magnetorheological finishing (MRF), demonstrate potential in managing complex surfaces. However, CCOS polishing has poor adaptability when processing high-curvature surfaces, affecting shape convergence efficiency [6]. Bonnet polishing can adapt to complex surfaces by using tool heads of different curvatures and hardness, but tool head wear may affect the determinacy of shape convergence [7]. MRF, an advanced non-contact polishing technology, offers nanometer-level precision and eliminates tool wear [8]. However, its D-shaped removal function introduces mid-spatial frequency ripples, restricting its use in high-precision complex surface optics [9]. To address these challenges, magnetorheological precession finishing (MRPF) technology, an innovative machining technique based on MRF, has been introduced. Compared to conventional MRF, MRPF incorporates a precession motion mechanism, improving the contact effect between the magnetorheological (MR) fluid and the workpiece, producing a Gaussian-like removal function, thereby achieving more uniform material removal [10]. However, the material removal characteristics of this processing method have yet to be studied.
Studies on traditional MRF have thoroughly investigated its removal characteristics, revealing that analyzing the MR fluid's interaction with the workpiece effectively predicts material removal model. Shorey used the Bingham fluid model and based on fluid lubrication theory, established a strong positive linear relationship between the MRR of glass and the shear stress in the wheel-based MRF process [11]. Building on this, Kordonski introduced a modified Preston coefficient according to the Preston equation to construct a material removal model [12]. Miao used a scanning tunneling microscope (STM) to study the processing forces on the workpiece during MRF, measuring normal and shear forces while polishing BK7 glass and aluminum oxynitride (ALON) ceramics under various conditions [13]. Bai et al., through the study of particle flow theory and MR fluid contact models, have elucidated the mechanism by which shear stress is generated during the MR fluid polishing process. They further proposed an innovative material removal rate (MRR) model. This model takes into account the effects of shear stress and pressure and introduces a pressure exponent parameter to create a removal function profile that more closely matches the actual polishing contact point [14]. However, existing removal function models are suitable for conventional MRF modes with only spindle rotation. In contrast, MRPF employs a unique “precession” motion mode, where the polishing tool orbits around the local normal of the polishing point while also rotating on its axis [15]. Therefore, it is necessary to develop a specialized TIF model for this combined rotation technique in magnetorheological polishing, including the synergistic effects of shear stress and pressure, accurately describing material removal characteristics under tilted and precession polishing [16].
In this study, the stress and velocity distribution in the contact area between MRPF tool head and the workpiece were first simulated using Multiphysics simulation software. TIF models, incorporating multiple influence coefficients, were developed based on velocity characteristics in tilted polishing, discrete precession, and continuous precession modes. Then, through a series of polishing experiments, morphological data of polishing spots under the same parameters were obtained. The optimal coefficients were then identified using a least fit error algorithm, fine-tuning the TIF model to closely match actual material removal characteristics. Additionally, it was demonstrated that the TIF in the precession mode exhibits lower surface roughness and a more uniform texture.
2. MRPF experimental setup
To analyze the removal function generated by MRPF, it is necessary to collect polishing spots from both tilted and precession polishing through a fixed-point polishing experiment. Figure 1 illustrates the self-developed MRPF machine tool, which primarily comprises two key systems: the MR fluid control and circulation system, and the mechanical movement system.
Fig. 1. Polishing experiment setup. (a) Machine tool movement; (b) Procession structure; (c) Polishing process.
As shown in Fig. 1(a), The MR fluid circulation system is designed to ensure the continuous renewal of MR fluids during the polishing process. This system comprises several key components: a centrifugal pump and a peristaltic pump, which are responsible for the efficient movement of the fluid. Nozzles are included to direct the flow of the fluid precisely where it is needed. Specialty recover plays a crucial role in the system, as they allow for the recovery and reprocessing of the MR fluid, ensuring its efficient reuse. Transfer hoses connect these components, facilitating the smooth and controlled transfer of the fluid throughout the system. This integrated approach allows for the effective management of MR fluids, maintaining their quality and consistency during the polishing process.
The mechanical motion system of the machine includes three linear axes and two rotary axes. Due to the intersection of the A-B-H axes at a virtual center, rotating the AB axis essentially keeps the position of the polishing head unchanged but alters its orientation, thereby creating a precession angle, as shown in Fig. 1(b). This design allows for the precise control of the polishing point (contact area) position solely through the movement of the X, Y, and Z linear axes. Therefore, in the tilted polishing experiment, it is only necessary to rotate the B axis to achieve the desired precession angle. The precession polishing is then completed by rotating the A axis on this basis. The accuracy of the machine tool is shown in Table 1.
Table 1. Accuracy parameters of MRPF machine
Inside the polishing head used in the experiments, there is an excitation device whose function is to establish a gradient magnetic field in the polishing area. This magnetic field magnetizes the magnetic particles in the magnetorheological fluid, forming a flexible ribbon that effectively captures abrasive particles, thus removing material from the surface of the workpiece, as shown in Fig. 1(c). The magnetorheological fluid is composed of deionized water, cellulose, cerium oxide abrasive particles, and carbonyl iron powder, with their respective mass fractions being 28.62%, 0.11%, 5.30%, and 65.97%. The average particle size of the carbonyl iron powder is 8 micrometers, and the average particle size of the abrasive particles is 10 micrometers.
3. Establishment of simulation model in polishing area
3.1 Finite element simulation model establishment
Finite element analysis (FEA) with COMSOL Multiphysics’ magnetic fields and laminar flow modules enables detailed simulation of complex phenomena in MRPF A 1:1 scale model of the polishing head, including both its exterior shell and internal magnetic field generator, was created. The polishing head's axis is angled at 30° to the workpiece surface normal, as shown in Fig. 2(a). Experimental measurements determined that a MR fluid ribbon, 2.47 mm thick, covers the exterior of the polishing head. As the head rotates, the ribbon circulates through the magnetic leakage gap and makes contact with the workpiece, sized 40 × 20 × 2 mm, with its flow direction shown in Fig. 2(b). The polishing area is contained within a 240 mm diameter air domain. The simulation mirrored the polishing experiments, with a 1.2 mm polishing gap and a 400-rpm rotation speed. The MR fluid's flow characteristics are described by the Carreau model, detailed in Eq. (1). This model precisely captures the magnetorheological fluids’ nonlinear viscosity over a wide range of shear rates, enabling detailed simulation of the fluid dynamics and magnetic interactions essential to MR polishing.
Fig. 2. Polishing simulation. (a) Finite element simulation model on XZ direction, (b) on YZ direction, (c) Mesh division, (d) Simulation result.
Parameters including ηi (infinite shear rate viscosity, Pa·s), η0 (zero shear viscosity, Pa·s), γ (shear rate, in s^-1), λ (time parameter, s), and n (a constant), were calibrated using rheometry data [17–23]. The MR fluid's magnetization characteristics are based on its measured B-H curve. The model for the polishing head's magnetic induction intensity uses the residual magnetic flux density, as outlined in Eq. (2).
In this context, μ0 signifies vacuum magnetic permeability (H/m), μrec denotes recoil permeability, H represents magnetic field strength (A/m), and Br is the residual magnetic flux density (T). The model shown in Fig. 2(c) employs tetrahedral elements for mesh division, particularly refining the mesh in the ribbon and removal areas, achieving a minimum mesh size of 10 micrometers. The steady-state solver in COMSOL was used to calculate the magnetic field distribution and magnetorheological fluid characteristics. The transient solver can analyze the effects of polishing time and speed increase on the stress and velocity distribution in the contact area, as shown in Fig. 2(d).
3.2 Results of stress field and velocity field in polishing area
Research shows that shear force, hydrodynamic pressure, magnetization pressure, and velocity distribution influence material removal rates in MRF [24]. Figure 3 displays simulation results for these four factors in the polishing zone between the ribbon and the workpiece. Figure 3(a) illustrates significant variation in hydrodynamic pressure along the magnetorheological fluid's flow direction. To the left of the centerline, magnetorheological fluid flow causes a sharp decrease in the gap between the polishing head and the workpiece surface. This squeezing action generates a maximum hydrodynamic pressure of 115 kPa on the workpiece surface. On the right side of the centerline, after flowing out of the polishing zone's center, the gap between the polishing head and the workpiece sharply increases. This leads to the formation of a vacuum zone and negative hydrodynamic pressure. When negative pressure occurs, surrounding MR fluid is drawn in to offset it. Thus, cavitation effects, which could disrupt the magnetorheological fluid's stable flow, do not occur in the actual polishing process.
Fig. 3. Polishing Simulation results. (a) Dynamic pressure, (b) Shear stress, (c) Magnetization pressure, (d) Polishing velocity.
Figure 3(b) shows that shear stress distribution forms an axisymmetric pattern around the projection line. The maximum shear stress at the polishing zone's center is 261 kPa. Figure 3(c) indicates an axisymmetric magnetization pressure distribution, with the maximum value near the projection line at the center. The maximum magnetization pressure is only 1.34 × 10^−4 kPa, which is six orders of magnitude lower than the maximum hydrodynamic pressure and shear stress, and can be neglected in simulation calculations.
4. Establishment of TIF model of MRPF
4.1 Modeling of TIF in tilted polishing mode
The TIF defines material removal properties as the distribution of material removal by the polishing tool at a fixed point per unit time, evaluated by area, peak removal rate (PRR), and volume removal rate (VRR) [25]. Ensuring a stable and predictable TIF is vital for the machining process's success [26]. Theoretically, the polishing process accumulates material removal as the tool moves across the workpiece surface. Thus, viewed macroscopically, Eq. (3) describes this process, where material removal rate is directly proportional to polishing pressure and relative velocity.
where dz(x,y) is the removal depth at the contact point (x,y), k is the Preston coefficient, P(x,y) is the pressure distribution, and V(x,y) is the relative velocity between the polishing wheel and the component.This study introduces a TIF model based on Eq. (3), accounting for the synergistic effects of pressure and shear stress as elaborated in Eq. (4). As previously stated, magnetization pressure from MRPF is negligible, so consideration is given only to fluid dynamic pressure and shear stress. Given the ambiguity surrounding the precise contributions of velocity, shear force, and fluid dynamic pressure to the material removal rate, Eq. (4) introduces the influence coefficients α, β, and σ to quantify these effects.
Shear stress (τ), polishing velocity (Vt), and fluid dynamic pressure (Pd) are calculated using finite element analysis. The influence coefficients α, β, and σ are then optimized using a parameter scanning iterative method (refer to Section 5.2). Fine-tuning these coefficients through multiple iterations is essential for achieving the targeted removal profile. This approach significantly enhances the precision and quantification of the effects of various factors in the polishing process, thereby improving predictability [24].
4.2 Modeling of TIF in continuous precession mode
In continuous precession mode, the polishing head rotates around the main spindle axis (ωS) and revolves around the contact area's local normal (ωa), generating linear velocities Vt and Va, as depicted in Fig. 4. Consequently, the velocity in the contact area is the sum of Vt and Va.
Geometrically, Va is expressed as:
The removal function for this mode is represented as:
4.3 Modeling of TIF in discrete precession mode
While continuous precession produces irregular surface textures, its high A-axis motor speed requirements have made discrete precession a more common alternative. discrete precession turns the A-axis's continuous rotation into stops at several symmetric positions within its rotation circle, typically 4, 6, or more. For instance, in the process known as “four-steps discrete precession,” there are four stopping positions, as shown in Fig. 5(a). This method produces a relatively chaotic texture effect within the core contact area due to the superposition of regular textures in four directions, as depicted in Fig. 5(b).
Fig. 5. Schematic diagram of four-step discrete precession. (a) Tool head pose distribution, (b) Texture in contact zone.
Discrete precession comprises tilted polishing from various positions, allowing its removal function to be built by superposing tilted polishing removal functions. In N-step discrete precession polishing, each step positions the polishing head at a consistent rotational angle difference, leading to the following velocity expression for each step:
Δi denotes the rotation direction matrix for each step. Given an initial velocity direction angle of 0 degrees, the velocity direction angle θi for the ith step can be expressed as:
The removal function for this mode is represented as:
5. Results and discussion
5.1 Influence of coefficient on surface topography of TIF polishing spot
To explore how the coefficients α, β, and σ affect the TIF morphology, simulations were conducted to generate two-dimensional profiles of the polishing spots for varying coefficients. In each simulation, a single coefficient was varied, with the remaining two fixed at 1. Considering the impact of varying coefficients on material removal depth, the two-dimensional profiles were normalized. Cross-sectional profiles through the deepest points were then extracted, as illustrated in Fig. 6 to 8.
Fig. 6. (a) Normalized surface topography of TIF polishing spot (Changing dynamic pressure coefficient α, β=σ=1), (b) corresponding cross-section profiles.
Figure 6 displays the normalized surface topography of the TIF polishing spot for varying dynamic pressure coefficients. With equal contributions from flow pressure and shear stress (α=β=1), the TIF profile's deepest point is slightly left of center. As α decreases, the deepest point shifts rightward, and the profile becomes steeper to the right of this point. At α=1/2, the profiles become symmetric on both sides of the deepest point. At α=1/8, the removal function's tail becomes extremely steep, highlighting dynamic pressure's significant impact on material removal. Figure 7 shows the normalized surface topography of the TIF polishing spot with varying shear stress coefficients. With increasing β, the profile's width remains largely unchanged, the deepest point shifts slightly rightward, and the slopes on either side of this point remain consistent. Figure 8 presents the normalized surface topography of the TIF polishing spot for different polishing velocity coefficients (σ). As σ decreases, the TIF profile's length significantly decreases, and the profiles steepen on both sides of the deepest point. Furthermore, the rightward shift of the deepest point gradually increases.
Fig. 7. (a) Normalized surface topography of TIF polishing spot (Changing shear stress coefficient β, α=σ=1), (b) corresponding cross-section profiles.
Fig. 8. (a) Normalized surface topography of TIF polishing spot (Changing polishing velocity coefficient σ, α=β=1), (b) corresponding cross-section profiles.
Furthermore, the rightward shift of the deepest point gradually increases. The analysis reveals that the two-dimensional profile shape is more sensitive to variations in α and σ, with β changes having a lesser impact. However, despite these observed influences, determining the specific values of the three coefficients remains challenging. Consequently, identifying the error between the actual and simulated removal function profiles is essential to determining optimal coefficients that minimize this discrepancy.
5.2 Establishment and verification of TIF model
The analysis indicates significant changes in fluid dynamic pressure and shear stress along the MR fluid flow direction, with flow rate also impacting the material removal process. Consequently, the cross-sectional profile of the polishing spot in the flow direction of the MR fluid most accurately represents material removal characteristics during polishing. Analysis of the cross-sectional profile variation enables investigation into the material removal morphology, leading to the proposal of four shape parameters to characterize this profile, as depicted in Fig. 9(a). HW represents the full width at half maximum of the TIF (FWHM, mm), Md denotes the maximum removal depth (μm), while Fs and Rs correspond to the tangent and front-rear slopes at the half-height width, respectively.
Fig. 9. (a) Schematic diagram of cross-sectional profile shape parameters, (b) The algorithm's flowchart.
To optimize the TIF simulation cross-sectional profile and minimize its discrepancy with the experimental TIF profile, we propose an iterative algorithm to precisely adjust the model's influence factors. The algorithm's flowchart is depicted in Fig. 9(b), where initial values are set based on prior experience: α=1/2, β=1, σ=1. Subsequently, using process parameters like rotational speed, working clearance, and polishing time for the simulation, we compute the distributions of Pd, τ, and V and substitute them into Eq. (2) to obtain Mdsim. Following this, polishing experiments using identical process parameters are conducted to measure Mdexp, compute a new k-value, and compare the post-end slope (Rssim) with the experimental rear-end slope (Rsexp). If the fitting error (η) exceeds acceptable limits, α is gradually reduced until the error is below η=1e−2. This method is similarly applied to calculate the fitting errors for the front-rear slope (Fssim) and FWHM (HWsim), continuing adjustments until errors fall below η, at which point the values of α, β, and σ are output.
5.3 Experimental and simulation results of TIF in tilted polishing mode
Figure 10(a) displays the experimental results of the polishing spot, with the process parameters listed in Table 2. The iterative algorithm (see Fig. 9(b)) yielded optimal coefficients α=0.14, β=1, σ=2.5, enabling the construction of the TIF model:
The three-dimensional surface topography of the TIF, derived from Eq. (11) and illustrated in Fig. 10(b), closely replicates the actual spot's slightly curved D-shaped morphology. In both the simulation and the actual spot, an increase in removal depth and width is observed along the magnetorheological fluid's flow direction, followed by a sharp decrease in removal depth towards the end. Figure 10(c) displays the experimental and simulated cross-sectional views of the MR fluid along the flow direction, with the cross-sectional curve sequentially passing through the foremost part, the deepest point, and the midpoint of the tail of the TIF. The curves demonstrate a significant congruence between the theoretical model and experimental outcomes. However, minor fluctuations at the TIF profile's tail end were observed in the experimental results, mainly attributed to ribbon fluctuations from the equipment's circulation system instabilities. The ribbon fluctuation is a common phenomenon, which may be caused by fluctuations in the magnetic field on the surface of the polishing wheel or by fluctuations in the magnetorheological fluid due to instability in the centrifugal pump [27].
Fig. 10. (a) Measured (b) Simulated TIF surface topography of polished spot in single step mode, (c) Experimental and simulated cross-sectional profiles.
Table 2. Processing parameters for experiment
5.4 Experimental and simulation results of TIF in precession mode
Under the same process parameters, by employing a discrete precession method, the A-axis dwells in 8 directions, rotating 45° each time to complete a fixed-point polishing experiment with rotational superposition. The superimposed TIF depicted in Fig. 11(a) forms the foundation for the TIF model constructed below:
Fig. 11. (a) Measured (b) Simulated TIF surface topography of polished spot in separated step precession mode, (c) Experimental and simulated cross-sectional profiles.
Figure 11(b) displays the 3D surface topography of polished spot calculated with Eq. (12), and Fig. 11(c) shows the cross-sectional profiles between diagonals. The TIF's three-dimensional profile closely matches the experimentally measured profile, exhibiting rotational symmetry around its center. The deepest point is at the center, resembling an eight-pointed star in shape. The angles between adjacent corners correspond to the A-axis's rotation angle for each move, suggesting the TIF's shape results from the precession motion. Besides, the TIF features two distinct types of removal areas. One type is a deeper removal area at the TIF center (defined as ‘zone 1’), while the other encompasses shallower, elongated areas at the 8 outer corners (defined as ‘zone 2’). These two TIF characteristics stem from two factors: first, during discrete precession, the TIF's rotation center, being at the deepest material removal point, leads to a higher removal rate near the center. Second, the discontinuity in TIF superposition during discrete precession means peripheral regions have less cumulative material removal, and some areas between the eight-pointed star's points may have no removal. Therefore, adopting a continuous precession mode could uniformize the TIF.
A continuous precession polishing experiment was conducted by uniformly rotating the A-axis from 0° to 360° under identical process parameters. Figure 12(a) displays the polishing spot's experimental data, followed by an outline of the constructed TIF model:
Figure 12(b) reveals the predicted 3D surface morphology, showing good consistency with the experimental result, depicting quasi-Gaussian type TIF. In comparison, the continuous precession mode yields a more uniform material removal across the TIF than the discrete precession mode, leaving no unpolished regions. The theoretical model's validity was confirmed by the X-Z and Y-Z profile curves’ measurement data at the deepest point, as illustrated in Fig. 12(c) and (d). The results show a correlation between the theoretical model and the experimental outcomes. However, discrepancies remain between the simulation and experimental results. The Y-Z experimental profile curve's asymmetry indicates a positional discrepancy between the rotation center and the TIF's deepest point. This discrepancy is due to geometric errors in the rotation axis. Theoretically, all rotation axes should intersect at a spatial virtual pivot (VP) and coincide with the center of the tool head. However, positional errors during assembly can result in minor shifts in the TIF's rotational superposition.
Fig. 12. (a) Measured (b) Simulated TIF surface topography of polished spot in separated step precession mode, (c) X-Z and (d) Y-Z cross-section profiles.
5.5 Micro-topography within the TIFs under different modes
The TIF profile typically spans low, medium, and high spatial frequencies. During surface form correction of optical components, the TIF's shape profile, low-frequency information, is used in convolution calculations. Numerous studies have indicated that the uniformity and randomness of the micro textures (high-frequency information) of the Tool Influence Function (TIF) significantly affect the microstructure and roughness of the surface after polishing [28–30]. This study employed a white-light interferometer to assess the local micro-topography of the TIF in tilted polishing and precession modes. In the tilted polishing mode, the TIF area exhibited distinct directional “plowing” textures, with a surface roughness reaching Sa 1.1797nm, as shown in Fig. 13(a). Under the continuous precession mode, the texture of the TIF appeared woven, with a more uniform distribution across the entire measured area, reducing the surface roughness to Sa 0.6357 nm, significantly lower than in the tilted polishing mode, as shown in Fig. 13(b). This demonstrates that the internal texture generated by the TIF in the precession mode is more uniform and significantly reduces surface roughness, forming a stark contrast with the tilted polishing mode.
Fig. 13. Measured surface micro-topography within TIFs under BRUKER Npflex (a) TIF in tilted polishing mode, (b) TIF in continuous precession mode.
6. Conclusion
This research introduced a new removal function model to clarify MRPF's material removal mechanism and quantitatively confirmed its accuracy and usability through experiments. It not only advances the understanding of MRPF but also sets a precedent for the development of more efficient and precise optical surface treatments.
MRPF material removal results from the combined action of dynamic pressure and shear stress, offering more accurate TIF predictions than models considering only shear or pressure. Influence coefficients for shear stress, dynamic pressure, and polishing speed are respectively 1, 0.14, and 2.5, highlighting the greater impact of shear stress and polishing speed on material removal. In tilted polishing mode, MRPF generates a D-shaped TIF, akin to conventional MRF, because fluid dynamic pressure is concentrated on one side of the polishing zone's center. This finding opens new avenues for exploring the removal mechanisms of other MRF methods. The precession mode enables superimposing the TIF into a rotationally symmetrical shape, showcasing MRPF's excellence in shape correction and convergence efficiency, suitable for freeform surface polishing. The TIF generated by MRPF in the precession mode exhibits uniform internal textures and results in lower internal surface roughness compared to the TIF in the tilted polishing mode.
Funding
National Natural Science Foundation of China (52075463); Science and Technology Planning Project of Shenzhen Municipality (JCYJ20210324122001003); Fujian Provincial Department of Science and Technology (2021H6008); Key Research and Development Program of Hunan Province of China (2023GK2069).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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