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Research on the influence of the non-stationary effect of the magnetorheological finishing removal function on mid-frequency errors of optical component surfaces

Bo Wang, Guipeng Tie, Feng Shi, Ci Song, and Shuangpeng Guo

Open Access Open Access

Abstract

With the continuous development of modern optical systems, the demand for full spatial frequency errors of optical components in the system is increasing. Although computer-controlled sub-aperture polishing technology can quickly correct low-frequency errors, this technology significantly worsens the mid-frequency errors on the surface of the component, which greatly inhibits the improvement of optical system performance. Therefore, we conducted in-depth research on the non-stationary effect of the removal function caused by the fluctuation in magnetorheological polishing and their influence on the mid-frequency errors of the component surface. We established a non-stationary profile model of the removal function and applied this model to simulate the distribution of mid-frequency errors on the surface of the processed component, considering the non-stationary effect. The simulation results showed that the non-stationary effect of the removal function weaken the mid-frequency ripple errors but increase other mid-frequency errors. Therefore, we first proposed the optimal single-material removal thickness corresponding to the non-stationary effect and experimentally verified the effectiveness of the optimal material removal thickness in suppressing mid-frequency errors. The experimental results showed that when the magnetorheological finishing single-material removal thickness is set to the optimal value, both the mid-frequency ripple errors and the mid-frequency RMS on the surface significantly decrease. Therefore, this work provides a basis for improving the existing magnetorheological finishing process and effectively suppressing the mid-frequency errors on the surface of processed components. It also provides theoretical and technical support for the magnetorheological processing and manufacturing of high-precision optical components. At the same time, the non-stationary effect and the corresponding analytical models has the potential to be extended to other polishing tools.

© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

With the continuous development of modern optical systems, the accuracy requirements for optical components used in high-power laser systems [1,2], ultraviolet lithography systems [3], and astronomical observation systems [4] have been constantly increasing. There is even a high demand for full spatial frequency errors [5–6]. The current method to achieve high-precision surfaces is commonly computer-controlled sub-aperture polishing. However, the size of the removal function in this technique is usually much smaller than the size of the workpiece. Under conventional scanning strategies, the removal function has a fixed line spacing feed motion in the direction perpendicular to the scanning motion. This fixed line spacing feed motion causes regular convolutional residual errors [7] (with a spatial errors period typically between 0.03 mm-1 and 8 mm-1 [8]), which are referred to as mid-frequency ripple errors (the value of its spatial frequency is the inverse of the fixed line spacing). Mid-frequency errors result in increased beam modulation, reduced image contrast, and even nonlinear self-focusing, severely degrading the performance of optical systems [5,7,9].

Magnetorheological finishing (MRF) is a deterministic and flexible sub-aperture polishing technique. By manipulating the rheological properties of the magnetorheological fluid flowing on the polishing wheel surface through a magnetic field, it enables the formation of a flexible polishing ribbon at the bottom of the polishing wheel to polish optical components [10]. MRF technology can achieve deterministic material removal, high surface shape accuracy, and low-defect surface quality of optical elements, making it widely used in the shaping and damage repair processes of high-precision optical components [11,12]. However, due to the fact that the removal function of MRF is much smaller than the outer dimensions of the processed components and the presence of non-stationary effect (the polishing ribbon is not stable but exhibits fluctuation), this leads to spatial variations in the removal function (non-stationary effect) [13]. The spatial variation of the removal function falls within the range of mid-frequency errors. Therefore, while rapidly repairing low-frequency surface shape errors and surface/subsurface damage of the components, in addition to introducing mid-frequency ripple errors, it usually introduces more small-scale manufacturing errors in the mid-frequency range [13,14]. Furthermore, once the mid-spatial frequency (MSF) errors are generated, they are difficult to be completely eliminated. Therefore, suppressing mid-frequency errors on the surface of optical components after sub-aperture polishing is of great significance [15].

In order to address the aforementioned issue, researchers have conducted extensive studies on the suppression of mid-frequency errors after magnetorheological finishing. They have identified that the main sources of mid-frequency errors generated during the polishing process are: the machining process, the regular polishing path, and the shape and stability of the removal function [15].

Regarding the research on machining processes, Shi et al. proposed a high-precision, low-defect combination process flow of ultra-precision grinding, CCOS (Computer Controlled Optical Surfacing) polishing, magnetorheological finishing, and ion beam cleaning. This combination process optimizes the surface shape accuracy, removes fractured defects, and suppresses mid-frequency errors [16]. Yan et al. conducted magnetorheological finishing experiments to suppress mid-frequency errors on a flat optical element by optimizing the MRF processing parameters and selecting appropriate processing strategies. After three iterations of processing, the mid-frequency PSD1 errors converged from 5.57 nm to 1.36 nm, and PSD2 changed from 0.95 nm to 0.88 nm, achieving convergence control of mid-frequency errors on large-aperture flat optical components through MRF processing [17]. Xu et al. studied the influence of MRF processing on mid-frequency errors and found that the tool path spacing, instability of the polishing spot, and removal depth significantly affect the mid-frequency errors on the surface of the component after MRF [18]. Hou et al. investigated the influencing factors of mid-frequency errors in MRF processing and explored the relationship between processing spacing, removal depth, and mid-frequency errors. They were able to modulate the mid-frequency errors on the surface of processed components by changing the processing spacing and material removal thickness [19].

In order to suppress periodic mid-frequency ripple errors on optical surfaces during magnetorheological finishing, Dunn et al. [20] developed a random path planning algorithm that avoids periodic trajectories. This algorithm effectively improves mid-frequency ripple errors. Hu et al. [21] proposed a step-adaptive Archimedean path that adjusts the step size of the path based on the material removal thickness to reduce the regular mid-frequency patterns on the surface of the component. Yang et al. [22] introduced the random pitch method to suppress the generation of mid-frequency errors in magnetorheological polishing. They theoretically analyzed the mechanism of mid-frequency errors generation in fixed pitch magnetorheological polishing and verified the effectiveness of the random pitch method in suppressing mid-frequency errors through simulations. Wang et al. [23] proposed a maze path that increases randomness while ensuring a uniform distribution of trajectories, thereby avoiding periodic surface errors.

Regarding the influence of removal function morphology and stability on mid-frequency errors, Pan et al. investigated the impact range of spatial frequencies on polishing methods, removal function size and shape, and analyzed the correction capability of different polishing methods for mid-frequency errors [24]. Zhou et al. studied the removal function and its corresponding spectrum in CCOS polishing, and found that both the size and shape of the removal function would affect the range of correctable frequency errors [25]. Zhang et al. used a dual-rotation magnetorheological polishing tool to generate Gaussian and W-shaped removal functions and estimated their power spectral density (PSD) in the frequency domain using 2D FFT. They compared the mid-frequency errors introduced by these two types of removal functions [26]. Liu et al. proposed a characteristic length coefficient for magnetorheological polishing removal functions to characterize their impact on mid-frequency ripple errors. Uniform removal experiments showed that reasonable characteristic length coefficients and step sizes can significantly suppress the generation of ripple errors [27]. Wan et al. used the Piecewise Path Convolution (PPC) method to study the characteristics and mechanisms of mid-frequency errors and demonstrated the coupling relationship between path type, removal function, and mid-frequency errors through filtering theory [28]. Based on this, Wan et al. proposed a new form of magnetorheological polishing tool that utilizes specific path angles and step sizes. With a bandwidth of only a few tens of micrometers (the “magic” angle step), it is possible to achieve a surface without noticeable path ripples stably, without affecting the convergence of other frequency errors [29].

Researchers have made rapid progress in suppressing mid-frequency errors in magnetorheological finishing and have developed a relatively complete processing system. However, previous studies have mainly focused on optimizing the path, processing techniques, and morphology of the removal function to suppress mid-frequency errors. Researchers have generally assumed that the removal function remains stable. However, according to our previous study, we found that there is a non-stationary effect of the removal function [13], which affects the mid-frequency errors distribution on the surface of the processed component. Therefore, in this study, we conducted an in-depth investigation into the impact of the non-stationary effect of the removal function caused by the fluctuation in magnetorheological polishing ribbon and their influence on the mid-frequency errors of the component surface. We established a non-stationary profile model of the removal function and applied this model to simulate the distribution of mid-frequency errors on the surface of the processed component, considering the non-stationary effect. We proposed the optimal single-material removal thickness corresponding to the non-stationary effect and experimentally verified the effectiveness of the optimal material removal thickness in suppressing mid-frequency errors. This work provides a basis for the process of magnetorheological finishing and the determination of the single-material removal thickness, effectively suppressing mid-frequency errors on the surface of the processed component, and providing theoretical and technical support for the magnetorheological finishing process and manufacturing of high-precision optical components.

2. Theoretical analysis

2.1 Remove function non-stationary profile model

The removal function is formed by the interaction between the magnetorheological polishing ribbon and the workpiece. Therefore, the state of the removal function depends on the morphological characteristics of the polishing ribbon, as shown in Fig. 1. Existing magnetorheological finishing theories assume that the removal function is constant during the manufacturing process. Experimental methods for measuring the actual removal thickness of materials are conducted in second-level time spans, ignoring the fluctuations at the millisecond level.

 figure: Fig. 1.

Fig. 1. Schematic diagram of removal function formation

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During the actual machining process, the polishing ribbon exhibits fluctuation, which result in dynamic fluctuation of the removal function, known as non-stationary effect. In our previous studies, in order to investigate the non-stationary effect of the removal function, real-time monitoring of the profile of the polishing ribbon during the polishing process was conducted. Polishing ribbon profile is tested by the magnetorheological ribbon data acquisition and analysis system as shown in Fig. 2. The principle is to use machine vision and linear array structured light scanning to measure the cross-sectional profile of the ribbon on the polishing wheel and collect the spectral information of the ribbon. The spectrum analysis of the profile changes in the ribbon cross-section revealed several dominant frequency peaks, corresponding to the causes of ribbon fluctuation, such as pulsations from the circulation system pump (related to the rotational frequency of the pump impeller), fluctuation from the polishing wheel (related to the rotational frequency of the wheel), and magnetic field fluctuation as shown in Fig. 2. The magnitude of these fluctuation ranges from 50µm to 200µm under different polishing parameters [13]. Typically, during magnetorheological finishing processing, the immersion depth of the ribbon into the surface of the component is between 0.15 mm and 0.25 mm. Therefore, the existence of fluctuations at this order of magnitude significantly affects the profile of the removal function, making it time-varying. Hence, it is necessary to establish a more accurate non-stationary model for the profile variations of the removal function.

 figure: Fig. 2.

Fig. 2. Schematic and spectral analysis of magnetorheological polishing ribbon test

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Firstly, assuming the ribbon shape is time-invariant, when the magnetorheological fluid flows into the magnetic field, it can form a polishing ribbon composed of iron powder, polishing particles, and base carrier liquid. The ribbon transforms into a Bingham fluid and becomes incompressible. It will diffuse and expand into shapes as shown in Fig. 3(a) and (b). We model its profile as shown in Fig. 3(c) and (d). The profile of the removal function can be simplified as the long axis L and the short axis N.

$${(R + D)^2} - {(R + D - H)^2} = {x_0}^2$$
x1 can be solved by the following integral equation:
$$\left\{ {\begin{array}{{c}} {Z = R + D - H - \sqrt {{{(R + D)}^2} - {x^2}} }\\ {\int_{{x_1}}^{{x_0}} {Zdx ={-} 2\int_{{x_0}}^0 {Zdx} } } \end{array}} \right.$$

 figure: Fig. 3.

Fig. 3. Schematic of profile modeling of magnetorheological polishing removal function. (a) Radial schematic diagram of the polishing wheel. (b) Axial schematic diagram of the polishing wheel. (c) Radial profile modeling of removal function. (d) Axial profile modeling of removal function. (e) Radial profile modeling under the non-stationary effect of removal function. (f) Axial profile modeling under the non-stationary effect of removal function.

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Thus the long axis L of the removal function can be calculated by Eq. (3)

$$L = |{{x_0}} |+ |{{x_1}} |$$
 The calculation method for the short axis N of the removal function is consistent with the method mentioned above:
$$N = 2 \times |{{y_1}} |$$

In the actual process, the thickness of the ribbon continuously changes as shown in Fig. 3(d) and (e). If we denote the fluctuation magnitude of the ribbon as ΔD, then the long axis Lv of the removal function at the same immersion depth H can be calculated by the following equation:

$$\left\{ {\begin{array}{{c}} {{{(R + D + \Delta D)}^2} - {{(R + D - H)}^2} = {x_0}^2}\\ {Z = R + D - H - \sqrt {{{(R + D + \Delta D)}^2} - {x^2}} }\\ {\int_{{x_2}}^{{x_0}} {Zdx ={-} 2\int_{{x_0}}^0 {Zdx} } } \end{array}} \right.$$
$${L_v} = L + \Delta L = |{{x_0}} |+ |{{x_2}} |$$

The calculation method for the short axis Nv of the removal function is consistent with the method mentioned above:

$${N_v} = N + \Delta N = 2 \times |{{y_2}} |$$

According to the above formula, when the polishing wheel rotation speed, magnetorheological fluid flow rate and viscosity, and initial immersion depth remain unchanged during the polishing process, the percentage change in the profile of the removal function can be solved by the following equation. Additionally, the profile variation relative to the unchanged removal function can be obtained based on the measured values of the ribbon fluctuation.

$$\left\{ {\begin{array}{{c}} {l = \frac{{\Delta L}}{L}}\\ {n = \frac{{\Delta N}}{N}} \end{array}} \right.$$

2.2 Analysis of the impact of non-stationary effect on frequency domain errors

During the actual magnetorheological polishing process, residual mid-frequency ripple errors are usually generated. The reason is that during the actual machining process, a certain-sized removal function moves along a determined trajectory with a fixed line spacing. The removal function has a material removal effect outside of the motion trajectory, resulting in periodic residual errors caused by the convolution effect of the regular path. Therefore, the regular path becomes an important source of mid-frequency errors. In addition, the non-stationary effect of the removal function will cause changes in the spatial scale of the removal function, and this scale falls within the range of mid-frequency errors, thereby affecting the mid-frequency errors, including mid-frequency ripple errors on the surface, as shown in Fig. 4.

 figure: Fig. 4.

Fig. 4. Schematic diagram of the impact of non-stationary removal function on mid-frequency errors (a) Magnetorheological polishing grating path diagram (b) Residual mid-frequency errors diagram after stationary removal function processing (c) Residual mid-frequency errors diagram after non-stationary removal function processing.

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Assuming that the one-dimensional graph of the removal function in the raster scanning direction is represented by f(x), its frequency domain representation is denoted as F(k). During the process of superimposing with a fixed line spacing of d along the line feed direction, the generated one-dimensional graph is denoted as g(x), and its function representation is as follows:

$$g(x) = \sum\limits_{n ={-} \infty }^\infty {f(x - nd)}$$

Its spectrum can be expressed as:

$$G(k) = \frac{1}{d}\sum\limits_{n ={-} \infty }^\infty {F(\frac{k}{d} - n)}$$

Due to the non-periodicity of f(x), F(k) is continuous in the frequency spectrum, and there are no obvious peaks. It is evident that G(x) exhibits a peak at k = 1/d. Therefore, there is periodic mid-frequency ripple errors at point x = d.

In the actual processing. There is a non-stationary effect in the removal function. The function corresponding to the one-dimensional graph of the above removal function is f(x) which is continuously scaled along line feed direction change, then its function is expressed as f(ax), where is the non-stationary factor a. Therefore, the corresponding spectrum is:

$$g(x) = \sum\limits_{n ={-} \infty }^\infty {f(ax - nd)}$$
$$G(k) = \frac{1}{{ad}}\sum\limits_{n ={-} \infty }^\infty {F(\frac{k}{d} - \frac{n}{a})}$$

Since a is time-varying with the non-stationary effect of the removal function, the peak of G(x) at this time is no longer concentrated at k = 1/d, but scattered at different frequencies with the variation of a. As a result, the regular mid-frequency ripple errors is weakened or even disappears, which also explains the unexplained disappearance of mid-frequency ripple in the study by wan et al. [29].

In the scanning direction, a similar phenomenon as described above exists. The variation of will introduce new mid-frequency errors in the mid-frequency range. Combining the above analysis, the non-stationary effect of the removal function will not only weaken the mid-frequency ripple errors but also introduce new mid-frequency errors on the surface after the machining process. The mid-frequency ripple errors are proportional to the material removal thickness in magnetorheological polishing [17]. Therefore, corresponding to different removal function non-stationary effect (different magnetorheological polishing ribbon fluctuation), there will be a range of removal thickness that allow the mid-frequency ripple to be suppressed without deteriorating the overall mid-frequency band errors on the component surface.

3. Simulation analysis

3.1 Simulation analysis of the impact of non-stationary effect on frequency domain errors

We use MATLAB simulation analysis, and use the experimentally obtained removal function to remove a larger thickness of material removal at a single time. In the experimental process of extracting the removal function, we use a 200 mm diameter magnetorheological finishing device. In terms of the finishing parameters, the rotation speed of the polishing wheel is 200 rpm, the flow rate is 125Lph, the viscosity is 220Pa·s and the depth of immersion of the polishing ribbon is 0.2 mm. In this condition, the length of the long axis of the removal function is 11.0 mm and the width of the short axis is 7.1 mm. According to the principle of a single variable, respectively to control its long-axis, short-axis profile and material removal rate of the percentage change of 10% (at this time, the corresponding ribbon fluctuation of roughly 100µm). The simulation of magnetorheological finishing uses raster scanning machining method. The line spacing during machining process is 0.6 mm. During the simulation, a sine signal is used to simulate the periodic change of the nonstationary effect of the removal function. The rate of change to the amplitude is 10% and the frequency is 21.99 Hz (the main frequency obtained from the spectrum analysis of the magnetorheological polishing ribbon test in Fig. 2). The simulation result is shown in Fig. 5.

 figure: Fig. 5.

Fig. 5. Simulated results of the surface shape and mid-frequency PSD curve when removing 10µm of material in a single magnetorheological polishing process. (a) Removal function without fluctuation. (b) Removal function width fluctuation. (c) Removal function length fluctuation. (d) Removal function removal efficiency fluctuation. (e) Mid-frequency PSD curve in the scanning direction. (f) Mid-frequency PSD curve in the line feed direction.

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Figure 5 Simulated results of the surface shape and mid-frequency PSD curve when removing 10µm of material in a single magnetorheological polishing process

As shown in Fig. 5(b), the variation in the long axis profile is more effective in suppressing mid-frequency ripple, reducing the ripple amplitude by more than half and even reducing the value of the mid-frequency RMS on the processed component surface. However, it has a certain deteriorating effect on the low-frequency range and other mid-frequency ranges. The variation in the short axis introduces new low-frequency and mid-frequency errors and has little inhibitory effect on the mid-frequency ripples, as shown in Fig. 5(c). The variation in material removal does not suppress the existing mid-frequency ripple and introduces new mid-frequency errors, as shown in Fig. 5(d).

From the mid-frequency PSD curve shown in Fig. 5(e), it can be observed that, in the scanning direction, compared to a stable removal function, fluctuation in the removal function will generate peaks at specific mid-frequencies corresponding to the spacing between scan lines, thus deteriorating the mid-frequency errors. When the width of the removal function fluctuates, the PSD curve in the low-frequency range overlaps with the PSD curve of the stable removal function. Therefore, width fluctuations have little deteriorating effect on the low-frequency range. However, at specific mid-frequencies corresponding to the fluctuation frequency, it shows high sensitivity to changes in width, generating significant characteristic peaks and worsening mid-frequency errors. The fluctuation in the length of the removal function is similar to the width fluctuation. The fluctuation in the removal efficiency of the removal function tends to smooth out small peaks in certain mid-frequency ranges, contributing to the homogenization of mid-frequency errors. However, it also generates large characteristic peaks at specific mid-frequencies corresponding to the fluctuation frequency. This is because in the scanning direction, due to the assumption that the removal efficiency fluctuation is a sine transformation, it has periodic characteristics. Therefore, there will be characteristic peak values in the PSD curve in the scanning direction, which to some extent affects mid-frequency errors. But the scale of the mid-frequency errors mentioned above is small, and in actual machining processes, there is not just one frequency range of fluctuations. Therefore, the peaks will be dispersed among various mid-frequency ranges, so there are no characteristic peaks in the mid-frequency range along the scanning direction during actual machining.

As shown in Fig. 5(f), in the line feed direction, due to the regular grating scanning path, significant characteristic peaks (mid-frequency ripple errors), occur at the frequencies corresponding to the fixed line spacing and its harmonics. According to the derivation of Eq (9), the fluctuation of the removal function efficiency is only multiplied by a coefficient on the removal function f (x), which does not change the spectral characteristics of the surface shape after convolution calculation. According to Eq (12), the mid-frequency ripple error is weakened only when the removal function is scaled along the line feed direction. In addition, since the efficiency fluctuation is periodic, the influence of the fluctuation will be homogenized during the convolution of the removal function along the path. Therefore, the efficiency fluctuation curve in Fig. 5 (f) is almost identical to the curve without efficiency fluctuation, causing minimal impact on the frequency range in this direction. It neither improves nor significantly deteriorates the mid-frequency errors in this direction. The fluctuations in the long and short axis of the removal function will significantly affect the low-frequency errors, with the short axis variation being more sensitive. The variation in the short axis has a small inhibitory effect on the mid-frequency ripples but causes the PSD curve of non-ripple frequency ranges to shift upward. On the other hand, the variation in the long axis significantly suppresses mid-frequency ripple errors but also affects errors in other frequency ranges. By sacrificing errors in other frequency ranges near the characteristic frequency, it homogenizes the mid-frequency ripple errors caused by the convolution effect of the removal function due to the regular polishing path. Therefore, the next section will mainly analyze the impact of the variation in the long and short axis of the removal function on mid-frequency errors.

3.2 Correspondence between non-stationary effect and mid-frequency errors

3.2.1 Correspondence between different ribbon fluctuation and mid-frequency errors

According to the theoretical derivation in section 2, the predicted profile changes of removing function corresponding to ribbon fluctuation of 50, 100, 150, 200, and 250µm are shown in Table 1. Taking this result into the simulation algorithm, the influence of ribbon fluctuation on mid-frequency errors is analyzed. During the simulation process, a uniform removal of 100 nm material from the surface is done. In order to avoid introducing significant mid-frequency ripple and considering the positioning accuracy of the machine tool, a line feed spacing of 0.6 mm will be used for both simulation and experiments. In addition, based on the limitation of the lateral resolution of the interferometer in the subsequent experiments, the mid-spatial frequency error range of our subsequent experiments and simulation analysis is 0.033mm-1-5mm-1. The results are shown in Fig. 6.

 figure: Fig. 6.

Fig. 6. Simulation results of the correspondence between different ribbon fluctuation and mid-frequency errors

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Tables Icon

Table 1. Predicted values of changes in the profile of the removal function

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As shown in Fig. 6, with the increase of ribbon fluctuation, the suppression effect on the amplitude of mid-frequency ripple PSD becomes stronger, effectively suppressing mid-frequency ripple errors. However, when the fluctuations increase, it deteriorates other mid-frequency errors on the surface. The ribbon fluctuation of 100µm is the point of maximum RMS value for the mid-frequency range, which is contrary to the common perception that larger fluctuations result in worse mid-frequency RMS on the surface.

3.2.2 Correspondence between different material removal thickness and mid-frequency errors

Based on the simulation analysis in the previous section, different material removal processes can cause a significant increase in mid-frequency ripple. However, due to the non-stationary effect of removal function caused by ribbon fluctuation, there may be differences in the existence of mid-frequency errors with the same fluctuations but different thickness of material removal. Therefore, it is necessary to study the corresponding relationship between different material removal thickness and mid-frequency errors when the fluctuation is the same.

According to the theoretical derivation in section 2, assuming a ribbon fluctuation of 100µm and considering the changes in the long axis and short axis, with a fixed line spacing of 0.6 mm, materials of 50, 100, 150, 200, 250, 300, 350, 400, 450, 600, and 800µm are removed respectively. These values are then input into the simulation algorithm, and the simulation results are shown in Fig. 7.

 figure: Fig. 7.

Fig. 7. Simulation results of the correspondence between different material removal thickness and mid-frequency errors (a) The correspondence between different material removal thickness and mid-frequency RMS (b) The correspondence between different material removal thickness and mid-frequency ripple PSD amplitude

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As shown in Fig. 7(b), under the same ribbon fluctuation, the suppression rate of mid-frequency ripple PSD amplitude decreases linearly with the increase of material removal. The red numbers in Fig. 7(a) represent the ratio of the mid-frequency error under 100µm ribbon fluctuation to the mid-frequency error without fluctuation when the material removal is less than 500 nm, and the growth rate of the mid-frequency error RMS amplitude decreases and then becomes larger. At a single material removal of 250 nm, the mid-frequency error growth rate is the smallest, with a value of 1.44 times that of the removal function in the steady state. Combining the suppression rate of the mid-frequency ripple PSD amplitude and the growth rate of the mid-frequency error caused by the nonstationary effect of the removal function, a single-material removal of 250 nm at 100µm ribbon fluctuation provides the best control of the mid-frequency error.

Therefore, there exists the optimal single-material removal thickness under different removal function non-stationary conditions, which allows the overall mid-frequency RMS on the surface to not deteriorate and effectively suppress mid-frequency ripple errors, as shown in Fig. 8. Therefore, in actual magneto-rheological finishing processes, the method of removing the optimal single-material removal thickness in each operation can be used to control the mid-frequency errors on the surface.

 figure: Fig. 8.

Fig. 8. Schematic of comparison of multiple material removal and single optimal material removal

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4. Experimental verification

4.1 Validation of removal function non-stationary model

To verify the accuracy of the non-stationary removal function contour model in magneto-rheological polishing, an extraction experiment of the removal function was conducted using a self-developed magnetorheological polishing machine tools with a 200 mm diameter polishing wheel in the laboratory, as shown in Fig. 9. During the experiment, in order to ensure the principle of a single variable and keep the magnetic field intensity, polishing wheel speed, magnetorheological fluid flow rate, and viscosity constant, an electromagnetic damping device was used to induce periodic fluctuations in the magnetorheological polishing ribbon. The basic principle of the electromagnetic damping coil is to wind an electromagnetic coil on the outer wall of the magnetorheological liquid guide tube. A sine alternating current is passed through the coil to generate a magnetic field damping along the direction of the guide tube inside the coil. The periodic change of the damping realizes the periodic fluctuation of the polishing ribbon. When no damping was applied, the ribbon fluctuated at 100µm. We allowed it to fluctuate at 150, 200, and 250µm, respectively, to extract the removal function and extract the contour dimensions. The experimental and predicted results are shown in Table 2.

 figure: Fig. 9.

Fig. 9. The MRF machine tools and polishing ribbon

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Tables Icon

Table 2. Predicted and experimental results of changes in the profile of the removal function

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The experimental results show that the errors between the predicted removal function contour and the experimental results is less than 4%. This verifies the accuracy of the non-stationary contour model for the removal function and confirms that it meets the requirements of the aforementioned simulation analysis.

4.2 Influence of removal function non-stationary effect on surface mid-frequency errors

Using magnetorheological finishing, uniform material removal is performed on two fused quartz components with similar mid-frequency errors on their surfaces. The processing parameters are as follows: the polishing wheel speed is 200 rpm, the flow rate is 115Lph, the viscosity is 110 Pa • s, the immersion depth of the polishing ribbon is 0.2 mm, the magnetic field strength is 7A/m, and the grating scanning spacing is 0.6 mm. Under these polishing process parameters, the fluctuation of the magnetorheological polishing ribbon is approximately 100µm. According to the above simulation results, the optimal single-material removal amount is 250 nm. During the experiment, each component is divided into two polishing areas. For the first component, 1µm of material is removed in a single pass on the upper half, and the lower half is divided into four removal steps, with each step removing 250 nm of material. For the second component, 250 nm of material is removed in a single pass on the upper half, and the lower half is divided into three removal steps, removing 100 nm, 100 nm, and 50 nm of material respectively. After processing, the mid-frequency errors on the surfaces of each region (the selected region is a circular region with a diameter of 30 mm) are tested, and the changes in mid-frequency ripple errors and mid-frequency RMS value are analyzed. The changes in the mid-frequency range PSD curve on the surface under different processes are also analyzed, as shown in Fig. 10.

 figure: Fig. 10.

Fig. 10. Experimental results of the influence of non-stationary effect of removal function on surface mid-frequency errors. (a) Surface mid-frequency error after removing 1µm of material. (b) Surface mid-frequency error after removing 4 × 250 nm of material. (c) Surface mid-frequency error after removing 250 nm of material. (d) Surface mid-frequency error after removing 100+100+50 nm of material. (e) Mid-frequency PSD curve in the line feed direction for the first set of experiments. (f) Mid-frequency PSD curve in the line feed direction for the second set of experiments. (g) Mid-frequency PSD curve in the scanning direction for the second set of experiments.

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The experimental results indicate that when 1µm of material is removed in a single step, the surface mid-frequency ripple is more pronounced, with a mid-frequency RMS reaching 2.783 nm. The PSD curve in Fig. 10(e) shows a higher peak value. However, when the material removal is divided into four steps, with each step removing 250 nm, the mid-frequency ripples on the surface are still present, but the mid-frequency RMS decreases to 1.575 nm. At the same time, the peak value of the PSD curve decreases by a factor of 5, significantly suppressing the mid-frequency errors on the component surface. This validates the effectiveness of the optimal single-material removal amount in suppressing mid-frequency errors. When 250 nm of material is removed in a single step, there is still some mid-frequency ripple on the surface, with a mid-frequency RMS of 0.570 nm. The PSD curve in Fig. 10(f) still exhibits characteristic peaks. After removing 100 nm, 100 nm, and 50 nm respectively, the mid-frequency ripple on the surface weakens, but the mid-frequency RMS increases to 0.665 nm. The PSD curve in the line feed direction shows a significant decrease, while the PSD curve in the scanning direction shows a noticeable increase. Under the combined effect, the mid-frequency errors on the surface is deteriorated. This demonstrates that when the single-material removal amount matches the optimal material removal amount, the mid-frequency errors on the surface is better suppressed.

5. Conclusion

This paper analyzes the change of mid-frequency errors on the component surface after magnetorheological finishing through the theoretical analysis model of the proposed removal function non-stationary effect. By combining the simulation and experimental analysis based on this model, the optimal single-material removal amount corresponding to the removal function non-stationary effect is obtained to suppress the mid-frequency errors. The main conclusions of this paper are as follows:

  • (1) By analyzing the contact situation between the magnetorheological polishing ribbon and the workpiece, a non-stationary modeling of the removal function profile was established to simulate the time-varying characteristics of the removal function profile. The prediction errors of this model were less than 4% compared with experimental results, verifying the accuracy of this non-stationary removal function profile model.
  • (2) Using the non-stationary removal function model to simulate the uniform material removal process, we verified that the non-stationary removal function effect can weaken the mid-frequency ripple errors and even reduce the characteristic peak value of the PSD curve by two orders of magnitude, but it will increase other mid-frequency errors and deteriorate the overall mid-frequency errors of the component surface.
  • (3) We first proposed the optimal single-material removal amount corresponding to the non-steady-state removal function effect and verified the effectiveness of the optimal-single material removal amount in suppressing mid-frequency errors through simulation and experiments.

In summary, due to the existence of removal function non-stationary effect, when the single-material removal amount of magnetorheological polishing is the optimal single-material removal amount, it can effectively suppress the mid-frequency errors on the component surface. This provides a new basis for existing MRF processes. It also provides theoretical and technical support for MRF processes and manufacturing of high-precision optical components.

Nomenclature

Symbols

L

The long axis of removing function

R

Radius of the polishing wheel

D

Thickness of polishing ribbon

Z

Distance between the polishing ribbon profile and the surface of the workpiece

y0

Contact point of the polishing ribbon with the workpiece in the y-axis direction

y1

Contact point of the polishing ribbon with the surface of the workpiece after being compressed in the x-axis direction

y2

Contact point of the polishing ribbon fluctuations in the y-axis direction with the surface of the workpiece after being compressed

Lv

The long axis of removing function under non-stationary effect

l

The percentage change in the long axis profile of the removal function

N

The short axis of removing function

H

Immersion depth of polishing ribbon

ΔD

Fluctuating value of polishing ribbon

x0

Contact point of the polishing ribbon with the workpiece in the x-axis direction

x1

Contact point of the polishing ribbon with the surface of the workpiece after being compressed in the x-axis direction

x2

Contact point of the polishing ribbon fluctuations in the x-axis direction with the surface of the workpiece after being compressed

ΔN

Change value of the short axis

ΔL

Change value of the long axis

Nv

The short axis of removing function under non-stationary effect

n

The percentage change in the short axis profile of the removal function

Funding

National Key Research and Development Program of China (No. 2021YFC2202403); Strategic Priority Research Program of the Chinese Academy of Sciences (No. XDA25020317); Natural Science Foundation of Hunan Province (2021JJ40673); Graduate Science and Technology Innovation Project of Hunan Prov. (CX20230019).

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.

References

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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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Figures (10)
Fig. 1.
Fig. 1. Schematic diagram of removal function formation

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Fig. 2.
Fig. 2. Schematic and spectral analysis of magnetorheological polishing ribbon test

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Fig. 3.
Fig. 3. Schematic of profile modeling of magnetorheological polishing removal function. (a) Radial schematic diagram of the polishing wheel. (b) Axial schematic diagram of the polishing wheel. (c) Radial profile modeling of removal function. (d) Axial profile modeling of removal function. (e) Radial profile modeling under the non-stationary effect of removal function. (f) Axial profile modeling under the non-stationary effect of removal function.

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Fig. 4.
Fig. 4. Schematic diagram of the impact of non-stationary removal function on mid-frequency errors (a) Magnetorheological polishing grating path diagram (b) Residual mid-frequency errors diagram after stationary removal function processing (c) Residual mid-frequency errors diagram after non-stationary removal function processing.

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Fig. 5.
Fig. 5. Simulated results of the surface shape and mid-frequency PSD curve when removing 10µm of material in a single magnetorheological polishing process. (a) Removal function without fluctuation. (b) Removal function width fluctuation. (c) Removal function length fluctuation. (d) Removal function removal efficiency fluctuation. (e) Mid-frequency PSD curve in the scanning direction. (f) Mid-frequency PSD curve in the line feed direction.

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Fig. 6.
Fig. 6. Simulation results of the correspondence between different ribbon fluctuation and mid-frequency errors

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Fig. 7.
Fig. 7. Simulation results of the correspondence between different material removal thickness and mid-frequency errors (a) The correspondence between different material removal thickness and mid-frequency RMS (b) The correspondence between different material removal thickness and mid-frequency ripple PSD amplitude

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Fig. 8.
Fig. 8. Schematic of comparison of multiple material removal and single optimal material removal

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Fig. 9.
Fig. 9. The MRF machine tools and polishing ribbon

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Fig. 10.
Fig. 10. Experimental results of the influence of non-stationary effect of removal function on surface mid-frequency errors. (a) Surface mid-frequency error after removing 1µm of material. (b) Surface mid-frequency error after removing 4 × 250 nm of material. (c) Surface mid-frequency error after removing 250 nm of material. (d) Surface mid-frequency error after removing 100+100+50 nm of material. (e) Mid-frequency PSD curve in the line feed direction for the first set of experiments. (f) Mid-frequency PSD curve in the line feed direction for the second set of experiments. (g) Mid-frequency PSD curve in the scanning direction for the second set of experiments.

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Tables (2)

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Table 1. Predicted values of changes in the profile of the removal function

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Table 2. Predicted and experimental results of changes in the profile of the removal function

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Equations (12)

Equations on this page are rendered with MathJax. Learn more.

( R + D ) 2 − ( R + D − H ) 2 = x 0 2
{ Z = R + D − H − ( R + D ) 2 − x 2 ∫ x 1 x 0 Z d x = − 2 ∫ x 0 0 Z d x
L = | x 0 | + | x 1 |
N = 2 × | y 1 |
{ ( R + D + Δ D ) 2 − ( R + D − H ) 2 = x 0 2 Z = R + D − H − ( R + D + Δ D ) 2 − x 2 ∫ x 2 x 0 Z d x = − 2 ∫ x 0 0 Z d x
L v = L + Δ L = | x 0 | + | x 2 |
N v = N + Δ N = 2 × | y 2 |
{ l = Δ L L n = Δ N N
g ( x ) = ∑ n = − ∞ ∞ f ( x − n d )
G ( k ) = 1 d ∑ n = − ∞ ∞ F ( k d − n )
g ( x ) = ∑ n = − ∞ ∞ f ( a x − n d )
G ( k ) = 1 a d ∑ n = − ∞ ∞ F ( k d − n a )
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