Investigations on process parameters of cluster magnetorheological polishing in a planet motion model

A planetary-type cluster magnetorheological polishing device with a rotating magnetic field was proposed to solve the problems of abrasive accumulation and low polishing efficiency caused by the untimely restoration of the conventional magnetic chain. Considering the microstructural deformation and squeeze-strengthening effect of magnetorheological polishing fluid, a material removal rate model was established based on the principle of fluid dynamic pressure and verified by experiments. The relationships between material removal rate or roughness and processing parameters were confirmed by multiple linear regression analyses, respectively. And the processing parameters optimization was made by linear weighting method under the premise of establishing the evaluation system. The results show that the eccentricity and angular velocity ratio are proportional and inversely proportional to MRR, respectively. When the polishing fluid is squeezed, the material removal rate can be significantly increased from 7 to 21 nm/min, but the roughness will be reversed at a gap of less than 0.9 mm. After the optimization of processing parameters, the workpiece roughness after rough and fine polishing was reduced from 1.079 μm and 1.083 μm to 0.346 μm and 0.184 μm, with a reduction of 67.9% and 83.01%.


Introduction
Magnetorheological polishing (MRP) is a new polishing technique with the polishing fluid consisting of magnetorheological fluid (MRF) and micron abrasives.MRF can accomplish the rheological reaction within a millisecond under the action of the applied magnetic field, which transports the polishing abrasive to polish the surface of the workpiece, and the rheological process features continuous adjustment, convenient control, etc. [1,2].Flexible control of abrasives based on magnetorheological technology can reduce surface micro-nano cracks and damage introduced by conventional processes.The MRP precision can be controlled with the magnetic field due to its unique processing principle.And it does not cause new surface damage to the workpiece surface, which has a broad application prospect in the field of precision machining [3][4][5].Shi et al. [6] analyzed the relationship between abrasive and workpieces in the MRP process based on the elastic-plastic deformation theory.And the polishing precision was improved by optimizing the MRF composition and polishing parameters.Liu et al. [7] established a tool influence function for aspheric workpieces based on the Reynolds equation to reveal the effect of gap thickness variation on the material removal rate.Ghosh et al. [8] experimentally analyzed the effect of the cross-feeding method of a wheeltype MRP machine on surface roughness, established the evaluation function of working gap thickness and wheel speed on the processing results, and provided a new basis for the processing parameters optimization.Ren et al. [9] proposed a Belt-MRF concept to expand the application of MRF to large mirrors and made a prototype with a large remove function, using a belt instead of a very large polishing wheel to expand the polishing length.A series of experimental results verified that the Belt-MRF has high material removal rates, stable removal function, and high convergence efficiency which makes it a promising technology for processing large aperture optical elements.Pan et al. [10] proposed a dynamic magnetic field magnetorheological effect (DMFME) polishing technology, tested the characteristics of polishing normal force of DMFME polishing pad, and researched the influences of the revolving speeds of the magnetic pole, the machining gaps, and the movement methods of workpiece on the polishing force.Found that compared with the static magnetic field, the magnetorheological polishing force and the torque presented significant dynamic behavior under the dynamic magnetic field formed by the rotating magnetic pole.Nie et al. [11] simulated the magnetic field for cluster MRP and analyzed that the magnetic poles are inverted and crossed to obtain a larger dynamic magnetic flux density.This research proposed an optimization scheme for the magnetic field arrangement of the MRP.
The use of dynamic magnetic fields can effectively reduce the centrifugal effect caused by the rotation of the polishing pad [12], reduce the restoration time of broken magnetic chains, and solve the problem of uneven magnetic field distribution, thus improving the effective polishing efficiency and material removal rate.However, the focus of current research on cluster MRP is generally on the magnetic field and process parameter optimization, without considering the effect of complicated magnetic chain shape and polishing fluid flow on the removal rate.In addition, the polishing process needs to match the appropriate process parameters to ensure polishing efficiency and quality [13].And to obtain the best process parameters, it is necessary to master the influence law of each parameter on the polishing effect.To address the above problems, a removal rate model considering multi-variable and magnetic chain shapes was developed in this article, and the effect of each process parameter on the processing precision was evaluated by linear weighted sum method (LWSM) to provide a reference for cluster MRP.

Mechanism of magnetorheological polishing
In general, the polishing fluid consists of nano-sized magnetic particles, polishing abrasive particles, base fluid, and additives (as shown in Fig. 1).The magnetic particles and abrasive particles are distributed randomly and uniformly in the base fluid without the application of a magnetic field.
As the magnetic field is applied, a rheological effect occurs in the MRF, and the magnetic particles are magnetized and gathered into chains [14].At this time, a certain number of the abrasive particles are adsorbed and dispersed around the magnetic chains.With the magnetic field further strengthening, the chain-like structure is further aggregated to form a cluster structure.Abrasive particles and MRF form a polishing pad under rheological action.When the workpiece is in contact with the polishing pad and moves relative to it, the abrasive particles collide and rub against the surface to be machined, completing the removal of surface material and thus achieving ultra-precision machining.Figure 2 illustrates the main process of material removal from the workpiece surface by abrasive particles.At first, under the pressure of abrasive particles along the vertical direction against the processing surface, the positive pressure makes more abrasive particles embedded in the workpiece surface.Next, as the MRP fluid flows, the abrasive particles cut along the horizontal direction against the workpiece surface, and the raised part of the workpiece surface is shaped and removed.
Figure 3 shows the magnetic flux density B at the workpiece surface when the magnetic field is rotating.As shown in Fig. 3a and b, the extreme value of the magnetic flux density at the workpiece surface is in the area where the pole centers are located and gradually decreases, with the two corresponding poles adjacent to each other forming a ribbon cutting area.Figure 3 c and d show the magnetic field distribution at the junction of the two rows of magnetic poles, where the cutting area is also in the shape of a ribbon, but the magnetic flux density decreases, by about 26.32%.Furthermore, the magnetic field in a part of the circular area is canceled out by the interaction between the multiple poles.
From the data in the figure, it is evident that the magnetic flux intensity varies widely due to the existence of gaps between the magnetic poles.Nevertheless, the variation shows periodicity when the magnetic poles rotate, which solves the problem of uneven magnetic field distribution.In addition, the magnetic field rotation drives the polishing fluid to circulate along the rotational direction of magnetic force lines.During the polishing process, the magnetic pole Fig. 1 Arrangement of the magnetic particles and abrasive particles rotation can accelerate the renewal rate of the magnetic chain within the polishing fluid, ensuring that the abrasive particles are always dispersed on the workpiece surface [15].Under the coupling effect of fluid pressure and magnetic force, the abrasive particles are alternately circulated in various regions, which can effectively improve the effective rate of material removal from the workpiece surface.

Modeling of material removal rate
The surface material removal of metal materials' MRP mainly depends on the mechanical friction effect of abrasive particles; the abrasive chips generated during processing and the oxidation effect on the workpiece surface will also have an impact on the polishing effect.To better analyze the material removal effect, we assume that the material removal is primarily based on the micro-cutting of nonmagnetic abrasive particles.Material removal rate (MRR) refers to the amount of material removed from the surface of the processed samples per unit of time, which is an important indicator to describe the polishing efficiency and processing difficulty.The material removal model can be derived from the Preston equation of the MRP model [16] where τ is the shear stress on the workpiece surface during processing, v is the relative motion speed between the polishing fluid and the workpiece, P is the grinding pressure generated by the polishing fluid, C p is the Preston coefficient, as determined by the experiment, C′ p is the modified Preston coefficient considering the relative friction and material deformation conditions, which can be expressed as where μ is the friction coefficient between the polishing fluid and the workpiece, which is constant when the magnetic flux density is invariable, E is the tensile modulus of the polished material, K is the fracture toughness, and H v is the Wechsler microhardness of the polished material.
From the geometric relationship in Fig. 4a, the total flow rate v ζ at an arbitrary point P on the workpiece is   where ω 2 and ω 1 are the workpiece's angular velocities along its own rotation and rotation along the polishing pad, respectively, α and β are the angles of point P with respect to circle centers O 1 and O 2 , and r 1 and r 2 are the radius of revolution and rotation.The relationship between them is where ρ 1 is the revolution radius of the workpiece, that is, the eccentricity, and ρ 2 is the radius of the workpiece.Then, define the angular velocity ratio g, eccentricity e and relative radius r′ about the planetary rotation as Bringing Eqs. ( 4) and (5) into Eq.( 3) we get that From Cosine Law, the circular angle at point P can be expressed as Similarly, the shear yield stress of polishing fluid can be divided into two parts according to the rotational center of the magnetic field and that of the workpiece.For analysis purposes, the modeling is performed in the cylindrical coordinate system (Z, θ, R), as shown in Fig. 4b.Then, the fluid flow rate u(Z) in the working gap along the -direction of the total velocity can be expressed as [17] where dP/dζ is the pressure gradient on the polishing fluid, η is the apparent viscosity of the polishing fluid, v 1 is the magnetic field rotation speed, and h is the thickness of the polishing pad before squeezing.
From the Reynolds equation, the pressure gradient in the ζ-direction can be expressed as where h ' 0 is the polishing fluid's actual thickness and its value changes with the shape of the workpiece surface.
Since the shear yield stress at each point of the polishing fluid is not easily measured during the experiment.By the definition of surface roughness R a (Fig. 5), the actual equivalent thickness of the working gap h ′ 0 is represented as where h 0 is the minimum distance from the polished workpiece reference surface to the polishing plate and L is the sampling length for roughness measurement.The total shear yield stress of the polishing fluid in the yielding state is where τ(B) is the magnetic shear stress, η 0 is the zero-field viscosity, and ̇ is the shear strain rate of the polishing fluid.( 8) As the polishing fluid volume fraction is high (ϕ ≥ 0.3), the magnetic particles form a body-centered tetragonal (BCT) structure along the magnetic field direction.Now the shear yield stress of the cluster MRP fluid can be expressed as [18] where χ is the susceptibility of the polishing fluid, B is the magnetic flux density, is the tilt angle of the magnetic chains, μ 0 is the vacuum permeability, which takes the value of 4π × 10 −7 H/m, and μ r is the relative permeability.k c and c are both the distance coefficients of the cells within the fluid, which can be expressed as The MRP fluid is compressed as it flows over the workpiece surface, the volume fraction of the magnetic particles rises, and the magnetic shear yield stress increases accordingly.Since the total volume of the magnetic particles V p before and after squeezing is constant, the compressive strain ε is where ϕ 1 and ϕ 2 are the volume fraction of the magnetic particles before and after squeezing.
When the flow rate of the polishing fluid u is constant, the relationship between shear strain γ and tile angle λ can be expressed as where t 1 and t 2 are the start and end times of the fluid sweeping through the workpiece.At last, substituting Eqs. ( 8), (11), and (12) into Eq.( 1), the material removal rate of cluster MRP fluid in yield state can be obtained as 4 Magnetorheological polishing device and test

Composition of polishing system
The planetary-type cluster MRP system can be divided into three parts: the controller part, the polishing stand, and the magnetic field generation device (as shown in Fig. 6). ( According to practical requirements, we used a bench drill instead of motor 1 and clamped the upper-pressure plate at the chuck.The magnetic field generating device was installed on the workbench of the bench drill, which has a ball screw mechanism at the bottom to facilitate the workbench moving left and right for eccentric purposes.The primary devices involved in the test system included an elastic coupling, a BG-518801 bench drill (power 800W, rated speed 200-2500 r/min, maximum spindle stroke 60 mm), a 7IK750RA-CF speed control motor (power 750 W, speed scope 90-1400 r/min), a 7GU-3K-RA-C right Angle gear reducer (transmission ratio i = 3), a frequency converter, and a ball screw sliding table.The workpiece is placed in the counterbore hole in the middle of the lower pressure plate, and the upper and lower pressure plates are coupled by bolts.The polishing fluid is formed into a polishing pad by rheology, and a splash shield is added to prevent the polishing fluid from spilling out during processing.The magnetic pole rotary motor and polishing table are fixed to the ball screw sliding table for easy adjustment of the eccentricity during machining, as shown in the specific assembly form in Fig. 7.
The workpiece material is 40Cr (ISO: 41Cr4) with r 1 = 5 mm.To improve efficiency and ensure that the roughness of each workpiece surface remains relatively consistent during the experiments, the workpiece is pre-treated by lathe machining so that the roughness R a = 1.08 μm (± 1.5%).And the MRP fluid is mixed by us, with a thickness of 1.5 mm.The material of magnetic particles is hydroxy iron powder with a radius of 5 μm (30%), the base fluid is silicone oil (50%), and the additive is oleic acid (10%).Added 2 μm of the carborundum to the well-stirred magnetorheological fluid as abrasive particles (10%).The performance parameters of the two materials are listed in Tables 1 and  2, respectively.The magnetic field generation device is a key part of the MRP system.The magnetic field generation device adopted for the magnetorheological polishing device designed in this article consists of a dotted magnet frame and several permanent magnet units (as shown in Fig. 8).
NdFeB (Nd2Fe14B) permanent magnets featuring simple structure, low manufacturing cost, high magnetic energy product, and coercivity are selected as the magnetic field unit, which has relatively high energy density, relatively  strong magnetic properties with high-cost performance as well as excellent mechanical characteristics, and the key parameters are shown in Table 3.
A few permanent magnet units are embedded in the support frame and placed on the bottom of the polishing table.The magnetic poles of adjacent permanent magnets are arranged opposite to each other with opposite polarities, so as to form a linear air gap, resulting in a high gradient magnetic field across the polishing plate.The magnetic flux density distribution is measured along the diagonal of the permanent magnet array using a Tesla meter.The average magnetic field intensity in the strong magnetic field area on the polishing pad surface is up to 0.6 T, and the magnetic flux density in the center area of the rotation axis of the polishing pad and in the weak magnetic field area on the edge of the polishing pad surface are both lower than 0.08 T. As the magnetic poles rotate, the polishing fluid can flow smoothly, enabling circulation renewal.

Polishing effect evaluation and testing
The polishing quality of the polished workpiece surface is assessed in terms of surface roughness values, material removal efficiency, and surface topography.The average roughness values of the 3 points are measured using a surface roughness profiler of Mahr model Marsurf PS 1 (measuring range 350 μm, maximum scan length 17.5 mm).The 2D morphological observation system prior to and after polished workpiece processing consists of a CX-2KCH microscope (imaging resolution 2048 × 1536, pixel dot size 3.2 μm × 3.2 μm, frame rate 8 FPS at the maximum width) and an observation stage (as shown in Fig. 9).The metallurgical morphology of the finished workpiece surface is taken by the microscope, and the electronic images are output and analyzed by S-EYE software.
The 2D/3D line laser measuring instrument of KEY-ENCE LJ-X8000 (sampling frequency 3200 kHz, measurement accuracy 0.1 μm) series is mixed with the belt tracks to build a surface profile inspection test bench, which is used to observe the 3D surface and ensure that the surface roughness before polishing is similar (as shown in Fig. 10).The polished workpiece was measured with a VK-X200 laser microscope (measurement accuracy 0.0005 μm), which is capable of quickly observing the 3D morphology of micron and submicron surfaces and measuring geometric data such as roughness, height, width, volume, and area.The obtained 3D point cloud data were processed and analyzed by Cloud Compare software, and the electronic images are output.
As it is not possible to accurately measure the MRR of polished workpieces in real-time during the experiment, it can be measured by the change in thickness before and after a single polishing.By comparing the calculated and experimentally obtained average values of the MRR over the entire workpiece surface, the correctness of Eq. ( 16) can be determined.
The experimentally measured MRR is defined as where h 0 ′ 1 and h 0 ′ 2 are the surface height of the workpiece before and after a single polishing process and Δt is the time used for a single polishing.

Polishing test scheme
Equation ( 16) reveals that the MRR is dependent on the workpiece's characteristics, the shear yield strength of the MRP fluid, the fluid pressure, and the relative velocity.When the applied magnetic field and the magnetic pole rotation speed are constant, the removal rate is mainly determined by the eccentricity, the angular velocity ratio, and the working gap thickness.When ω 1 = 400 r/min, B = 0.6 T, h 0 = 1.0 mm, the MRR is shown in Fig. 11.As can be summarized from the data, the MRR of the workpiece reaches its maximum value away from the rotation center of the magnetic pole.To facilitate comparison with the experimental data, we chose the average value of the entire surface as the MRR under current conditions.In summary, there are two types of factors affecting the polishing quality in the MRP test: (a) MRP parameters: magnetic field strength, polishing gap, workpiece rotation speed (spindle speed), polishing disk rotation speed (magnetic pole speed), polishing time; (b) polishing fluid parameters: hydroxy iron powder particle size, abrasive particle size and mass fraction [19].A few factors can affect the final polishing effect, to minimize the complexity of the test, four factors with relatively high impact on the surface finish quality results are selected for analysis: workpiece rotation speed (spindle speed), abrasive mass fraction, polishing gap, and polishing time.In the polishing test, the specific processing parameters are shown in Table 4.

Fig. 8 Magnetic field generation device
In order to ensure the accuracy of the experiment, L 26 (6 5 ) orthogonal experiment was designed [20,21].And the test scheme is shown in Table 5.

Analysis of test process parameters
To ensure the accuracy of the experiment, each group of experiments was conducted three times, and the arithmetic mean was used as the final experimental value.Specifically, the error bars indicated the standard deviation (SD) of the data.The calculated and experimental results of the orthogonal experiment designed as in Table 5, in which only a single factor was changed while ensuring the same other conditions, are shown in Fig. 12.
From the experimental data, it can be concluded that the MRR is inversely proportional to the angular velocity ratio g, while it is the opposite of the eccentricity e.However, it should be noted that when the magnetic poles rotate too fast, the fluid shear strain is larger and a certain degree of shear thinning is produced, which is the reason why the experimental value deviates from the calculated value.Similarly, due to the low magnetic flux density B at the edge of the polishing pad, the MRR did not reach the expected level when the rotation radius ρ 1 was greater than 40 mm, which led to a large difference in the workpiece roughness obtained from the three experimental values.Meanwhile, we also conducted a comparison experiment for magnetic pole fixation.But along with the increase in processing time, the abrasives accumulated toward the edge severely, resulting in lower polishing quality.As for the polishing fluid without squeezing, i.e., h 0 = 1.5 mm, the experimentally measured MRR was 7 nm/min, and the roughness change after 30 min of polishing was negligible.
With the combination of Figs.12c and 13, as the machining gap becomes smaller, the MRP fluid is squeezed, more abrasive particles are embedded in the workpiece surface, and the MRR increases [22].However, along with the increase in positive pressure, the plowing effect intensifies, resulting in a reverse increase in surface roughness.Therefore, it is necessary to use the above two parameters as evaluation indexes to determine the optimal processing parameters since the MRR and surface roughness are not linearly related.In the following, we will analyze the effect of each process parameter on roughness and MRR from SD (as shown in Table 6).The three parameters that have a significant effect on the MRR are revolving speed, revolution radius, and machining gap.In other words, the eccentricity and the angular velocity ratio have a great effect on the MRR.As for the roughness, it is affected by the machining time, but also more sensitive to the workpiece revolving speed and machining gap, the former affects the planeness, while the latter affects the positive and shear stresses during polishing.
Using the data from the orthogonal experiments and the fitted function for a single factor, we can obtain the linear regression equation for MRR and roughness as follows [23] (8) MRR = 25.9207− 19.846h 0 + 0.047 1 − 7.6906 To verify the accuracy of the fitted regression equation, the experimental data were compared with the calculated data from Eq. ( 18) (as shown in Fig. 14).Overall, the linear fit accuracy was not high due to the strong nonlinear relationship between the process parameters and the roughness or MRR, and the variables were correlated with each other.Again, we tried the nonlinear multivariate coupling method, but the accuracy was also not high.Therefore, in this paper, an evaluation system was used to assess the processing quality.
In the actual machining process, the polishing is often divided into two stages: rough polishing and fine polishing, and the evaluation indexes of the two stages are different.To select the optimal process parameters for each stage, the following composite grade method (CGM) is used to score the machining quality.The standard of scoring criteria is as follows: MRR ≤ 3 nm/min is 0, MRR ≥ 25 nm/min is 100; R a ≥ 1.0 μm is 0, R a ≤ 0.4 μm is 100, and the rest are discounted proportionally.The MRR score Y 1 and the roughness score Y 2 are shown in Table 7.Among these, each group of five experiments corresponds to a factor.
The linear weighted sum method is used to superimpose the weighted scores of the two evaluation indicators, and the final score can be expressed as [24] where Γ 1 and Γ 2 are the weight (Γ 1 + Γ 2 = 1), respectively.At the rough polishing stage, Γ 1 = 0.7, Γ 2 = 0.3, and at the fine polishing is Γ 1 = 0.1, Γ 2 = 0.9.Bringing the data of Table 7 into Eq.( 19) and the final score at the two stages can be obtained as Table 8.
Two polished workpieces of the same size with comparable surface morphology and quality are taken to perform the verification test of the optimal process parameters.The average surface roughness of the workpieces prior to polishing is 1.079 μm and 1.083 μm respectively, and upon completion of polishing, the average surface roughness of the workpieces measured after cleaning and drying is 0.346 μm and 0.184 μm, which is 67.9% and 83.01%lower compared with that before processing.We take either workpiece for surface topography scanning, to obtain a two-dimensional local microscope scanning comparison of the workpiece surface before and after polishing (Fig. 15), and three-dimensional local topography scanning comparison of the workpiece (Fig. 16).
As shown in Figs. 15 and 16, MRP can improve the surface quality of the workpiece to a certain extent.Nevertheless, from the surface topography scanned images, the effect of the local surface processing area of 40Cr is not uniform enough.The workpiece surface before polishing has relatively deep grooves, after polishing obviously becomes shallow, and presents a bright unevenly distributed vertical strip texture, which is due to the small magnetic flux density and gradient distribution at the bottom of the polishing pad, and the uneven pressure distribution on the workpiece surface during polishing also affects the surface processing quality.Therefore, under the conditions of the existing equipment system and the optimal process parameters that have been adapted, a workpiece can be polished several times, with different abrasive particle sizes selected for each trial for variance analysis, and in-depth studies are still needed in terms of magnetic field distribution, polishing fluid viscosity, and positive pressure variation of the spindle.

Conclusion and discussion
This article established the planetary MRP device based on the cluster principle and the dynamic magnetic field magnetorheological effect of rotating poles, analyzed, and investigated the principles and key technologies related to the MRP process based on the consideration of magnetic pole rotation by using the self-configured MRP fluid and 40Cr as the test material.The research further established related mathematical models for material removal, performed material removal experiments, and surface roughness experiments.The actual influence of each process parameter on the processing was investigated, and a linear regression function of MRR and roughness were established based on the experimental values.The optimal process parameters were also obtained based on the scoring system and linear weighting method, and were experimentally verified.The results of the research are as follows.
(1).We developed a shear model of MRP based on the body-centered cubic structure while considering the fluid dynamic pressure.As the machining gap becomes smaller, the MRP fluid is squeezed and intensified, more abrasive particles are embedded in the workpiece surface, and the MRR is significantly increased (up to 21 nm/min, a twofold improvement).However, with the machining gap less than 0.9 mm, the plowing effect intensifies, resulting in a reverse increase in surface roughness.(2).For planetary MRP devices, the eccentricity and angular velocity ratio are proportional and inversely proportional to MRR, respectively.As for roughness, it is influenced by machining time, but it is also more sensitive to the rotation speed of the workpiece and the machining gap.(3).The presence of multicollinearity in multiple independent variables leads to a certain error in the prediction accuracy of the established linear regression functions for MRR or roughness, especially in the case of changing eccentricity radius.(4).The analysis by building a scoring system and linear weighting method shows that changing the eccentricity, machining gap, and machining time had a greater effect on the MRR and surface quality than changing the angular speed ratio in the rough and fine polishing stages (when the optimal machining parameters were determined).The workpiece roughness after rough polishing and fine polishing was reduced from 1.079 μm and 1.083 μm to 0.346 μm and 0.184 μm, respectively, a reduction of about 67.9% and 83.01%.
In addition, planetary MRP leads to unavoidable concentric scratches in the polished workpiece due to its characteristics (e.g., Fig. 15d).How to improve this phenomenon will be the focus of the next research. y

Fig. 4
Fig. 4 Schematic diagram of cluster magnetorheological polishing under moving magnetic field: (a) diagram in cartesian coordinates; (b) diagram in cylindrical coordinates

Fig. 5
Fig. 5 Gap thickness equivalent model considering surface roughness

Fig. 6 Fig. 7
Fig. 6 Schematic diagram of the cluster MRP system

Fig. 13
Fig. 13 Tendency of the MRR

Table 2
Performance parameter of the MRP fluid

Table 3
Performance parameter of the permanent magnet