Precision Engineering 74 (2022) 69–79

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Precision Engineering
journal homepage: www.elsevier.com/locate/precision

Experimental study on vibration-assisted magnetic abrasive finishing for

internal blind cavity by bias external rotating magnetic pole
Liaoyuan Wang, Yuli Sun *, Fayu Chen, Guiguan Zhang, Peng Zhang, Dunwen Zuo
College of Mechanical and Electrical Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, China

A R T I C L E I N F O A B S T R A C T

Keywords: The work aims to solve the finishing problem of blind cavities and grooves in the internal channel of 3D printed
Blind cavity parts. Based on the principle of vibration-assisted magnetic abrasive finishing (VMAF), a novel scheme of the bias
Bias external rotating magnetic pole (BERMP) external rotating magnetic pole (BERMP) was proposed, where the spherical steel grit (SG) was used as the
Steel grit (SG)
magnetic abrasive to extrude, impact and rub the surface of the workpiece to realize finishing. And then, its
Response surface method (RSM)
Surface roughness
feasibility was verified by theoretical and experimental methods. The response surface method (RSM) was used
to establish a quadratic regression equation model to reveal the influence of processing clearance, magnetic pole
rotation speed, vibration frequency and amplitude on surface roughness and obtain the optimum processing
parameters. After processing the specimen under the conditions of 1 mm of processing clearance, 800 r/min of
magnetic pole speed, 15 Hz of vibration frequency and 0.2 mm of vibration amplitude, the Ra decreased from
original Ra12.60 μm to Ra1.25 μm with the reduction by 90%, improving the surface quality of the workpiece
remarkably. The research results can provide a theoretical basis and technical reference for finishing irregular
parts with complex cavities.

1. Introduction developed the mathematical model of material removal in the MAF

process, then validated it by experiment with the spherical magnetic
With the development of ultra-precision technology, 3D printed abrasive powder (MAP) developed by an atomization technology.
parts with complex cavities such as shaped waveguides have been Mosavat et al. [10] investigated the crack propagation and material
widely used in radio communication, aerospace, medical and other removal mechanism of the surface polishing for the brittle material sil­
high-tech fields, which usually require very smooth and delicate sur­ icon wafers by the coupled simulation of the finite element method
faces [1,2]. However, it is difficult to grind and polish the concave (FEM) and smooth particle hydrodynamic (SPH). The obtained results
surface of a tiny internal channel with traditional processing procedures indicate that the smaller the processing clearance, the higher the mag­
[3]. Magnetic abrasive finishing (MAF) is an advanced surface finishing netic field strength and the higher the material removal rate. Kala et al.
process that can remove the fine asperities from the internal free-form [11] provided a device that arranged two opposite N–S poles and placed
surface and even a special-shaped surface to yield a delicate surface a paramagnetic workpiece between them to obtain the influence of
that meets the performance requirements [4,5]. process parameters on surface finish. Moreover, adding lubricant to
As it turns out that the MAF process can solve traditional technical MAPs can hardly improve the processing efficiency, which is consistent
dilemmas to some extent, thus it has been one of the focuses of inter­ with the conclusion drawn by Mulik [12].
national academic research. Many efforts have been made to probe into As the study on the MAF progresses, the introduction of multiple
its processing mechanism and extend its utility, significantly improving physical fields into the MAF has given it new advantages. Recently,
the processing performance [6,7]. Nagdeve et al. [8] developed a novel vibration-assisted magnetic abrasive finishing (VAMAF), an improved
MAF setup attached to the lathe to achieve the inner and outer cylin­ version of MAF, has been introduced, which has proved to enhance the
drical surface finishing. They reported that as the relative speed between processing performance of MAF. Misra et al. [13] added the ultrasonic
the workpiece and magnetic abrasive brush (MAB) increases, the vibration in MAF to obtain the fine surface in the nanometer range and
roughness decreases until it reaches a critical value. Gao et al. [9] established the material removal rate (MRR) model consisting of a

* Corresponding author.
E-mail address: sunyuli@nuaa.edu.cn (Y. Sun).

https://doi.org/10.1016/j.precisioneng.2021.11.007
Received 31 August 2021; Received in revised form 11 October 2021; Accepted 2 November 2021
Available online 5 November 2021
0141-6359/© 2021 Elsevier Inc. All rights reserved.
L. Wang et al. Precision Engineering 74 (2022) 69–79

Fig. 1. The structure diagram of the waveguide.

Fig. 2. The structure diagram of the single cavity. (a) The construction principle details. (b) The overall structure system.

steady-state term and a time-varying term. Zhang et al. [14] investigated placed inside the workpiece, which can cleverly realize the complex
the polishability of selective laser melted parts with different slope an­ inner surface finishing of the tube. Yamaguchi et al. [21] presented a
gles by VAMAF and characterized the surface roughness and topog­ classic internal surface polishing scheme, placing the alumina ceramic
raphy. The results show that most unmolten particles can be removed tube and the toroidal magnetic yoke coaxially. The yoke has four mag­
after processing for 75 min. Henga et al. [15] studied the effects of netic poles evenly distributed along its circumference, efficiently fin­
different magnetic pole shapes on the surface finish of ZrO2 bars with ishing complicated bend tubes. Li et al. [22] arranged spirally multiple
mesoscale diameters. The results indicate that the most considerable electromagnets outside the irregular elbow. The magnetic abrasive in
improvement in surface roughness was obtained with a 2 mm the tube can obtain the spiral finishing track by controlling the power
square-edge magnetic pole. Zhou et al. [16] and Ma et al. [17] attached supply sequence, avoiding the massive mechanical structure. The
the ultrasonic vibrator to the machine tool to produce a vibration workpieces mentioned in the above literature are all pipes with a cir­
perpendicular to the surface of the workpiece in MAF. The study shows cular cross-sectional area, but the cross-section of the waveguide is
that the surface microcracks were effectively lessened, and the stress rectangular. If the above processing methods were adopted, the cavity
state changed from residual tensile stress to compressive stress to bottom might be processed into a curved surface.
improve the fatigue performance of the workpiece after polishing. Many experts have carried out in-depth explorations of MAF tech­
Many scholars have also been committed to the research of complex nology, improved processing performance and solved some engineering
internal surface finishing. Kim [18] established an internal finishing problems of finishing complex parts. However, the parts used for in­
system for the rectangular pipe. The outer magnetic poles move along ternal surface finishing often have a circular cross-section [23–26]. The
the pipe axis, driving magnetic abrasives to polish. A significant sliding literature on the finishing for blind cavities and blind grooves in com­
stroke can hardly be obtained due to the small axial dimension of the plex internal channels was rarely reported. This kind of workpiece is
cavity in the waveguide. So, this solution may not be suitable for fin­ commonly represented by complex-shaped waveguides, with tiny tracks
ishing the cavity. Amnieh et al. [19] proposed a new solution for fin­ arranged inside to realize the reflection and guidance of the light wave.
ishing internal spiral grooves of a cylindrical tube. The magnetic pole Due to the internal cavity size limitation, it is difficult for the abrasive to
rod with abrasives (carbon steel grits) is placed in the tube in this obtain a sizeable sliding stroke. It is not suitable for arranging the
scheme and moves along the spiral groove on the inner surface, thor­ magnetic pole inside waveguides, but the standard coaxial outer
oughly polishing the periphery and bottom surface of the groove. rotating magnetic pole scheme tends to produce a curved surface at the
However, due to the tiny space in the waveguide, it may not be suitable bottom of the cavity [27].
to place the magnet inside it. In this paper, a novel scheme of bias external rotating magnetic pole
Zhang et al. [20] developed a novel polishing tool that applies (BERMP) was proposed and conducted the experimental investigation
external magnetic poles to drive the spherical magnet with abrasives on a single-blind cavity artificially constructed according to the actual

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2. Mechanism

Fig. 1 is the structure diagram of the waveguide. The raw material of

the workpiece is AlSi10Mg powder, which is made by the selective laser
melting (SLM) process. Fig. 1(a) is the front view of the waveguide. Cut
along the waveguide centerline A-A, and it can be seen that the wave­
guide is not a conventional long rectangular tube but a special-shaped
piece with a meandering inner surface profile. Generally, the polishing
of the inner surface can be achieved by abrasive flow machining (MAF).
For example, the convex surface can be finished by abrasives flowing
through the channel, but it is difficult for abrasives to reach the cavity
bottom and rub against it [28]. The MAF can provide an effective so­
lution for the finishing of the cavity bottom due to the adaptability of the
magnetic brush to the freeform surface. Fig. 1(b) is the side view of the
waveguide. Cut along the waveguide centerline B–B, and it can be
known that several cavities are arranged at intervals. In order to shorten
the finishing time and quickly characterize the surface quality, one of
the cavities, C, is taken as the subject of the investigation to facilitate
obtaining the best process parameters. Cut along the line D-D, and it can
be shown that the cross-section of the cavity is rectangular. The single
cavity structure can be easily constructed according to the cavity.
Fig. 2 is the structure diagram of the single cavity. It can be seen from
Fig. 2(a) that a rectangular notch is milled in the aluminum block, and
the geometric dimensions of its cross-section E-E are consistent with the
cross-section D-D of the waveguide. The upper cap and the specimen are
Fig. 3. The schematic diagram of the principle of BERMP.
attached to the top and side surfaces of the aluminum block respectively
to form an enclosure space. A part of the specimen surface becomes the
size of the waveguide. Based on determining the use amount of steel grit bottom of the cavity for polishing. Fig. 2(b) is the overall structure of the
(SG) and reasonable processing time, the response surface method single cavity system, consisting of the aluminum block, the upper cap,
(RSM) was used to construct a quadratic regression model to reveal the and the specimen, and screws are used to connect the three.
influence of processing gap, magnetic pole rotation speed, vibration Fig. 3 is the schematic illustration of BERMP. There is a suitable
frequency and amplitude on the surface roughness. Finally, the best amount of Q235 SGs inside the groove, and SGs subjected to the mag­
processing parameters were obtained and verified by experiments. The netic field force are pressed against the bottom surface of the groove and
research conclusions can provide theoretical bases and technical support arranged along the direction of the magnetic force line to form a mag­
for further improving the finishing effect of complex cavities. netic brush with a certain degree of flexibility. Establish a coordinate
system regarding the centroid of the bottom face of the cavity as the

Fig. 4. The simulation of static magnetic field (a) Contour map of magnetic induction intensity distribution (b) Magnetic induction intensity curve.

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Table 1
Parameters and conditions for the magnetic-field simulation.
Conditions Parameters

Big magnet Nd-Fe-B magnet N35 (20 mm × 20 mm × 10 mm)

Small magnet Nd-Fe-B magnet N35 (10 mm × 10 mm × 5 mm)
Processing clearance d 2 mm
Magnetic polo plate Ø180 mm × 20 mm, aluminum

origin. While the magnetic pole plate rotates clockwise, the single cavity
system vibrates at a high frequency in the x-direction. In order to avoid
producing a curved surface at the cavity bottom, the magnetic pole plate
slowly reciprocates in the y-direction.
Because only when the magnetic pole passes near the cavity can the
MAF be realized, Increasing the number of magnetic poles within a
specific range can theoretically improve processing efficiency. However,
up to 4 magnetic poles can be arranged in the circumferential direction
of the magnetic pole plate due to the size limitation. Since the magnetic
pole is very close to the specimen during the processing, the size of the
magnetic pole tip should not be too large to avoid collision between the
two. However, to ensure considerable magnetic field strength in the
processing area, the size of the magnet in the magnetizing direction
should be as large as possible. The slender monolithic magnet is very
fragile and susceptible to be broken. Therefore, several tiny cubic Fig. 6. The schematic diagram of a single SG pressing against the surface of
magnets are used to form the magnetic poles for cost-saving and easy the specimen.
replacement.
The distribution and magnitude of magnetic induction intensity were
simulated by finite element analysis based on ANSYS Maxwell software.
As shown in Fig. 4, the magnetic field is generated by four magnetic
poles, each composed of five big magnets and four small magnets. The
farther away from the magnetic pole, the smaller the magnetic induction
intensity. The specific simulation parameters are shown in Table 1. Take
point A at the processing clearance d = 2 mm as the starting point to
make clockwise the auxiliary circle. The distance from the point on the
circle to point A is denoted as S. The analogue values of B were
extracted. Then the actual values of B were also recorded with a hand­
held digital tesla meter (TD8620). The results are shown in Fig. 4(b). It
can be seen that the simulation results are almost in agreement with the
actual of that, and the maximum deviation between the two is only
3.7%, the simulation results are valid.
Fig. 5 is the simulation of the time-varying magnetic field. Because
the magnetic poles are symmetrical, a quarter model can be used for
simulation to save calculation costs. Fig. 5(a) shows the initial magnetic
field, the magnetic pole faces the processing area, boundaries 1 and 2 are Fig. 7. The processing trajectory of a single SG.

Fig. 5. The simulation of the time-varying magnetic field.

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Fig. 8. The experimental setup of BERMP.

Fig. 9. Change in surface roughness with the processing time and weight of SGs. (a) Change curve of Ra vs T: M = 0.1 g, d = 2 mm, n = 540 r/min, f = 15 Hz, l = 1
mm (b) Change curve of Ra with M: d = 2 mm, n = 540 r/min, T = 40 min, f = 15 Hz, l = 1 mm.

the x-direction. As a result, the SGs can produce reciprocating motion in

Table 2 the x and y directions relative to the specimen.
Process parameters and their levels. The magnetic force in the z-direction Fz experienced by a single SG
A-PC (mm) B-MPS (rpm) C-VF (Hz) D-VA (mm) Y-Response (μm) can be determined by equation (1) [29,30]:
1 400 5 0.2 Ra B2 3π(μr − 1)ω d2 π
3 600 15 0.6 Fz = × × a (1)
5 800 25 1
4μ0 3(2 + μr ) + π(μr − 1)ω 4

where μ0 is the absolute permeability of free space (μ0 = 4π × 10− 7H/m),

given symmetrical properties, and the motion domain is given a rotation μr is the relative permeability of the SG (μr≈4000), da is the diameter of
speed of 800 r/min. When the magnetic pole rotates 45◦ to position A, the SG (da = 1 mm), ω is the content of the ferromagnetic substance in
the other magnetic pole adjacent to it reaches position B. If the magnetic SG (ω ≈1 when ignoring impurities), B is the magnetic induction in­
pole plate continues to rotate 45◦ , it will revert to the initial state and tensity of the end of magnetic poles (B = 0.43T). Substituting these
complete a processing cycle. At the same time, the specimen vibrates in values in equation (1) gives Fz = 0.044 N.

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Table 3 is not deformed. Fig. 6 is a schematic diagram of an SG pressing against

Experimental parameters and responses. the surface of the specimen under the action of magnetic forces.
Exp. No. A-PC B-MPS C-VF D-VA Y–Ra Based on the measurement principle of Brinell hardness, the rela­
tionship between the indentation diameter dinden and the Brinell hard­
1 3 600 15 0.6 2.239
2 5 600 15 1 3.958 ness HHBW can be expressed as follows [33]:
3 3 600 15 0.6 2.228
2Fz
4 3 400 25 0.6 2.965 HHBW = 0.102 × ( √̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ ) (3)
5 3 800 5 0.6 3.764 π da da − d2a − d2inden
6 3 600 25 1 1.758
7 3 400 15 1 2.918
8 5 600 25 0.6 3.765 where HHBW is the Brinell hardness of the specimen, dinden is the diameter
9 3 600 5 1 3.768
of indentation. HHBW is measured in kgf/mm2. For HHBW = 120 kgf/mm2
10 3 600 15 0.6 2.235
11 5 800 15 0.6 5.669 (measured value), the value of dinden = 6.7 × 10− 3 mm.
12 3 400 5 0.6 3.538 The indentation depth hinden can be derived from the Pythagorean
13 1 600 15 1 1.695 theorem, as shown in equation (4), and then hinden = 1.11 × 10− 5 mm
14 3 800 15 0.2 2.761 can be obtained.
15 1 400 15 0.6 3.198
16 3 600 15 0.6 2.239
√̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
da 1
17 3 400 15 0.2 2.543 hinden = − d2a − d2inden (4)
2 2
18 1 600 5 0.6 2.093
19 3 800 15 1 2.766 From Fig. 4, the cross-sectional area Sinden is equal to the sector area
20 3 600 5 0.2 2.322 Ssector-OAB minus the triangle area Striangle-OAB, which can be given by
21 5 600 5 0.6 5.179
equation (5), and then Sinden = 4.91 × 10− 14 m2.
22 3 600 25 0.2 2.815
23 3 800 25 0.6 2.808 ( ) ( )
d2 dinden dinden da
24 1 600 15 0.2 1.335 Sinden = a arcsin − − hinden (5)
25 5 400 15 0.6 3.444
4 da 2 2
26 3 600 15 0.6 2.219
When the SG moves in the direction of the magnetic field gradient,
27 1 800 15 0.6 1.281
28 5 600 15 0.2 3.932 the impact and rolling of the SG on the specimen cause the surface of the
29 1 600 25 0.6 1.988 specimen to undergo plastic deformation. In turn, the specimen material
also has a force named the processing resistance Fresist on the SG to resist
deformation, and the Fresist can be given by:
Table 4 Fresist = σ s Sinden (6)
The ANOVA for the surface roughness (Ra).
Source SOS DF MS F-value p-value where σs is the yield strength of the 3D printed specimen. For σ s = 200
Model 30.12 14 2.15 1382 <0.0001 MPa, the value of Fresist = 9.82 × 10− 6 N.
A 17.25 1 17.25 11086 <0.001 Due to the small size of the end of magnetic poles, dB/dy can be
B 0.014 1 0.01 11 0.009 approximately expressed by:
C 1.74 1 1.74 1116 <0.0001
D 0.111 1 0.112 71 dB
(7)
<0.0001
≈ Bmax − Bmin
AB 4.19 1 4.19 2690 <0.0001 dy
AC 0.428 1 0.428 275 <0.0001
AD 0.028 1 0.028 17 0.0008 For Bmax = 0.43 T and Bmin = 0.06 T, the value of dB/dy = 0.37 T.
BC 0.037 1 0.037 23 0.0003 Substituting equation (7) in equation (2) gives Fy = 0.265 N.
BD 0.034 1 0.034 21 0.0003
In the MAF process, there are three possible situations, which are
CD 1.566 1 1.566 1006 <0.0001
A2 2.102 1 2.102 1351 <0.0001
discussed in detail as follows [34]:
B2 2.212 1 2.212 1422 <0.0001
C2 1.411 1 1.411 906 <0.0001
a. Fy = Fresist
D2 0.022 1 0.022 13 0.0023
Residual 0.022 14 0.002
It is an equilibrium situation. The SG only presses on the surface of
LOF 0.004 10 <0.001 the specimen, but no relative motion occurs.
PE <0.001 4 <0.001
b. Fy > Fresist
Std. Dev. Adjusted Predicted R2 Adequacy
R2 R2 Precision It means that the specimen can be finished.
0.0394 0.993 0.989 0.998 156.649
c. Fy < Fresist
SOS-Sum of Squares; DF-Degree of Freedom; MS-Mean Square; LOF-Lack of Fit;
PE-Pure Error; Std. Dev.-Standard Deviation. Under this condition, the indentation depth is adjusted by rotation of
the SG until the processing conditions are met.
When the magnetic pole rotates at high speed, a time-varying mag­ It can be known that Fy > Fresist in this scheme by comparing the
netic field is generated in the processing area, which drives the SG to above calculation results. It indicates that the finishing scheme proposed
move in the y-direction. The driving force Fy is given by [31]: in this paper is theoretically capable of finishing for the specimen.
V χ dB Assuming that the SG moves synchronously with the magnetic pole, the
Fy = B (2) motion governing equations of an SG are derived as follows:
μ0 dy
⎧ x = l sin(2πft)
⎨
where V is the volume of the SG, χ is the Magnetic susceptibility of the SG ( )
(8)
and χ = μr − 1. ⎩ y = k sin 2πnm t
The surface hardness of the Q235 SG after strengthening is much 2 60
larger than that of the original 3D printed part [32]. Therefore, the SG
where l、f are the amplitude and frequency in the x-direction,
can be regarded as a rigid body compared with the specimen. That is, it

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Fig. 10. The effects of linear terms on the response.

respectively, k is the length of the cavity in the y-direction, n is the realize the axial feed.
magnetic pole rotation speed, m is the number of magnetic poles, t is the
time. For l = 0.2 mm, k = 7 mm, m = 4, t = 1 s, the trajectory of a single 3.2. Experimental procedure
SG can be drawn in Fig. 7. When no vibration in the process, the SG only
reciprocates in the y-direction. The motion trajectory is a straight line, The initial surface roughness Ra of specimens is about 12 μm. In the
resulting in producing unprocessed areas due to the gap between two experiment, the specimens are cleaned with ultrasonic equipment and
SGs. When the vibration in the x-direction is introduced, the motion absolute ethanol. Five points are uniformly selected on the processing
trajectory of a single SG is densely grid-like shape covering a more surface to record the roughness value, and their average value is
extensive processing area, which can effectively improve processing regarded as the roughness of the specimen. The roughness Ra is
efficiency and quality [35]. Due to the diameter limitation of SGs, there measured with a roughness tester (Mitutoyo). Scanning electron mi­
is a problem that SGs cannot reach the corner of the cavity. However, it croscopy (Quanta FEG 250) is used to observe surface topography. The
can be further solved by reducing the diameter of SGs or using SGs with distance represented by d between the magnetic tip and the specimen is
mixed diameters in future research [36]. the processing clearance. A precision electronic scale with the accuracy
Through the above-mentioned mathematical modelling and theo­ of 0.001g measures the weight of SGs M used in each experiment.
retical analysis, the feasibility and rationality of the scheme of BERMP
combined with VAMAF finishing for the blind cavity of the internal
4. Experimental verification
channel are demonstrated.
In order to verify the above theoretical analysis, two sets of experi­
3. Experimental conditions
ments with and without vibration were carried out, respectively. Fig. 9
(a) shows the variation curve of Ra with time. It indicates that the sur­
3.1. Experimental setup
face roughness reduces after processing for a period, proving the feasi­
bility of the processing scheme. Compared with the situation without
Fig. 8 shows the self-developed experimental setup of BERMP, and its
vibration, the processing with vibration can obtain lower roughness,
detailed introduction is as follows: The servo motor fixed on the bracket
which is consistent with the analysis result of equation (8). That is, the
is used to drive the magnetic pole plate to rotate, and the bracket can be
motion trajectory of SGs covers a larger processing domain after intro­
positioned on the linear guides to realize the adjustment of the pro­
ducing vibration. As observed in this Figure, the Ra curve first drops
cessing clearance. The linear actuator unit drives the lifting platform
rapidly and then becomes flat with increased processing time T. In the
equipped with linear guides to make the entire magnetic pole plate
early processing stage, due to the low bonding strength between the
reciprocate up and down. The single cavity system is fixed on the upper
unmelted metal powder and the substrate, the SGs can quickly reduce
end of the transition block attached to the vibrator. The vibrator consists
the roughness by continuously impacting and rolling on the unmelted
of a voice coil motor (VCAR0436-0187-00A, SUPT Motion Co., LTD), a
metal powder. However, after the roughness drops to a certain level, the
sliding table and a frame. When the voice coil motor is enabled, high-
volume of SGs is much larger than that of asperities, limiting the further
frequency vibration can be introduced into the processing domain.
reduction of the roughness and may even damage the processed surface.
The vibrator installed on the slider of the synchronous belt guide can
Accordingly, the processing time of 40 min is reasonable.

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Fig. 11. The combined effects of factors on Ra.

The details are shown in Table 2.

Table 5
The commercial software Design-Expert generated 29 sets of design
The results of optimization and experimental verification.
tests, each with 0.05 g SGs and 40 min processing time. The specific
Parameters Ra(μm) Relative error results are shown in Table 3. Further analysis of the input and output
(%)
Predicted Experimental data can establish the functional relationship between the two, which is
A = 1 mm, B = 800 r/min, C = 15 1.20 1.25 4.0 helpful to determine how to adjust these parameters to control the
Hz, D = 0.2 mm response.
A = 1 mm, B = 600 r/min, C = 15 1.42 1.44 1.4
Hz, D = 0.2 mm 5.2. Analysis of variance
A = 1 mm, B = 600 r/min, C = 15 1.79 1.86 3.8
Hz, D = 1 mm
The experimental results were analyzed with an analysis of variance
(ANOVA) to evaluate the contribution rate of the input parameter to the
Fig. 9(b) is the relationship curve between the use amount of SGs and response change. The various tests were applied to choose the adequate
Ra. It indicates that with the increase of the weight of SGs, the Ra curve model that fits the response. The “Sequential model sum of squares” test,
shows a trend of first decreasing and then increasing with M = 0.05 g as the “Model summary statistics” test and the “Lack of fit” test were car­
the turning point. This is because too many SGs reduce the sliding ried out in sequence. The results show that the quadratic model is sig­
stroke, resulting in congestion of SGs in the tiny space. Nevertheless, nificant, while the cubic model is not. Therefore, the quadratic model is
when M = 0.025 g, the SGs are too little to achieve uniform finishing, so applicable to fit the experimental data. The ANOVA for the surface
the mean and range of Ra are relatively large at this point. roughness (Ra) is given in Table 4. Fisher’s F-test and P-value can
evaluate the importance of every coefficient, and the higher the value of
5. Experimental investigation with RSM F-value and the lower the P-value, the more critical the relating coeffi­
cient. The F-value of the model is 1382, and the corresponding p-value is
5.1. Experiment design less than 0.05, which means that the proposed model is significant.
Similarly, all input parameters are significant terms. The difference
In order to investigate the influence of various process factors on the between “Predicted R2” and “Adjusted R2” is less than 0.2, which is in
roughness and obtain the optimal process parameters, the experimental good consistency. The “Adequacy Precision” used to characterize the
plan was designed based on the Box-Behnken of RSM. This plan regards signal-to-noise (S/N) ratio is 156.649, indicating an adequate signal.
the surface roughness (Ra) as the output response and involves four This model can be used to navigate the design space.
input process parameters: A-processing clearance (PC), B-magnetic pole
speed (MPS), C-vibration frequency (VF), D-vibration amplitude (VA).

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Fig. 12. SEM photographs of the surface before and after processing.

5.3. Individual effects of linear terms on Ra VF. Equation (8) can account for this. Increasing the VF can make the
trajectory of SGs cover a more extensive processing area, which effec­
F-value can sort the influence of linear and quadratic terms on the tively improves processing efficiency and quality. However, too large VF
response as A>C>D>B, B2>A2>C2>D2. The effects of linear terms on can aggravate the collision between SGs, causing a part of SGs to hit the
the response are illustrated in Fig. 10. Fig. 10(a) shows that the Ra in­ processed surface and gradually deteriorate the surface quality. Fig. 10
creases as the PC increases. It can be explained by the above theoretical (d) shows that Ra gradually increases with the increase of VA. This is
model, in which the larger the PC is, the smaller the indentation depth because the probability of SGs hitting the cavity walls increases with the
becomes and the bigger the roughness is in the same processing time. It increase in VA. The rebounded SGs have considerable kinetic energy and
can be seen from Fig. 10(b) that as the MPS increases, the Ra decreases leave a visible mark on the surface after hitting the specimen.
rapidly and then gradually slows down and finally even shows a slight
upward trend. This is because the speed of SGs in the y-direction in­ 5.4. Combined effects of factors on Ra
creases as the MPS increases, which improves the processing efficiency.
However, excessively increasing the MPS can cause SGs to stagnate at Based on the conclusion of ANOVA, the combined effects of factors
the limit position in the y+ direction due to inertia, thereby reducing the on Ra were obtained. The effects of four items with greater F-value,
sliding stroke and limiting the further reduction of roughness. Fig. 10(c) namely AB, CD, AC, and BC, on Ra are visualized in Fig. 11. It can be
shows that the Ra first decreases and then increases with the increase of seen from Fig. 11(a) that when the PC is less than 3 mm, the Ra first

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L. Wang et al. Precision Engineering 74 (2022) 69–79

The crucial part of the experimental investigation is to decide the

optimal process conditions where the minimum Ra can be obtained.
Therefore, applying the built-in numerical algorithm of Design-Expert
software, several optimal parameter combinations were obtained. The
top three were recommended by considering the actual processing
conditions, and corresponding experimental verifications were also
carried out. The details are shown in Table 5. The results show that the
best processing conditions are: A = 1 mm, B = 800 r/min, C = 15 Hz, D
= 0.2 mm. In this condition, the predicted and experimental Ra are 1.20
μm and 1.25 μm, respectively, and the two have good agreement. The
relative error is only 4.0%, indicating that the prediction model is
reasonable.

5.6. Surface topography analysis

In order to further investigate the processing mechanism, the surface

morphology of the specimen before and after processing was observed
by the scanning electron microscope (SEM). The R-Profile graphs
Fig. 13. The specimen photos of before and after processing.
depicting the peak-to-valley height were received from the surface
roughness tester (Mitutoyo). Fig. 12(a) is the SEM image of the unpro­
decreases and then becomes flat as the MPS increases, which is consis­ cessed specimen. It shows that a large number amount of unmelted
tent with the results of the analysis as mentioned above. However, when metal powders adhered to the original surface. The peak-to-valley height
the PC is greater than 3 mm, the Ra increases with increasing of MPS. of the R-Profile is relatively large, and the measured Ra is 12.60 μm.
Especially when PC is close to 5 mm, and the MPS reaches 800 r/min, the Fig. 12(b) is the SEM image of the processed specimen with the opti­
Ra increases sharply. This is because when the PC is large, the magnetic mized processing parameters. It indicates that the surface with unmelted
force acting on SGs is small, but the speed of SGs is very high so that SGs metal particles have flattened, and there are few marks of plastic
continuously hit the walls and rebound in the cavity, which intensifies deformation left on it, and the entire finishing surface is relatively uni­
the deterioration of Ra. form and smooth. The peak-to-valley height diminishes significantly,
From Fig. 11(b), when the VA is 0.2 mm, the minimum Ra is and the minimum Ra is 1.25 μm. Next, to discuss the role of high-
approximately obtained at a VF of 15 Hz. As mentioned above, too high frequency vibration in processing, a comparative experiment was car­
a VF can aggravate the collision of SGs, but too low a VF can not make ried out, in which the only difference of the processing parameters used
the specimen uniformly processed. Also, the Ra tends to decrease when this time from the above-optimized parameters is that there is no vi­
the VF and VA are both large. This may be because the SGs, in this case, bration. The SEM image of the specimen, in that case, is shown in Fig. 12
have considerable kinetic energy, and the collisions between the SGs are (c). It shows that although the unmelted metal powders on the surface
very violent, which increases the complexity and diversity of motion have disappeared, many plastic deformation marks are left, which is
trajectory to a certain extent but also increases the uncertainty of the SGs consistent with the theoretical prediction. The peak-to-valley height is a
motion. Therefore, the processing parameters, in this case, are not bit large, and the measured Ra is 2.02 μm. In summary, when processing
suitable for application in processing practice. with vibration, the mark of plastic deformation is weakened, which
It can be seen from Fig. 11(c) that when the PC is 1 mm, the mini­ effectively improves the surface quality. After processing with optimized
mum Ra is approximately obtained at the VF of 20 Hz. However, as the parameters, the Ra can be reduced from the original Ra12.60 μm to
vibration frequency rises, the Ra exhibits a slight upward trend, which is Ra1.25 μm with a reduction of 90%.
consistent with the above analysis results. Fig. 11(d) indicates that when The specimen photos of before and after processing are shown in
the PC and VF are both 1 mm, the minimum Ra can be obtained by Fig. 13. It can be seen that nothing shows on the rough surface before
increasing the MPS to about 800 r/min and adjusting the VF to about processing, while the reflections appear on the processed surfaces. This
15Hz. illustrates that the surface becomes smooth and flat by the finishing
process. Moreover, a brighter reflection can be shown on the sample
5.5. Parameter optimization surface processed with vibration, while the reflection on the sample
surface processed without vibration can be seen vaguely. This is because
The above research shows that each term of the quadratic model vibration can effectively improve the processing quality, which is
affects the roughness to varying degrees. In order to comprehensively consistent with the analysis above.
evaluate the contribution rate of each term to the change of Ra, a
quadratic regression model was established by Design-Expert software 6. Conclusion
as shown in equation (9):
This paper proposed a novel scheme of BERMP to finish the blind

Y = +9.66167 − 1.47063A − 0.023733B − 0.007353C + 4.01698D + 0.002519AB

− 0.015401AC − 0.105906AD − 0.000048BC − 0.001155BD − 0.156388CD (9)
+0.141649A2 + 0.000015B2 + 0.004592C2 − 0.351045D2

where A is the processing clearance (PC), B is the magnetic pole speed cavity and proved to be feasible through the theory and experiment. And
(MPS), C is the vibration frequency (VF), D is the vibration amplitude then, the RSM was used to conduct experiments to investigate the effects
(VA). of the processing clearance, magnetic pole speed, vibration frequency,

78
L. Wang et al. Precision Engineering 74 (2022) 69–79

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finishing using a multiple pole-tip system[J]. Precis Eng 2012;36(3):510–6.
of China (Grant No. 52075254); the Joint Funds of the National Natural [24] Yun H, Han B, Chen Y, Liao M. Internal finishing process of alumina ceramic tubes
Science Foundation of China (Grant No. U20A20293); and the Graduate by ultrasonic-assisted magnetic abrasive finishing[J]. Int J Adv Manuf Technol
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