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Investigations on response time ofmagnetorheological elastomer under compressionmodeTo cite this article: Mi Zhu et al 2018 Smart Mater. Struct. 27 055017
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Investigations on response time ofmagnetorheological elastomer undercompression mode
Mi Zhu, Miao Yu , Song Qi and Jie Fu
Key Lab for Optoelectronic Technology and Systems, Ministry of Education, College of OptoelectronicEngineering, Chongqing University, Chongqing 400044, People’s Republic of China
E-mail: [email protected]
Received 25 December 2017, revised 27 February 2018Accepted for publication 13 March 2018Published 20 April 2018
AbstractFor efficient fast control of vibration system with magnetorheological elastomer (MRE)-basedsmart device, the response time of MRE material is the key parameter which directly affects thecontrol performance of the vibration system. For a step coil current excitation, this paperproposed a Maxwell behavior model with time constant l to describe the normal force responseof MRE, and the response time of MRE was extracted through the separation of coil responsetime. Besides, the transient responses of MRE under compression mode were experimentallyinvestigated, and the effects of (i) applied current, (ii) particle distribution and (iii) compressivestrain on the response time of MRE were addressed. The results revealed that the three factorscan affect the response characteristic of MRE quite significantly. Besides the intrinsic importancefor contributing to the response evaluation and effective design of MRE device, this study mayconduce to the optimal design of controller for MRE control system.
Keywords: magnetorheological elastomer, normal force, response time, compression mode
(Some figures may appear in colour only in the online journal)
1. Introduction
As a versatile intelligent material, magnetorheological elastomer(MRE) has attracted considerable attention due to its unsur-passed magnetic-control properties [1–6]. Also, being a solidderivative of magnetorheological fluid (MRF), the smart MREmanifests the superiority over MRF for its no settlement, goodsealing performance, simple application structure [7–9], etc. Assuch, MRE holds immense potential in various areas of noisereduction [10], intelligent sensing [11–13], biomimetic fabrica-tion [14, 15], electromagnetic shielding [16, 17], especiallyshock mitigation and vibration attenuation [18–26]. In thepractical applications, magnetorheological material generallyacts as a smart kernel element in the semi-active or active controlsystems. Nevertheless, time delay frequently arises in the controlsystems, which often degrades the control performance or evencauses system instability [27, 28]. Therefore, for realizing thereal-time and effective control, research on the switching time ofa magnetorheological smart system is critical [29, 30].
Tremendous amount of research has been carried out toexplore the transient response characteristics of MRF devices.View from the existing literatures [31–35], the time responseof MRF devices have five main factors: (a) response time ofMRF, (b) inductance of devices’ coil, (c) eddy currents in thecoil core, (d) response time of the driving electronics, and (e)geometry of the device. Among them, as a functional core inMRF device, the response characteristic of MRF has a pro-found effect on the overall response time of the smart devices.Chooi et al demonstrated that the response time of MRF wasbetween 28 and 67.5 ms, and which was significantly affectedby applied current, shear rate and particles volumetric con-centration [36]. Goncalves et al investigated the response timeof MRF at high shear rate, the results showed that the MRFcan achieve 63.2% of the expected yield stress for dwell timeas low as 0.6 ms [37]. However, Laun et al pointed out thatthe response time in [37] ignored the shear rate and residencetime distribution across the slit, and thus it represented ahydraulic time constant of the device rather than the practical
Smart Materials and Structures
Smart Mater. Struct. 27 (2018) 055017 (11pp) https://doi.org/10.1088/1361-665X/aab63e
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response of MRF [38]. Meanwhile, Laun et al reported anexperimental study of the transient response of MRF, andswitching time of 2.8 and 1.8 ms were obtained to describethe startup and shutdown transient. In short, MRFs with dif-ferent raw materials and particle properties have differentresponse performance, and which also greatly depending onthe testing method. Likewise, the time delay of MRF deviceswith different materials and different geometry of magneticcircuit, are not the same [39–42].
Other than liquid MRF in which the ferromagnetic par-ticles present from a homogeneous distribution to a chain-likestructure under a transient magnetic field, solid MRE pos-sesses a fixed structure in which the particles are solidified inthe matrix, they can barely move in the nonmagnetic elasticmatrix under an external magnetic field. Additionally, the
intelligent control of MRF is realized by changing its apparentviscosity and yield stress with an application of magneticfield, while the real-time intelligent control of MRE isimplemented by changing its modulus with an adjustablemagnetic field. So the MRF actuator is mainly designed as adamping-variable device, while MRE device usually man-ifests as a variable-stiffness device. Taken together, theresponse mechanism of MRE and MRE devices are quitedifferent from MRF and MRF devices.
Deserved to be mentioned, although there have beenhandful publications on the preparation and mechanicalcharacterization of MRE [43–46], key problems such as theresponse time of MRE material, has not yet been properlyaddressed. Gu et al tested the response time of a MRE isolatorand explored two feasible approaches to minish the response
Figure 1. Preparation of MRE. (a) Preparation process of isotropic MRE. (b) Preparation process of anisotropic MRE. Insets 1, 2 and 3 showthe SEM images of anisotropic MRE, isotropic MRE and CIPs (type CN), respectively.
Figure 2. Experimental set-up of response time test. (a) Schematic illustration of testing part for rheometer: I is the control current. The insetshows the experimental photo. (b) Schematic illustration of the test under a transient magnetic field: H is the magnetic field intensity, F is theadditional normal force.
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Smart Mater. Struct. 27 (2018) 055017 M Zhu et al
time-delay, it was found out that the modified approachreduced the force response time from 421 ms to 52 ms in therising and from 402 ms to 48 ms in the falling edges,respectively [47]. Bai et al investigated the response time of aMRE actuator and analyzed it theoretically, the experimentalresults showed that its force responses were almost 350 msboth in rise and fall edges. Besides, the author pointed out thatthe isolator mechanical structure is the main influencing factoron the response time of MRE actuator [48]. However, both ofthe studies are simply concerned with the response time ofMRE devices, rather than that of MRE itself and they havenot considered the impact of MRE transient properties on thedevice’s response. Moreover, in the application of MREdevices, besides working in shear mode as discussed in[47, 48], MRE is often designed for working in compressionmode, or working in shear mode but bearing compressionload, so its characteristics under compression status is unique[49]. Taking the two points into consideration, the responsetime of MRE material under compression mode is animportant matter worthy of investigation.
Under compression mode and for a transient magneticfield, MRE would become magnetized and the magneticparticles in MRE have a tendency to form or further form achain-like structure parallel to the direction of magneticfield. The tendency of the movement would make thedeformation of MRE and generate an additional normalforce. So by detecting the transient response of magnetic-induced normal force, the response time of MRE can bedetermined. In this paper, a set of experiments were pre-sented to study the transient response of MRE under com-pression mode, and a Maxwell behavior model with timeconstant l was proposed to describe the transient responseof normal force by separating of the coil response. Besides,
the relative factors such as applied current, particle dis-tribution and compressive strain were taken into account.
2. Experimental section
2.1. Fabrication of MRE samples
2.1.1. Raw materials. Soft magnetic carbonyl iron particles(CIPs, type: CN, size distribution: 1–8 μm, provided by BASFCorporation, Germany) (see inset 3 of figure 1) were used asthe filler particles. Castor oil (CO, purchased from SinopharmChemical Reagent Co. Ltd, China) and diphenylmethanediisocyanate (MDI: 4,4≈50%, 2,4≈50%, purchased fromYantai Wanhua Polyurethanes Co. Ltd, China) were selected asthe necessary materials for the preparation of polyurethane(PU) matrix. Dibutyl phthalate (DBP, purchased from TianjinBodi Chemical Holding Co., Ltd, China) was used as theplasticizer. Stannous octoate (purchased from SinopharmChemical Reagent Co. Ltd, China) was used as the catalyst.
2.1.2. MRE preparation. The fabrication process is shownschematically in figure 1. Firstly, CO was mixed with MDI intoa beaker and the temperature remained at 70 °C for 10min toobtain the polyurethane (PU) matrix. Secondly, iron particles,DBP and stannous octoate were added into the mixturesuccessively with vigorous stirring for 30min and thetemperature was increased to 80 °C. Finally, as soon as theviscosity of reactant increased obviously approximately after2 h, placed the mixture into vacuum oven to remove the airbubbles and then poured them into an aluminum mold. Duringthe curing process, there are two types of MRE were prepared:isotropic (figure 1(a)) and anisotropic MRE (figure 1(b)), the
Figure 3. Coil response. (a) Schematic illustration of magnetic circuit part. (b) Electric schematic of coil in rheometer. (c) A square wavecontrol signal. (d) Definition of coil response time at rise and fall edge.
3
Smart Mater. Struct. 27 (2018) 055017 M Zhu et al
former one was cured under natural conditions without an appliedmagnetic field, the latter one was cured at a constant magneticfield (0.7 T) for about 30min. After several days’ placement, thefinal MRE samples were obtained. As shown in the inserts 1 and2 of figure 1, the CIPs in anisotropic and isotropic MRE wereexhibited patterned chain like structure and uniformly distributedstructure, respectively. In the following tests, there were twoready-prepared MRE samples with different particles distributionand their mass fractions of CIPs were all set to 70%.
2.2. Experimental set-up
Measurements were made in an advanced commercial rhe-ometer (Modal: Physica MCR301, Anton Paar GermanyGmbH) equipped with a magneto-rheology device (MRD/1 T), which was described in greater details by Laeuger et al[50]. The schematic of experimental setup is shown infigure 2(a), and the inset shows the test photo. The stimulatedmagnetic field was adjusted by controlling the current appliedto the electromagnetic coil. A two-part metal cover was usedas the magnetic bridge to close the magnetic circuit and toensure a homogeneous magnetic field generated in the MREsample. The rounded MRE sample was placed between theparallel plate and the pedestal. The size of each testing samplewas 20 mm in diameter and 2 mm in thickness.
The process of the response time test is as follows. First, theservo motor was instructed to maintain a constant compressivestrain on the parallel plate. Then a step current I was applied tothe electromagnet coil to create a uniform magnetic field H inthe MRE, as shown in figure 2(b). In this step, the CIPs in MREhave a tendency to further form a chain-like structure parallel tothe direction of magnetic field. The tendency of the movementmakes the deformation of MRE and provide an additionalnormal force F to the parallel plate. So by examining the tran-sient response of this delivered normal force, the response timeof MRE can be obtained. The tests were undertaken under
different applied currents (1, 2 and 3 A), particles distribution(anisotropic and isotropic), and compressive strains (0.01%, 1%,3%, 5% and 10%). Each sample was tested three times underthe same condition and take the mean value as the final value.Significantly, in order to avoid large data fluctuations and to getvalid data, the minimum time interval of MCR301 rheometer islimited to 5ms, so here we use 5ms as the sampling time tocollect the effective and reliable data.
3. Theoretical analysis
The test normal force response time in this method means theswitching time during a short period from controlled transientcommand to the force response. However, because of theinductance existed in the magneto-cell, the response time ofnormal force actually contains the coil response time, whichshould be separated so as to get the response time of pureMRE material. Specifically, the weak time-delay betweenapplied current and generated magnetic flux density is over-looked in the next discussions.
3.1. Coil response time
To facilitate understanding, the magnetic circuit part isschematically depicted in figure 3(a), and in order to study itsresponse to a step coil current, the electric schematic infigure 3(b) is used. The electro-magnetic coil can be modeledby an inductance L in series with a resistance R [47]. For astep-up current (I=0 to I=Is) at t 0= and step-downcurrent (I=Is to I=0) at t 0,¢ = the respective currentresponse in rise and fall edge follows single exponentialfunction with L R:t =
I t I 1 e , 1st= - t-( ) [ ] ( )( )/
I t I e . 2st¢ = t- ¢( ) [ ] ( )( )/
Figure 4. Response time in rise and fall edge under different currents. (a) 1 A. (b) 2 A. (c) 3 A.
4
Smart Mater. Struct. 27 (2018) 055017 M Zhu et al
In order to observe both the rise and fall edges of theresponse characteristic, as shown in figure 3(c), a square wavecontrol current will applied to the circuit. According toequations (1), (2) and as shown in figure 3(d), t represents thetime constant required to accomplish 63.2% of the transitionchange, and t 3c t= is the time required to accomplish 95%of the transition change. In this paper, tc is defined as the coilresponse time.
3.2. Normal force response time
For ease of experiment and analysis, here we neglect eddycurrents in the magnetic circuit and magnetic saturation effectsof the metal material. In the test, once the square wave controlcommand is activated, the electric current is applied to the coiland the current delay is appeared. It is possible to subtract thecurrent delay time from the total normal force response time.However, as the closed magnetic circuit system is nonlinear,the simple rule of superposition cannot be worked.
At a constant compressive strain, we consider the normalforce F of MRE is dependent on the interparticle distance dand applied magnetic field H.
F kH
d, 31
2
4= ( )
where k1 is a constant, the derivation process of the relation isaddressed in the appendix.
For a certain closed magnetic circuit as depicted infigure 3(a), the H generated in MRE can be expressed as [51]
HNI
SL u uk I, 4
total 0 MRE2= = ( )
where S is the cross-sectional area of MRE, Ltotal is the totalmagnetic reluctance of the closed magnetic circuit, N is thewinding number of electromagnetic coil, u0 and uMRE are thepermeability of vacuum and relative permeability of MRE,respectively. Here uMRE is considered as a constant, then theH is proportional to the coil current I with a constant of k2.Therefore, for mathematical simplicity, the normal force F isassumed to be proportional to I d ,2 4 where k is the slope ofthe linearity.
F kI
d. 5
2
4= ( )
Assume that the effect of a transient current change onthe normal force response is described by a simple Maxwellbehavior model in which the time constant l plays the role ofthe material specific parameter
F F kI
d. 6
2
4l+ = ( )
For a step current, either from I=0 to the steady-stateI=Is at t=0 or from I=Is to I=0 at t′=0, substitutingequations (1), (2) into equation (6) yields analytic expressionsfor the transient normal force response (ds respects the steady-state interparticle distance)
F t kI
d1
2
2e
2e
2e , 7
s
s
t
t t
2
4
2
2
ll t l t
tt l
tt l
= -- -
+-
--
l
t t
-
- -
⎡⎣⎢
⎤⎦⎥
( )( )( )
( )
( )
( ) ( )
/
/ /
F t kI
d
2
2e
2e . 8s
s
t t2
42l
l tt
t l¢ =
-+
-l t- ¢ - ¢⎡
⎣⎢⎤⎦⎥( ) ( )( ) ( )/ /
In reality, the off-state normal force has an initial valueF ,0 it can be considered as an additional offset, soequations (7), (8) can be expressed as follows with steadynormal force F kI ds s s
2 4=
F t F F 12
2e
2e
2e , 9
st
t t
0
2
2
ll t l t
tt l
tt l
= + -- -
++
--
l
t t
-
- -
⎡⎣⎢
⎤⎦⎥
( )( )( )
( )
( )
( ) ( )
/
/ /
F t F F2
2e
2e . 10s
t t0
2ll t
tt l
¢ = +-
+-
l t- ¢ - ¢⎡⎣⎢
⎤⎦⎥( ) ( )( ) ( )/ /
As a consequence, if the time constant t is obtained byfitting curves of coil response according to equations (1), (2),it would be feasible to determine the time constantl (the timerequired to accomplish 63.2% of the transition change) ofnormal force by equations (9), (10). Therefore, as with thedefinition of coil response time in section 3.1, the responsetime t 3m l= (the time to reach 95% of the transition change)of MRE can be obtained.
Figure 5. Dependence of the response time on applied current. (a) Rise time. (b) Fall time.
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Smart Mater. Struct. 27 (2018) 055017 M Zhu et al
4. Results and discussion
4.1. Dependency on applied current
In the intelligent control of MRE system, different controlalgorithms will generate different control currents, and whichmay lead to its different response characteristics. Here wesimulated an off and on state with different current amplitudesto investigate the effect of applied current on the responsetime of MRE. At constant compressive strain of 0.01%, theapplied current was varied from 1 to 3 A in steps of 1 A. Byfitting of experimental data with equations (1), (2) and (9),(10), the responsive results of anisotropic MRE at rise and falledges are displayed in figure 4. Due to the coil inductance, asteep current ramp was observed after conducting the controlcommand. The transient normal force also showed a steepincrease, but somewhat delayed compared to the current, andit did exhibit an overshoot (due to the feedback characteristicsof the control device) and then gradually reached a steadyvalue. The fitting parameters are given in table 1. Withincreased current amplitude, the current response time (3t) atrise edge were 32.08 ms, 32.13 ms and 32.19 ms, respec-tively. The current fall time were 31.47 ms, 31.53 ms and
31.59 ms, respectively. It can be seen that the current rise andfall time have little difference with increased applied current.The normal force responsive time (3l), seen in figure 5,showed a decreased and increased trend in rise and fall edgewith the increment of current. For example, the rise responsetime of MRE was 19.17 ms at 1 A, while decreased by around36.78% to 12.12 ms at 3 A. Nevertheless, the fall response
Table 1. Fitting parameters for the response characteristic according to equations (1), (2) and (9), (10).
1 A 2 A 3 A
Fitting parameters Rise edge Fall edge Rise edge Fall edge Rise edge Fall edge
IS (A) 0.99 1.02 2.00 2.04 2.99 3.05t(ms) 10.69 10.49 10.71 10.51 10.73 10.53F0 (N) 0.005 0.008 0.006 0.005 0.006 0.008FS (N) 0.21 0.22 0.79 0.81 1.28 1.28l(ms) 6.39 4.39 4.9 5.97 4.04 6.23Response time (ms) 19.17 14.07 14.7 17.91 12.12 18.69
Figure 6. Response time in rise and fall edge under different particles distribution. (a) Anisotropic MRE. (b) Isotropic MRE. (c) Dependenceof the response time on the particle distribution.
Table 2. Fitting parameters for the response characteristic accordingto equations (1), (2) and (9), (10).
Aniso MRE Iso MRE
Fittingparameters Rise edge Fall edge Rise edge Fall edge
IS (A) 2.99 3.05 3 3.05t(ms) 10.73 10.53 10.71 10.52F0 (N) 0.006 0.008 0.005 0.008FS (N) 1.28 1.28 1.12 1.13l(ms) 4.04 6.23 10.08 8.96Responsetime (ms)
12.12 18.69 30.24 26.88
6
Smart Mater. Struct. 27 (2018) 055017 M Zhu et al
time increased by more than 32.84% from 14.07 ms at 1 A to18.69 ms at 3 A. Beyond that, as shown in figure 4 andtable 1, the magnitude of normal force to a higher appliedcurrent was larger.
As analyzed in section 3.2, the magnetic-induced normalforce increases with increasing applied current, whence ahigher normal force is obtained to a larger current. In addi-tion, with a higher step-up applied current, more power isdelivered to the CIPs to generate interaction energy, then thereluctance of MRE can be overcame more easily and thus therise process can be completed in a shorter time. However,with a higher step-down change in current, the bigger inter-action energy existed in the particles needs more time torecover to zero-current state. In consequence, for a MRE
control system, different delay compensation should be con-sidered for different control current ranges.
4.2. Dependency on particle distribution
Based on former researchers’ work [52, 53], MRE withaligned structure has a better magnetorheological behaviorcompared to isotropic one. While which one has a betterresponsive performance is worth exploring. Figure 6 shows acomparison of response time curves for isotropic (Iso) andanisotropic (Aniso) MRE, table 2 shows the fitting para-meters. The tests were performed at a step current of 3 A, andthe compressive strain was maintained at 0.01%. As shown inthe figure as well as the table, it is obvious that anisotropic
Figure 7. Response time in rise and fall edge under different compressive strains. (a) 0.01%. (b) 1%. (c) 3%. (d) 5%. (e) 10%. (f) Dependenceof the response time on the compressive strain.
7
Smart Mater. Struct. 27 (2018) 055017 M Zhu et al
MRE possessed a better responsive characteristic. The riseand fall time of isotropic MRE were respectively 30.24 msand 26.88 ms, which are 2.5 and 1.44 times of anisotropic onewith respective 12.12 ms and 18.69 ms. Besides, a largernormal force amplitude was observed in the anisotropic MRE.
With the same concentration of CIPs, the average dis-tance between iron particles in anisotropic MRE is muchsmaller than that in isotropic one [54], and the insets 1 and 2in figure 1 clearly confirm the conclusion. The smaller par-ticles distance leads to a larger normal force according to theanalysis in section 3.2, and thus the anisotropic MRE has alarger normal force than that of isotropic MRE. Besides, thesmaller particles distance in the chain-like structure makes theparticles more like a magnetic path. Consequently, the chainstructure speeds up the magnetic-flow time, especially whenthe transient rise and fall current are applied to the MRE.Hence, in the part of MRE device design, the anisotropicMRE can be given greater priority over isotropic one (if theapplied current is parallel to the chain direction of CIPs).
4.3. Dependency on compressive strain
In the application of MRE devices which working in com-pression mode or standing compression load, MRE will bearvarious compressive strains, and it may also results in variousresponse characteristics. In this case, the response times ofanisotropic MRE on five compressive strains of 0.01%, 1%,3%, 5% and 10% were investigated under the same currentexcitation of 3 A. The results are revealed in figure 7 andtable 3. As observed, there were no substantial changes incurrent response time. However, with the increment of com-pressive strain, the rise response time of MRE at each strainwere 12.12 ms, 17.37 ms, 20.25 ms, 21.96 ms and 24.39 ms,respectively. The fall response time were 18.69 ms, 21.39 ms,36.33 ms, 43.68 ms and 48.09 ms, respectively. That is, itwent up gradually in a nonlinear fashion from 12.12 ms and18.69 ms with the 0.01% strain to about 24.39 ms and48.09 ms with 10% strain. It can be concluded that the sampleat small compressive strain tends to have a better responseperformance. Besides, the initial and transient change ofnormal force to a higher strain were larger.
Due to the compression of rubber matrix, the initialnormal force increases with increment of compressive strain.With increasing compressive strain, the interparticle distance
decreases. The smaller interparticle distance leads to a greatermagnetic interaction of CIPs in MRE, and this brings about alarger magnetic-induced change of normal force. Besides, asdiscussed in the section 4.2, smaller particles distance canspeed up the magnetic-flow time with a step current excita-tion. Whereas, due to the stress relaxation property of MRE[55], with an increased compressive strain, it needs more timeto get an equilibrium value between particles interaction forceand binding force the matrix apply against the particles, andthis cause plays the leading role in the MRE response.Therefore, the MRE sample with a smaller compressive strainshows a better response property. In conclusion, for a MREcontrol system, larger displacement would need greater timedelay compensation. If without delay compensation, MRE ismore suitable for being used in precision or ultra-precisionvibration system.
5. Conclusions
This paper investigated the response time of MRE undercompression mode to a step current excitation. A Maxwellbehavior model with time constant l was proposed to fit thetransient response of MRE by separating the coil response. Inaddition, three factors such as applied current, particle dis-tribution and compressive strain were taken into account forthe response tests. From the results it can be concluded that:(1) the anisotropic MRE possessed a better responsive per-formance over isotropic one (as long as the applied current isparallel to the chain direction of CIPs). (2) MRE sampleunder smaller compressive strain had a better responsiveproperty. (3) At the rise edge, the larger step-up currentbrought a faster response time, while at the fall edge, thesmaller step-down current presented a better response. That is,for anisotropic MRE in this investigation, the shortest time atrise edge was 12.12 ms under the applied current of 3 A andcompressive strain of 0.01%. The shortest time at fall edgewas 14.07 ms under the applied current of 1 A and com-pressive strain of 0.01%. The results proved that the responsetime of MRE is at about several milliseconds level. Other thanthe intrinsic importance for contributing to the responseevaluation and effective design of MRE devices, this researchis in the hope of contributing to the optimal design of con-troller for MRE control system.
Table 3. Fitting parameters for the response characteristic according to equations (1), (2) and (9), (10).
0.01% 1% 3% 5% 10%
Fittingparameter Rise edge Fall edge
Riseedge Fall edge
Riseedge Fall edge
Riseedge Fall edge
Riseedge Fall edge
IS (A) 2.99 3.05 2.99 3.05 2.99 3.03 3.01 3.02 3 3.05t(ms) 10.73 10.53 10.72 10.52 10.73 10.51 10.69 10.53 10.71 10.5F0 (N) 0.006 0.008 0.27 0.26 5.59 5.68 16.08 16.39 31.51 32.49FS (N) 1.28 1.28 2.35 2.37 6.7 6.72 11.6 11.7 14.92 14.51l (ms) 4.04 6.23 5.79 7.13 6.75 12.11 7.32 14.56 8.13 16.03Responsetime (ms)
12.12 18.69 17.37 21.39 20.25 36.33 21.96 43.68 24.39 48.09
8
Smart Mater. Struct. 27 (2018) 055017 M Zhu et al
Acknowledgments
This research is funded by the National Natural ScienceFoundation of China (Grant No. 51775064) and the Equip-ment Research Joint Fund Project of Chinese Ministry ofEducation (Grant No. 6141A02022108). The authors aregrateful for the support.
Appendix. Relation between normal force andmagnetic field strength
In order to analyze the normal force response time of MRE,the mechanism of normal force generated by MRE should beteased apart firstly. Here a possible theoretical model is pro-posed based on magnetic dipole theory. To simplify, all CIPsin the matrix are assumed to be uniform, homogeneousspheres that can be treated as identical dipoles. It alsoassumed that the distances between particles are equal andonly the interactions between adjacent particles within a chainare considered. Besides, the compressive strain is assumed tobe uniformly distributed over the thickness of MRE, as thefigure A1 shows. Under a uniform magnetic field, the inter-action energy of the two magnetic dipoles is [56, 57]
Eu u
m
d
1
4
2, A1
m12
0
2
3p=
- ( )
where m is the magnetic dipole moment of each iron particle.um and u0 are the relative permeability of the rubber matrixand the permeability of vacuum, respectively. d is the dis-tance of two adjacent particles.
A composite MRE material consists of many sphericalCIPs imbedded in the rubber matrix, the total number ofparticles n, each having volume Vp, can be expressed as
nV
V
V
r
3
4, A2c
p
c3
f fp
= = ( )
whereVc is the total volume of MRE, f is the volume fractionof iron particles in MRE, and r is the particle radius. There-fore, the total energy density U (energy per unit volume) canbe obtained by:
Un
VE
m
u u r d
3
8. A3
c m12
2
20
3 3
fp
= =- ( )
The compressive strain of the MRE sample can beexpressed as [49]
d d
d, A40
0e =
- ( )
where d0 is the initial distance of adjacent particles. Themagnetic-induced stress can be expressed as [56]
U U
d
d U
dd
d m
u u r d
9
8. A5
m0
02
20
3 4s
e ef
p=
¶¶
=¶¶
¶¶
=¶¶
= ( )
Then the magnetic-induced compression force (normalforce) is
F SS d m
u u r d
9
8, A6
m
02
20
3 4s
fp
= = ( )
where S is the compression area of the MRE sample, and m∣ ∣can be approximately as u u r H
4
3m0
3p c in an uniform magn-
etic field [58]. Therefore the magnetic-induced normal forcecan be expressed as
FS d u u r H
d
2. A7m0 0
3 2 2
4
f c= ( )
Note that the normal force in equation (A7) is dependentof interparticle distance and magnetic field. Therefore thenormal force can be assumed to
F kH
d, A81
2
4= ( )
where k S d u u r2 m1 0 03 2f c= is a constant for a certain MRE
sample.
ORCID iDs
Miao Yu https://orcid.org/0000-0002-8547-6432
References
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Figure A1. Schematic diagram of magnetic dipoles model.
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