Modelling and experimental characterisation of a compressional adaptive magnetorheological elastomer isolator

Abstract

This article proposes a simple physical-based model to describe and predict the performance of axially compressed magnetorheological elastomer cylinders used as vibration and shock absorbers. The model describes the magnetorheological elastomer macroscopic stiffness changes because of an externally applied magnetic field from a microscopic composite cell of silicone rubber and carbonyl iron particle. Despite neglecting the material hyperelasticity, anisotropy and adjacent magnetic interaction, the model describes effectively the effect of the magnetic field on the macroscopic modulus of elasticity. The changes in the mechanical properties with the induced magnetic field are measured on samples of different particle concentration based on volume percentage, that is, 10 and 30 percent concentration of iron particles in a silicone rubber matrix. The manufacturing process of the samples is detailed, as well as the experimental validation of the effective stiffness change under a magnetic field in terms of transmissibility and mobility testing. However, the prediction seems to be limited by the linear elastic material model. Predictions and measurements are compared, showing that the model is capable of predicting the tunability of the dynamic/shock absorber and that the proposed devices have a possible application in the reduction of mechanical vibrations.

Keywords

Magnetorheological elastomer vibration isolation smart materials

1. Introduction

Magnetorheological elastomer (MRE) is a class of material that consists of a polymeric matrix with embedded ferromagnetic particles such as carbonyl iron particles (CIPs). Since its mechanical properties change very rapidly by the application of a magnetic field, the MRE became a widely used material when it comes to controllable features applied to active and semiactive vibration control and suppression. A comprehensive review of MRE has been presented by Li et al. (2014) discussing the fundamental theory of these materials, manufacturing methods, configurations and applications. Some of the most relevant works are briefly explained here.

Modelling of MREs has focussed on predicting how the MRE elastic properties change with the induced magnetic field. Dipole–dipole magnetic particle interaction model was used firstly by Jolly et al. (1996) to describe the increase in shear modulus in a suspension of ferromagnetic particles in a non-magnetic matrix as a magnetic field is applied perpendicular to the strain direction. Overall, under small strain condition, the dipole–dipole magnetic particle interaction model is efficient and mathematically concise. The same model has been used over the years by Davis (1999), Shen et al. (2004), Chen and Jerrams (2011) and Li and Sun (2014) to describe the magnetic field dependency of MRE shear modulus. These studies only differ on the approaches for computing the zero-field shear stiffness, namely finite element model, hyper elastic rubber model, standard linear solid and viscoelastic interfaces, respectively. Subsequently more complicated models, which are based on magnetic-mechanical microstructure, have been developed (Borcea and Bruno 2001; Bustamante 2010; Castañeda and Galipeau 2011; Danas et al., 2012; Dorfmann and Ogden 2004; Han et al., 2013; Wang et al., 2003; Yin and Sun 2005; Yin et al. 2002, 2006) to consider both zero-field and magnetic-field dependency on the shear modulus of the MRE composites. These models provide the shear and Young’s modulus of MRE as a magnetic field is applied perpendicular to the load surfaces.

Experimental work has been carried out concurrently by separate research groups. Brigadnov and Dorfmann (2003) showed that the effect of the magnetic field is to stiffen the shear response of the material. They conducted mathematical modelling of MRE by using the concept of the strain-energy functions. The derivation of the equations was based on a model of MRE confined by parallel top and bottom plates, which are subjected to shear deformation under the influence of a magnetic field normal to the plates. An acceptable agreement was achieved between the numerical simulation and experimental results. Blom and Kari (2005) indicated that the magneto-sensitive effect of MR rubber is also dependent on its amplitude and frequency. From experimental results, rubber samples in the audible frequency range showed that both the magnitude and loss factor of the shear modulus are strongly amplitude-dependent. These results also concluded that introducing iron particles into the rubber increase its amplitude dependence over the entire frequency range. Wang et al. (2006) investigated the effects of interactions between CIPs and a silicone rubber matrix, on the performance of MRE related to MR effect and mechanical properties. By using silane coupling agents during the mixing process, it was seen that improvement in dispersion of the iron particles and the interaction between iron particles and the matrix are possible, thus increasing the tensile strength of the MRE. Stepanov et al. (2007) conducted elongation, static and dynamic shear experiments to study MRE behaviour in the presence of a homogeneous magnetic field. It was found that the elastic modulus increased as an external homogeneous magnetic field increases up to 0.3 T (tesla).

Other researchers mainly focussed on the design of MRE smart devices. Deng and Gong (2008) developed an adaptive tuned vibration absorber (ATVA) by using MRE as its smart spring elements. From the theoretical and experimental results, the developed ATVA performed a frequency shift from 27.5 Hz to 40 Hz and it showed vibration absorption capability up to 25 dB. Blom and Kari (2012) used the MR material in the form of rubber bushing and investigated its property with the influence of frequency, amplitude and magnetic field. The frequency and amplitude-dependent stiffness of the bushing was modelled based on geometric dimensions and shear modulus. From the results presented, it can be seen that the stiffness of the bushing can be controlled over a large range of the applied magnetic field strength.

An interesting application of MRE is the control and suppression of vibrations using switchable stiffness because of their characteristics of being able to rapidly change between low and high stiffness states. This can be particularly suitable for on–off switching stiffness strategies for shock and vibration isolation and suppression (Ledezma-Ramirez et al, 2011, 2012). More recently, MRE have been applied in sandwich beams using rectangular control functions leading to faster dissipation of transient vibrations (Eshaghi et al., 2016; Dyniewicz et al., 2015; Landi et al., 2019).

Based on the previous literature overview, the main contribution of the study is to provide a simple physical-based model to predict the compressional stiffness changes because of a magnetic field aligned with the MRE element, following the model proposed by Jolly et al. (1996). First, the material samples and their manufacturing process are presented because this will be used to present analytical and numerical results. Then the stiffness of the composite in the absence of a magnetic field is predicted from the properties of the matrix and of the particles, which in this particular application are silicone rubber and CIPs, respectively. The effect of the magnetic field is quantified and a nonlinear constitutive equation for the magnetic field given. To calculate the latter, the magnetic induction in the MRE element is computed. Eventually, a relationship for the linearised stiffness around the equilibrium position given as a function of the magnetic field, and the vibration isolation and absorption properties are obtained from the transmissibility and impedance functions. The second part of the study presents the experimental characterisation of the MRE samples to show the usefulness of the model by assessing the stiffness change under the influence of a magnetic field and the applicability of the material into vibration control devices. The natural frequency and damping are obtained by shaker excitation for different configurations, and transmissibility and impedance functions are presented. The shock isolation properties are also briefly considered. The study finishes with a general discussion of the theoretical and experimental findings, highlighting the possibilities for further developments of semiactive stiffness control strategies for vibration and shock control.

2. Material samples manufacturing

The MRE samples manufactured and tested in this study have been made using silicone rubber (MM288 by ACC Silicones Ltd) as matrix and carbonyl iron powder (CIP Type SQ by BASF SE) as ferromagnetic particles. Type SQ CIP consists of fine, grey soft iron powder particles which are spherical in shape with an average diameter of 5 μm and with a 99.5% iron content purity. Density of the CIP, ρ_CIP, and the silicone rubber, ρ_SR, are 7.874 g/cm³ and 1.12 g/cm³, respectively. Silicone rubber was chosen because it is easy to use and possesses good properties such as very good chemical and UV resistance in a wide temperature range (Schubert 2014). Iron particles were chosen because of their low remnant magnetisation, high permeability and high saturation magnetisation (Lokander and Stenberg 2003). Because of these properties, high magnetorheological effects can be obtained as a result from better inter-particle interaction (Carlson and Jolly 2000).

The MM288 silicone rubber kit comprises two components, namely the rubber and the hardener which have been mixed together with a 10:1 ratio. The desired percentage of CIP was added first to the silicone rubber, and then the hardener was added as the last component. As recommended by the supplier, no additive was required for the MM228 kit since the viscosity of the silicone rubber is relatively low in comparison to the one of other materials used in MRE (Schubert 2014). The mixture was placed into a vacuum chamber for 10 min to remove any trapped air inside the mixture. After this degassing process, the MRE mixture was poured into an aluminium mould. All surfaces of the mould in contact with the mixture were sprayed with silicone release agent for ease of removing the MRE from the mould. The rubber was cured at room temperatures for 24 h although the curing process can be accelerated by heating (curing at 100°C takes only 30 min).

Each sample is in the form of a cylinder with a 30 mm diameter and a 12 mm height. Samples with a CIP volume percentage of 0, 10 and 30% have been manufactured and tested. Table 1 shows the material properties of the silicone rubber and of the CIP.

Table 1.

Material parameters.

	Description	Value
E _SR	Young’s modulus silicone rubber	1.013 MPa
ν _SR	Poisson ratio silicone rubber	0.45
E _CIP	Young’s modulus carbon iron powder	211 GPa
ν _CIP	Poisson ratio carbon iron powder	0.33

3. Modelling

In this section, a constitutive model of the MRE, able to predict the change in the effective compressional modulus of elasticity because of exposure to an aligned external magnetic field, is proposed. First, the effective modulus of elasticity for the composite MRE, assumed to be a homogeneous and isotropic suspension of metal spherical particles into the elastomeric matrix, is considered. Next, following the approach proposed by Jolly et al. (1996) for magnetorheological fluids, the magnetic induction in the metal spherical particles and in the elastomeric matrix is calculated taking into consideration an elemental cell of the MRE. Consequently, the magnetic constitutive response of the MRE is obtained. Finally, since the obtained stress–strain curves are highly nonlinear, the linearised effective elastic modulus around an equilibrium position is calculated for the subsequent linear vibration response predictions.

3.1. Elasticity of the MRE composite

To calculate the MRE composite effective elasticity in absence of magnetic field, it was assumed that the ferromagnetic spherical particles are evenly and homogeneously distributed in the matrix, hence that the macroscopic response can be obtained from the microscopic model. Assuming each ferromagnetic particles is separated by a distance r₀ from any other ferromagnetic particle in the matrix suspension, the elementary cell representing the material is given by a cube of size r₀ including a ferromagnetic sphere of diameter d. It is also assumed that CIP are evenly distributed in the rubber matrix, no voids are present in the matrix and both of the materials work as linear elastic materials. Figure 1 shows the elemental cell considered for the microscopic model, where d is the diameter of the iron particles and r₀ is the distance between them when no magnetic field is applied.

The volume fraction of the iron particles in the rubber matrix, φ, is defined as the ratio of the volume of the ferromagnetic particle over the volume of the elementary cell

φ = \frac{π d^{3}}{6 r_{0}^{3}} = \frac{π}{6 h^{3}}

(1)

where h = r₀/d is an inverse indicator of the gap between two adjacent particles. When there is no ferromagnetic particle, φ is zero and h = ∞. When spherical particles are considered, the maximum admissible φ is π/6 ≈ 0.52 for which h = 1.

A mixture law can be used to find the effective macroscopic properties of the MRE material. The effective mass density, ρ_eff, is simply related to the CIP volume fraction by

ρ_{eff} = φ ρ_{CIP} + (1 - φ) ρ_{SR}

(2)

where ρ_CIP and ρ_SR are the densities of the CIP and of the silicone rubber, respectively.

To find a mixture law for the effective Young’s modulus, it is important to understand how the load is shared between the matrix and the particles. Given the large number of simplifications already adopted and wishing to avoid unnecessary and unjustified complication, the parallel and series connections have been considered and taken into account. Figure 2 shows the variation of effective modulus of elasticity as a function of the volume ratio of CIP in MRE φ for the two mixture laws proposed. The material properties of the two materials are given in Table 1. The predictions for series and parallel model are compared in Figure 2 to a mechanical finite element method (FEM) model and to some experimental measures obtained from the samples later described. From the plot, it is clear that the series connection provides an approximate estimation of the macroscopic modulus of elasticity with the same order of magnitude as predicted by the FEM and experimental tests. Hence, it was decided to use the series formula as prediction of the effective stiffness

E_{eff} = \frac{E_{CIP} E_{SR}}{(1 - φ) E_{CIP} + φ E_{SR}}

(3)

where E_CIP and E_SR are the Young’s modules for CIP and silicone rubber, respectively. Furthermore, the validity of the series model is also supported by the fact that higher ranges of particle content are not practical and may not satisfy the following magnetic field interaction model.

3.2. Induction curves

The relationship between the magnetic induction B (T = N/Am) and the magnetic intensity H (A/m) is found following the procedure suggested by Jolly et al. (1996) which assumes that the polarisation of the iron particles begins at the polar region and increases towards the centre as the applied field increases. Figure 3 shows a schematic of the unit cell where the matrix is represented in grey and the saturated and unsaturated region in the ferromagnetic particles are indicated by the dark and light blue, respectively. In addition to this, the reluctance of the unsaturated region is assumed negligible (μ_p ≫ 1), the relative permeability of the matrix and saturated portion of the particle are assumed unity (μ_m = μ_s = 1) and the flux lines are assumed perpendicular to the secants that define the saturated region. Finally, assuming that all the polarisation is because of the iron particles only

B = μ_{0} H + φ 〈 J_{p} 〉

(4)

where ⟨J_p⟩ is the average polarisation density of the iron particles and μ₀ is the vacuum permeability. ⟨J_p⟩ is calculated integrating over the saturated and unsaturated regions with the assumption that the magnetic induction in the saturated portion of the particles is

B_{s} \approx μ_{0} H + J_{s}

(5)

where J_s is the saturation magnetisation (for pure iron J_s = 2.1 T). Such integration gives an induction relationship

μ_{0} H = \frac{B - φ J_{s} (1 - α^{3})}{1 + (3 / 2) φ (α - α^{3})}

(6)

where α = 2s/d is a measure of the non-saturated region. As shown in Figure 3, s measures the unsaturated region and d the diameter of the iron particle. α will be 1 when there is no saturation and at very low applied fields, and it will be 0 when all the iron particles are fully saturated.

To find α, the magnetic flux circuit through the particles has to be considered. It is assumed the reluctances of the matrix R_m and particles R_p are in parallel and that the magnetic flux (BA) is the same through the saturated and unsaturated region (series connection): B_pA_p = B_sA_s = B_uA_u. The magnetic induction flux across the particles (B_sA_s) can be defined as function of the total magnetic flux as

B_{s} A_{s} = \frac{1}{1 + (R_{p} / R_{m})} B A

(7)

where B_s is the magnetic induction in the saturated particles,

A = r_{0}^{2}

is the area of a cell and A_s = π(r² − s²) is the area at the saturation threshold. The area ratio can be also expressed as function of the volume ratio and the saturation ratio

\frac{A_{s}}{A} = π (1 - α^{2}) {(\frac{3}{4} \frac{φ}{π})}^{2 / 3}

(8)

whereas the ratio of reluctances in the matrix and in the particle can be approximated as

\frac{R_{p}}{R_{m}} \approx k \frac{2}{π (1 + α)} [{(\frac{3}{4} \frac{φ}{π})}^{- 1 / 3} - π {(\frac{3}{4} \frac{φ}{π})}^{1 / 3}]

(9)

where k is a factor added by Jolly et al. (1996) to accommodate for magnetic interaction between adjacent columns and packing arrangements and obtain a better fit to experimental data.

Substituting equations (8), (9), (5) and (6) in (7), an implicit function of α is obtained

\begin{array}{l} (μ_{0} H + J_{s}) {π (1 - α^{2}) {(\frac{3}{4} \frac{φ}{π})}^{2 / 3}} \\ {1 + k \frac{2}{π (1 + α)} [{(\frac{3}{4} \frac{φ}{π})}^{- 1 / 3} - π {(\frac{3}{4} \frac{φ}{π})}^{1 / 3}]} \\ + - μ_{0} H [1 + \frac{3}{2} φ (α - α^{3})] - φ J_{s} (1 - α^{3}) = 0 \end{array}

(10)

The root of equation (10) gives α as function of the magnetic intensity H. Figure 4(a) shows the polarisation ratio α for three values of the volume ratio φ. Hence, equation (6) can be used to find the magnetic induction B, whereas equation (4) gives the average polarisation. Figures 4(b) and (c) show the average polarisation ⟨J_p⟩ and the magnetic induction B as function of the magnetic intensity H.

3.3. Magnetic stress–strain relationship

Filler particles are considered to be homogeneously distributed in the silicone rubber matrix. When a magnetic field is applied, the ferromagnetic particles are polarised in magnetic dipoles which tend to attract each other. The microscopic individual cell used to study the effect of the magnetic field and the macroscopic constitutive behaviour of the material has been represented by two attracting dipoles as it is shown in Figure 1. The interaction energy E between two dipoles with equal strength m is given by Jolly et al. (1996); Rosensweig (1985)

E = \frac{| m^{2} | (1 - 3 \cos^{2} θ)}{4 π μ_{0} μ_{1} {| r |}^{3}}

(11)

where μ₀ = 4π × 10⁻⁷NA⁻² and μ₁ are the permeability of free space and the relative permeability of the rubber matrix, respectively. r is the distance between the two dipoles, and θ is the angle between the two dipoles which it is assumed to be zero hereafter. Assuming that the shear deformation is negligible, the distance r between the dipoles can be expressed as a function of the axial strain ɛ = Δr/r₀

r = r_{0} + Δ r = r_{0} (1 + ε) = d h (1 + ε)

(12)

Hence, E can be simplified and rewritten in terms of scalar strain ɛ

E = \frac{- {| m |}^{2}}{2 π μ_{0} μ_{1} d^{3} h^{3} {(1 + ε)}^{3}}

(13)

Assuming that the particles are aligned in long chain and there is no multipole interaction, the total energy density U (energy per unit of volume) associated with the one-dimensional strain can be calculated by multiplying the particle-to-particle energy by the total number of particles (N) and dividing by the total volume $(N (π d^{3} / 6) / φ)$

U = \frac{E φ}{(π d^{3}) / 6} = \frac{- 3 {| m |}^{2} φ}{π^{2} μ_{0} μ_{1} d^{6} h^{3} {(1 + ε)}^{3}}

(14)

The stress induced by the magnetic field application can be obtained by differentiating U with respect to scalar strain ɛ

σ_{magnetic} = \frac{\partial U}{\partial ε} = \frac{9 {| m |}^{2} φ}{π^{2} μ_{0} μ_{1} d^{6} h^{3} {(1 + ε)}^{4}}

(15)

Since dipole moments are difficult to measure microscopically, it is more convenient to express dipole moments in terms of the average particle polarisation ⟨J_p⟩ as $| m | = 〈 J_{p} 〉 (π d^{3} / 6)$ . Hence, the nonlinear magnetic stress–strain relationship can be expressed as

σ_{magnetic} = \frac{3 {〈 J_{p} 〉}^{2} φ^{2}}{2 π μ_{0} μ_{1} {(1 + ε)}^{4}}

(16)

where ⟨J_p⟩ can be obtained from equation (4).

Figure 4(d)–(f) shows the stress–strain curves for different values of magnetic induction B and volume ratio φ. The force of interaction reduces, increasing the distance between the magnetic dipoles.

3.4. Linearised magnetorheological stiffness

The MRE is composed by CIP in a silicone rubber matrix. An applied magnetic field will cause the CIP magnetic attraction. This is equivalent to an external force acting on the material which will compress the matrix as detailed in the previous section. At the same time, the elastic response of the composite will oppose such magnetic attraction force until the equilibrium state is reached. Considering small displacement around the equilibrium position, the combined linearised stiffness can then be calculated. The force equilibrium equation can be written as

\frac{3 {〈 J_{p} 〉}^{2} φ^{2}}{2 π μ_{0} μ_{1} {(1 + ε)}^{4}} + E_{eff} ε = 0

(17)

where it has been assumed a linear constitutive equation for the mechanical stress. E_eff can be estimated using equation (3) or it can be measured experimentally from a material sample.

Solving the fifth order polynomial obtained rearranging equation (17), the equilibrium strain for a given average polarisation can be found

ε^{5} + 4 ε^{4} + 6 ε^{3} + 4 ε^{2} + ε + \frac{3}{2 π} \frac{{〈 J_{p} 〉}^{2} φ^{2}}{μ_{0} μ_{1} E_{eff}} = 0

(18)

Only the one root of the polynomial which falls within the range −1 ≤ ɛ ≤ 0 will be considered as the static equilibrium position. Figure 4(g) depicts the variation of equilibrium position against applied magnetic field for different volume ratios φ. Experimental values of E_eff obtained from the samples have been used to facilitate comparison with the experimental campaign results. Larger magnitude changes of the equilibrium position can be observed as the applied magnetic field is increased.

The relation between σ_magnetic and ɛ can be linearly approximated within a small strain range around the equilibrium position ɛ_eq using Taylor’s series expansion. Therefore, the factor (1 + ɛ)⁻⁴ of σ_magnetic in equation (16) can be expanded using Taylor’s series

{(1 + ε)}^{- 4} \approx {(1 + ε_{e q})}^{- 4} - 4 {(1 + ε_{e q})}^{- 5} Δ ε

(19)

where Δɛ = ɛ −ɛ_eq is the strain from the equilibrium strain ɛ_eq that accounts for the further MRE deformation after the static equilibrium state is reached. Hence, an expression for the linearised magnetic Young’s modulus can be obtained

E_{magnetic, lin} = \frac{6 {〈 J_{p} 〉}^{2} φ^{2}}{π μ_{0} μ_{1} {(1 + ε_{e q})}^{5}}

(20)

where the sign has been changed because externally applied forces have to be considered here. Figure 4(h) shows the linearised magnetic Young’s modulus as a function of volume concentration and induced magnetic field. Figure 4(i) shows the dependency of the total linearised stiffness of the MRE material, E_tot = E_eff + E_{magnetic, lin}, by the induction magnetic field.

3.5. Predicted impedance and transmissibility

The MRE samples have been manufactured in a cylindrical form. It is assumed that the MRE cylinders are subject to compressional deformation only and their dynamical behaviour can be mainly represented as that of compressional springs. The total complex stiffness of the sample from the linearised stress equation as $k = (1 + i η_{SR}) (E_{SR} + E_{magnetic, lin}) (A / L)$ , with η_SR is the loss factor of the silicone rubber, A and L the cross-sectional area and height, respectively. When used as adaptive isolators or dynamic vibration absorbers, additional masses representing payload, permanent magnets and electromagnets are added to them. For such applications the MRE sample can be modelled as a stiffness element between two masses m₁ and m₂ as shown in Figure 5. Hence, the predicted point impedance for an input force F and velocity ${\dot{X}}_{1}$ at the shaker table can be computed as

Z = \frac{F_{1}}{{\dot{X}}_{1}} = \frac{(- ω^{2} m_{1} + k) (- ω^{2} m_{2} + k) - k^{2}}{i ω (- ω^{2} m_{2} + k)}

(21)

and the transmissibility between base displacement X₁ and mass displacement and X₂ as

T = \frac{X_{2}}{X_{1}} = \frac{k}{- ω^{2} m_{2} + k}

(22)

Figure 6 shows the predicted point impedance and transmissibility for the case of 10% and 30% CIP volume ratio and 1 T change in the magnetic induction field. The resonances shift 4.7% for the 10% samples (from 93.7 Hz to 98.17 Hz) and 7.7% for the 30% samples (from 170.6 Hz to 183.7 Hz). Despite the small change in the resonance behaviour of the samples, such adaption can still be effectively used in suitably designed devices. Such results will be compared with the experimental results in the next section giving an insight in the advantages and limitations of the adopted model.

4. Experimental procedure

This section describes the tests performed to assess the properties of the MRE samples. First, vibration transmissibility and mobility were measured. Then shock isolation properties were measured. An electromagnet (Magnetech opposite pole electromagnet OP-1212) was placed on top of the sample, acting both as an isolated mass of 141 g and as the source of the magnetic field. The electromagnet is cylindrical in shape, and both the length and height are 31.75 mm. Although the electromagnet shows an internal resonance at about 350 Hz, this is highly damped and it can be effectively considered, in the frequency range of interest, as a rigid mass. It is however important to take the dynamical response of the electromagnet in account for the further design of vibration control devices. The electromagnet was supplied with a DC voltage from a high current source. According to the manufacturer data sheet (Magnetech Corporation 2020), the electromagnet produces an average nominal flux density of 0.075 T when driven at 18 V DC drawing a current of 2.2 A (10% duty cycle). To increase the magnetic range of investigation, one or two permanent magnets (with a mass of 18 g) were placed at the bottom and top of the sample. A limitation of this setup is that because of the size of the MRE sample, the magnetisation may not be uniform. Three different configurations of the sample have been studied, and they can be seen in Figure 7. For each configuration, driving conditions at different supply voltages were considered. For clarity, only a total of five different conditions are going to be reported. Figure 7(a) represents the first configuration of the MRE samples with the electromagnet at the top as payload mass. Condition (a) refers to this configuration with no magnetic field applied. The condition with the electromagnet turned off is considered as the reference basis for other measurements and comparisons. The second configuration, that is, Figure 7(b), sees the MRE sample with one permanent magnet at the bottom of the MRE and the electromagnet at the top. Conditions (b), (c) and (d) refer to this configuration when a supply voltage of 0, 18 and −18 V is provided to the electromagnet, respectively. The largest voltage supplied to the electromagnets was ±18 V to avoid overheating. Figure 7(d) shows a picture of the real sample in this configuration. The final configuration, that is, Figure 7(c), sees the MRE sample with a permanent magnet at the bottom and one at the top. Both magnets were oriented with the same polarity. Condition (e) refers to this configuration when no voltage is supplied to the electromagnet.

4.1. Magnetic field measurements

A preliminary measurement of the magnetic field of the permanent magnet and of the electromagnet has been carried out in air. The permanent magnet’s magnetic field was measured at 70 mT with a Gaussmeter PCE-MFM 3000. The magnetic field at the electromagnet top was measured with a SS49 E linear Hall effect sensor. The measurement was repeated for two available electromagnets. Both electromagnets showed an almost linear relationship between the voltage and the magnetic field with a constant of proportionality of 0.0032 T/V.

4.2. Frequency response function results

The MRE samples have been tested as indicated in Figure 7. They were secured to the shaker table via an impedance head and with the electromagnet resting on them. This configuration can be used to measure both their isolation (considering the electromagnet as the payload) and their absorption capabilities (considering the MRE sample and the electromagnet to act as a dynamic vibration absorber). Hence, to assess the isolation and absorption properties of the 10% and 30% MRE samples, the impedance and transmissibility were measured for the different conditions described before. The samples were excited by an electrodynamic shaker LDS V406, and the input force and acceleration were acquired with an impedance head sensor PCB Piezotronics model 288D01, whereas the transmitted acceleration on top of the electromagnet was measured with a miniature accelerometer PCB Piezotronics 352C22. The samples were excited by a broadband random excitation up to 500 Hz through the shaker. The amplitude of the input signal was kept constant during the tests at about 1.5 g (rms) to assure linearity which was confirmed by the measured coherence function. The signals from the impedance sensor and the accelerometer were acquired through dynamic signal analyser Dataphysics QUATTRO. It is important to mention that the tests were performed at constant room temperature to avoid changes in stiffness and damping because of viscoelastic effects. It is also known that stiffness and damping in rubbers change with frequency, but because only a relatively short frequency range is analysed in the test under random excitation and not cyclic loading, this effect was not taken into account. The tests were repeated twice swapping the face of the permanent magnet. To observe the largest frequency changes, some examples of the impedance and transmissibility functions are presented in Figure 8. For those voltages, the largest frequency changes are obtained, whereas when the electromagnet is turned off, the natural frequency falls in the middle. When the bottom permanent magnet is completely removed, the resulting natural frequency is the lowest registered, effectively measuring the properties of the sample with no magnetic effect.

In addition, the properties of the silicon rubber sample with no CIP, that is, 0%, were measured, resulting in a natural frequency of 90.83 Hz and viscous damping ratio of 0.0743. On the other hand, Table 2 presents the calculated values of natural frequencies and loss factor for the 10% and 30% samples, respectively. The data were processed using a circle fit of the impedance measurements, for the different conditions considered, that is, case (a) for the samples with no permanent magnet (PM) and the electromagnet (EM) turned off, cases (b) with the permanent magnet at the bottom and voltages of +18 V and −18 V supplied to the electromagnet, and lastly two permanent magnets with the electromagnet turned off acting only as payload. These scenarios are summarised on Table 2 together with the percentage change in the stiffness and measured natural frequency. The stiffness change was estimated considering case (a) as reference for comparison, that is, the electromagnet turned off without the permanent magnet at the bottom.

Table 2.

Values of natural frequency in Hz, damping ratio and stiffness change measured for different configurations of the MRE sample with 10% and 30% of particles concentration (PM: permanent magnet; V_EM: voltage supplied to the electromagnet).

	B (T)	f_n (Hz)		ζ		% Δk
(a). No PM, V_EM = 0 V	0	98	168	0.1148	0.1309
(b). 1 PM, V_EM = 0 V	0.07	150	180	0.1105	0.1551	57.315	12.888
(c). 1 PM, V_EM = 18 V	0.127	164	214	0.0945	0.0961	64.292	38.370
(d). 1 PM, V_EM = −18 V	0.013	143	174	0.1065	0.0858	53.034	6.777
(e). 2 PMs, V_EM = 0 V	0.14	175	247	0.1016	0.0801	68.640	53.737
Particle concentration φ (%)		10	30	10	30	10	30

Assuming the magnetic field is because of the superposition of the magnetic fields, generated in vacuum from the permanent magnet and the electromagnet, it is possible to estimate the variation of the Young’s modulus of the MRE samples. In other words, it has been assumed that the ferromagnetic particles in the sample do not alter the magnetic field and that the stiffness of the sample is related to the Young’s modulus as k = EA/L where A and L are the cross-sectional area and height of the sample. The results of the Young’s modulus estimation are presented in Figure 9. Comparing Figure 9 with Figure 4(i), it is clear that the model suggested here describes the change of the effective Young’s modulus because of the magnetic field only up to about 0.08 T. For larger magnetic field, the magnetic compression seems to produce a stiffening in the samples, which is probably because of the hyperelastic response of the composite. This also explains why the model predicts that the resonance shift would happen over a larger change in the magnetic induction field.

The variation of isolation properties was also evaluated for different values of the supplied voltage. For this experiment, the voltage was varied from −18 V to +18 V in 9 V steps, for two conditions of the permanent magnet at the bottom, that is, for condition 1 the positive pole (or face 1) of the magnet was facing up, whereas in condition 2, the face was swapped, that is, negative pole facing up. The change of natural frequency as a function of the supplied voltage is presented in Figure 10, for the two samples considered and the two conditions of the permanent magnet. Thus, it can be seen that when the polarity of the permanent magnet is inverted by swapping the face, the trend in the variation of the natural frequency is also reversed as a result.

4.3. Shock response results

Shock response was evaluated experimentally by applying a velocity compensated half-sine acceleration pulse in the base of the system. The pulse was generated by the data acquisition software and applied through the electrodynamic shaker. The rest of the experimental setup was the same as the previous stage. Pulse durations of 1 ms, 5 ms, 10 ms, 15 ms and 20 ms were considered to cover the different areas typically found in the shock response of a linear system, that is, pulses shorter than the natural period, around the natural period and larger than the natural period. The amplitude of the pulse ${\ddot{y}}_{p} = {\ddot{x}}_{1}$ in all cases was kept constant to 1 g acceleration. For the shock response, only the 30% particle volume concentration sample was studied because it gives the largest stiffness changes for the different configurations. Figure 11 presents the time response for the different configurations of the MRE studied, where each subplot gives a particular pulse duration. The response amplitude ${\ddot{x}}_{2}$ is normalised by the shock amplitude ${\ddot{y}}_{p}$ .

The maximum value of the absolute response is compared to the maximum input amplitude and presented as a function of the period ratio between the duration of the pulse τ and the natural period of system T in Figure 12. For this plot, the natural period of the MRE sample with no permanent magnet and the electromagnet turned off (case a), that is, approximately 5 ms for the 30% sample was considered as a reference. This effectively represents the shock response spectra, and it is presented for the different configurations of the MRE.

5. Discussion of experimental findings

The mobility function and transmission ratio presented in Figure 8 show the effect of change in the natural frequency of the isolator sample. First, the effect of the particle percentage is clearly seen. Compared to the sample with 0% CIP concentration which has a natural frequency of 90.83 Hz, the stiffness change because of the addition of the particles is 14% and 71% for the 10% and 30% samples, respectively. When comparing the 10% and 30% samples without any magnetic effect (case a), the natural frequencies are 98 Hz and 168 Hz, respectively, resulting in a change of stiffness of approximately 66%. This obeys to the fact of increased stiffness because of the higher concentration of particles as expected. However, the main objective is to compare the effect of different magnetic fields applied to the sample. The initial value for comparison as a reference is considered when the bottom permanent magnet is in place and the electromagnet is turned off effectively acting as a payload, that is, condition (b). The effective natural frequency for this state is 150 Hz and 180 Hz for the 10% sample and the 30% sample, respectively. The frequency is increased to 164 Hz and 214 Hz in each sample when a positive voltage is applied, resulting in a stiffness increase of 64% and 38%. On the other hand, a negative voltage produces a decrease in the frequency, corresponding to 143 Hz and 174 Hz, respectively. The actual stiffness changes from minimum to maximum stiffness for each sample results in 53% and 7% for the 10% and 30% samples, respectively, when considering condition (a) as a baseline. However, for practical implications, if condition (b) is taken as a reference, then by switching on and off the electromagnets in reverse polarities, the stiffness can be increased and reduced from the baseline, which can be useful for further control strategies where it is important to have the largest stiffness change possible from a minimum to a maximum value.

A practical limit of the electromagnets used on the experiments was the maximum voltage, current and time of operation. Because of the high current drawn at 18 V, they experienced a quick overheating. Thus, it was not possible to sustain the tests for higher voltages and longer times. In addition, this will also affect the total mass of the system because increasing the magnetic field, that is, by using a stronger electromagnet will result in a larger and heavier system, thus limiting practical applications. However, the feasibility of quickly changing the effective stiffness of the system was shown in the tests. To observe larger stiffness changes, permanent magnets were used. In this case, the maximum natural frequencies recorded were 175 Hz and 247 Hz for the 10% and 30%, respectively, and considering the initial sample with no magnetic effects as reference, the actual stiffness increase was 68% and 53% for each sample.

A comparison with the predicted stiffness values is presented in Table 3 for the 10% and 30% samples. Theoretical stiffness is calculated from the Young modulus values presented in Figure 9 as k = EA/L, whereas experimental stiffness is estimated from the natural frequencies discussed in Table 2 and the payload mass. Only three values of applied magnetic field are included for brevity, that is, 0 T, 0.07 T and 0.14 T. It can be seen that the predictions provide acceptable results useful for later implementations. However, the difference becomes large with higher magnetic fields, probably because of the limitations in the experimental testing, that is, magnetisation not uniform, and previously discussed simplifications in the model.

Table 3.

Comparison of the theoretically predicted stiffness and the experimental results.

B (T)	k_theory (N/m)		k_exp (N/m)		Error %
0	58,905	170,824.5	53,081	155,993	10.4	9
0.07	129,591	188,496	124,357	179,074	4.1	5.1
0.14	200,277	412,335	169,263	337,195	16.8	20
φ (%)	10	30	10	30	10	30

The shock isolation results also agree with previous findings and are not surprising as a system with lower stiffness results in a better shock isolation for short pulses compared to the duration of the pulse, that is, $(τ / T) < 2$ . Only the 30% sample was studied in these experiments because it is the one with the largest stiffness change considering case (b) as a reference, and as a consequence larger differences in shock response can be appreciated. In this case, the higher effects in the response were observed for the 5 ms, 15 ms and 20 ms pulses. For example, in the case of the 5 ms pulse considered short compared to the natural period, the shock response decreases approximately 6% for the lower stiffness value corresponding to a supply voltage of −18 V and increases 8% for the larger stiffness at 18 V, taking case (b) as baseline. For longer pulses, $(τ / T) > 2$ the opposite effect is observed as a lower stiffness increases the response and a higher stiffness decreases the response. This effect is because of the system approaching the amplification region, that is, by reducing the stiffness, the relative shock duration to system natural period ratio decreases and it is moved towards the amplification region in the shock response spectra. Increasing the stiffness moves the effective period ratio towards the quasi-static zone resulting in a longer pulse compared to the duration of the shock. When using both permanent magnets, the effects discussed are observed at a larger magnitude. However, this proves useful for further use in a switchable stiffness strategy as previous studies have suggested, or even the implementation of a two stage mount with switchable stiffness.

6. Conclusions

The design and modelling of simple vibrations isolators using MREs was presented and discussed. Even though the use of MRE is relatively well understood for vibration isolation and suppression, this study attempts to present a simple yet functional model for axial configurations. Based on the properties of the individual components, that is, the silicon rubber matrix and the carbonyl iron powder, the Young modulus of the mixture was established. Then the magnetic effects were investigated, finding polarisation curves as a function of the volume ratio between the particles and the matrix and stress–strain relationships under the effect of the magnetic field for the same particle concentrations. Finally, a linearised stiffness was predicted for the MRE samples based on the Young modulus of the mixture and the magnetic Young modulus. The model is limited to the linear elastic response and does not model the hardening which happens for large magnetic induction fields. A compensation for this limit has been obtained increasing the simulated induction field. Vibration transmissibility and mobility were calculated for different particle concentrations and magnetic field values. Then MRE samples were manufactured in different particle concentrations, and the change of stiffness was assessed under different magnetic field levels in shaker testing. Vibration transmissibility and mechanical impedance were experimentally measured, finding that it is possible to obtain a significant stiffness change of up to 64% which is controllable by changing the magnitude and polarity of the voltage supplied to the electromagnets. Shock response was also evaluated, finding better isolation for low stiffness states. The proposed procedure is simple, but it allows us to estimate the stiffness of a MRE sample for different percentages of CIP volume concentration as a function of the applied magnetic field. Estimating theoretically, the minimum and maximum stiffness of MRE samples can prove useful in designing switchable stiffness strategies. In addition, the properties of the experimentally developed samples, that is, a large change in stiffness in a very short time, makes them suitable for use in further stiffness switching strategies for vibration isolation and suppression.

Footnotes

Acknowledgements

The authors would like to acknowledge the ‘Mexico-UK Visiting Chair’ mobility funding between the University of Southampton and the Universidad Autonoma de Nuevo León for the support provided in the development of this project.

Declaration of conflicting interest

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iDs

Emiliano Rustighi

Neil Ferguson

Azrul Zakaria

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