A review on dynamic characteristics of magnetorheological elastormer sandwich structures

Ming LONG, Lin-sen LI, Gang LI, Guo-liang HU

(Key Laboratory of Conveyance and Equipment, Ministry of Education, East China Jiaotong University, Nanchang 330013, China)

Abstract: In recent years, more and more researchers focus on the magnetorheological elastomer (MRE) sandwich beam structures. MR elastomers have a better control performance than that of typical MR fluid. it has extensive applications in the semi-active control of structural vibration. In this paper, the studies on dynamic characteristics of the MRE sandwich beam at home and abroad are reviewed. The review focuses on constitutive models of material, methods of theoretical modeling and experimental analysis of MRE sandwich beam, researches and findings. The latest research results show that the natural frequency of MR sandwich beam can be increased or decreased by controlling the intensity and location of the applied magnetic field. At last, the content of this paper is summarized, and the future research direction in this field is prospected.

Key words: MRE, Sandwich structure, Dynamic characteristics

1 Introduction

Among the many smart materials, sandwich beam structure containing the magnetorheological (MR) materials has received wide attention in the semi-active control of structural vibration. MR materials are new smart materials which have MR effects and many unique properties under exerted magnetic field [1]. The properties of MR materials, such as stiffness, frequency of vibration, damping and other mechanical properties can be changed quickly, controllably and reversibly under the varying magnetic field. The process of change is instant, which can be completed within milliseconds [2]. Since sandwich structure has the rigidity-weight ratio, it can conveniently combine and optimize different surface material and the core layer material. And on the basis of small increase in weight, the stiffness of the structure can be greatly improved, so it can be used in vibration absorb in aerospace, transportation, civil structure or else. The MR material is used as the core layer in the traditional structure, the characteristic that the properties of MR materials will change with the applied magnetic field which is used to achieve the vibration control of the structure. This kind of smart material structure has many outstanding advantages, the sandwich structure can not only achieve the variable stiffness and damping, but also can ensure the rigidity of the composite structure and the ability to resist deformation, taking into account the effect of vibration and noise reduction. Therefore, more attention has been obtained in the semi-active control of flexible structures [3].

Since their discovery by Rabinow in 1948 [4], MR materials have developed into a family with MR fluid (MRF), MR foam and MR elastomer (MRE). At present, MR fluid is the most widely studied MR material. A lot of applications based on MRF benefit from the properties that the dynamic yield stress can be continuously, rapidly and reversibly controlled by the applied magnetic field. So MRF has been widely used in industrial engineering. But MRF also exhibits some shortages, such as deposition, poor stability, environmental contamination and sealing problems. Also, particle residue can degrade the performance of MR device, which hinder their wide application. MRE, the structural solid analogs of MRF, may be a good solution to overcome these disadvantages.

MRE is composed of magnetic polarizable particles (in the 1～100 μm range) dispersed in a polymer elastomer or gel-like matrix. Matrix can prevent particles from settling down. Typically, magnetic field is applied to the polymer composite during cross linking so that particles form chainlike or columnar structures, which is fixed in the matrix after curing [5]. MRE has two working modes: shear mode and squeeze mode, as shown in Fig.1[2]. The difference between MRE and MRF is that the MR effect of MRE is the field-dependent modulus within the pre-yield regime. This difference makes the operating mechanism of these two kinds of materials totally different, and the field-dependant property of MRE is more stable than that of MRF. Therefore, MRE has extensive application in the semi-active control of flexible structures. A review of studies reporting analyses of dynamics of MRE sandwich beam structure is presented in this paper. This article mainly exposits about four parts: constitutive models of materials, methods of theoretical modeling and experimental analysis of MRE sandwich beam.

Fig.1 Modes of operation for MRE

2 Constitutive models of materials

The constitutive equation and form of the MR elastomer have an important influence on the dynamic analysis of the system. The constitutive equation of the material describes the stress-strain relationship of the material. The rheological characteristic of MRE material can be described by elastic plastic material model. In order to identify the storage and loss moduli of the MRE in terms of applied field and frequency, some models have been developed. It should be known that there are no comprehensive model has been developed to describe pre-yield behavior of these smart materials. So, different models should be employed to describe the rheological properties of MR materials because of the complex constitutive relation. Viscoelastic models are the most common models to account for rheological properties of the smart elastomer in the pre-yield regime. The form of dynamic constitutive equation of viscoelastic materials has decisive influence on process of dynamic analysis of elastic-viscoelastic composite structure. Some typical viscoelastic models are described in detail.

2. 1 The basic form of constitutive equations

Suppose the viscoelastic material is linear, isothermal, homogeneous and isotropic, and it can be expressed in the form of Boltzmann solid integral [6], that is:

(1)

wheretis time,σ(x,t) is stress,ε(t) is strain,χ(t,τ)=E(τ)+Q(t,τ) is a function aboutτ(t≤0),E(τ) is young’s modulus andQ(t,τ) is relaxation function, ift<0,Q(t,τ)=0,ε(t)=0. Relaxation modulus is generally continuous monotone non-increasing function because of the characteristics of “attenuation forgets” of viscoelastic materials.

For non-aging viscoelastic materials, the equation (1) can be simplified as:

(2)

Where:

(3)

H(t) is unit step function.

Fourier transformation is applied to equation 2, and the result is:

(4)

2.2 Model of complex constant modulus

G*(ω)=G′(ω)+iG″(ω)

(5)

whereG′(ω) andG″(ω) are storage modulus and loss modulus of viscoelastic materials, respectively [7].

(6)

(7)

Application of complex modulus model, in the steady-state harmonic vibration, the viscoelastic sandwich structure forced vibration differential equation can be written to the complex form:

(8)

That is:

(9)

Thus we can use the complex arithmetic to find the solution of the complex form. Whereηis the damping loss factor of structure; [M], [K*] are the mass matrix and stiffness matrix of the system respectively; {q} is displacement matrix;f0is coefficient matrix of the excitation.

The basic characteristics of complex modulus model are non frequency variation and simple form, it can better describe the mechanical properties of viscoelastic materials under the harmonic excitation. At present, many literatures adopt complex constant models. Actually the viscoelastic modulus of the material is frequency dependent, so complex constant model is only applicable to analysis of steady-state harmonic vibration in finite frequency ranges.

2.3 General linear viscoelastic model

Weiss et al[8] theoretically studied rheological properties of MR materials and ER materials. The results showed the stress-strain relationship of MR material divided into three regions (pre-yield region, yield region and post-yield region), in the pre-yield region, MR materials performance as the viscoelastic properties. The viscoelastic properties can be described by the linear viscoelastic theory, that is:

τ=G*γτ<τy

(10)

Whereτrepresents the shear stress of the material,γis shear strain of the material,τyis yield strength, and the complex shear modulusG*is:

G*(B)=G′(B)+iG″(B)

(11)

WhereG′ is storage modulus, it is proportional to the average value of the energy stored of material in the unit volume in a certain deformation;G″ is loss modulus, it is proportional to the energy consumption of material in the unit volume after a period of deformation. The ratio between the storage modulus and the loss modulus is called the loss factor of the material, that is:

(12)

Whereωis the external excitation frequency when material withstands the shear deformation.

Qing Sun et al obtained a set of data through the experiment, and the nonlinear relationship between the complex modulus and magnetic field strength of the magneto rheological material was fitted [9]:

G′(B)=3.11×10-7B2+3.56×10-4B+5.78×10-1

(13)

G″(B)=3.47×10-9B2+3.85×10-6B+6.31×10-3

(14)

WhereBis magnetic flux density (unit: Gauss). The research showed that the shear stress of the MR sandwich beam is relatively small, and the shear strain of the MR layer is less than 0.1%, MR materials work in the pre-yield region. So the stress-strain relationship of the MR materials can be described by linear viscoelastic model. In addition, Ginder provided the relationship between shear modulus of MR fluid and the applied magnetic field [10], the relationship is as follows:

G′(B)=3φμ0MsB

(15)

Whereφis the volume fraction of suspended particles (such as ferromagnetic particles) of MR materials,μ0is permeability of free space,MSis saturation magnetization of ferromagnetic particles,Bis magnetic flux density.

2.4 Four parameter viscoelastic model based on Kelvin-Voigt model

Standard rheological model is more accurate, and has been widely used in structural dynamic analysis and other aspects [11]. There are two main kinds of analysis models to describe the MR elastomers: Maxwell model and Kevin-Voigt model [12-13]. However, Maxwell model is difficult to describe the creep behavior of the material, and the Kelvin-Voigt model is lacking in description of the stress relaxation behavior. Li et al extended the Kelvin-Voigt model, and proposed a four parameter viscoelastic model to describe the MR elastomers, the model can be described better the viscoelastic behavior of MR elastomers, the specific model is shown in Fig.2[14].

Fig.2 Four-parameter Viscoelastic Model

Assumeγ*is the complex shear strain of input,τ*is the complex shear stress of output andG*is the complex shear modulus. So the stress-strain relationship of the model is shown below:

τ*=G*γ*=(G1+iG2)γ*

(16)

WhereG1andG2are the real and imaginary parts of the complex modulus, respectively, they can be obtained by linear viscoelastic theory [15]:

(17)

(18)

Whereωis the excited frequency.

And Li used the Least Square Theory and the optimization toolbox of MATLAB to estimate the value of the parameters, and made a comparison between the result and the experimental data, shown in Table 1.

Table 1 Related parameters of four parameter viscoelastic model

3 Methods of theoretical modeling

The method of dynamic theoretical modeling of MRE sandwich beam is the basis of the analysis of the composite beams with viscoelastic materials. Due to the MR viscoelastic behavior occurs in the yield region, all the models of the vibration characteristics of the viscoelastic sandwich structure can also be applied to the MRE sandwich structure. The first model of viscoelastically damped structure in shear configuration was developed by Ross et al [15], and the model is known as Ross-Kerwin-Ungar (RKU). David studied the RKU model and pointed out that the model is based on a modified Euler-Bernoulli beam equation and is expressed as:

(19)

Wherem(x) is the mass per unit length of beam,EIis flexural rigidity of structure. This model is based on the assumption that the structure is subjected to sinusoidal motion, and it can be applied to the real structure only when the structure is subjected to simply supported boundary conditions.

Coulter and Duclos first employed the RKU model to the MRE sandwich beam with simply-supported structure, through rheological experiment, they obtained related data of the complex shear modulus [16]. The results showed that the theoretical calculation results of RKU model are in good agreement with the experimental results. Choi et al studied the rheology mechanism of ER materials using ASTM standard G756-83 in 1992 [17]. Experimental results showed that the rheological data does not match ASTM standards, thus, cantilever structure under the low order, the RKU model assumptions (the composite structure in simple harmonic motion) does not hold true.

But too large calculation amount of RKU model bring restrictions in application. So in practical application, the finite element method is the most reported approach in dynamic analysis of the MRE sandwich structures. The finite element method is a more accurate approximation method, which can transform a continuous unlimited freedom problem into a discrete finite degree of freedom problem. There are too many analyses and experiments show that the finite element method is an effective method which cannot be replaced. Jacques introduced a numerical analysis of the nonlinear vibration characteristics of viscoelastic sandwich beam by using finite element method [18]. Another application of the finite element method was made by Nayak in a rotating sandwich with MRE core [19]. The governing equation of motion is expressed in a matrix form:

(20)

Where[m], [k], and {f} are the mass matrix, stiffness matrix, and force vector, respectively. Besides, Hamilton energy method and Ritz method also have been widely reported. Sun [20] developed governing equations of motion of MR-based sandwich beams by using Hamilton principle, as follow:

(21)

Whereρis density of the beam,Efis Young’s modulus of each surface layer,G*is complex shear modulus of the core layer andIfis moment of inertia at the centroid of elastic layer. Besides,bis the beam width andh2is core layer thickness. Nayak analyzed the dynamic characteristics of MRE sandwich beam, and developed the equation of vibration characteristics by using the extended Hamilton energy method. At the same time, the influences of the parameters such as the magnetic field strength, the ratio of the ferromagnetic material, the length of the sandwich beam and the center layer thickness of the magneto rheological elastomer on the natural frequency of the system are analyzed [21]. Aguib,S. researched the dynamic behavior of the magnetorheological elastomer sandwich plate by Ritz method[22].Ritz method uses the basis function to describe the displacement of the structure, and the structure match the geometry boundary conditions. The governing equation expressed in this method is as follow:

(K-ω2M)C=0

(22)

WhereCis the vector of arbitrary coefficient, used to define displacement field,Kis the stiffness matrice andMis the mass matrice. The normal stress in the core layer, the shear strain and stress components in the elastic layers are neglected in all of these methods, and there are no slipping between the elastic layers and viscoelastic layer.

4 Experimental analysis of MRE sandwich beams

The theoretical analysis methods of the calculation and modeling of the characteristics of the MRE and the dynamic characteristics of the sandwich structure are mostly carried out by the simplified object. However, it is necessary to carry out the experiment to verify the theoretical results. In the experimental study of the vibration characteristics of the MRE, the fabrication of the MRE is very important.

The fabrication of MRE first appeared in 1995, Shiga from Research and Development Laboratory of Japan’s TOYOTA center fabricated the prototype of MRE, a kind of magnetic gel mixed with silicon rubber and iron powder. In the fabrication process, typically the natural rubber and synthetic rubber as the elastic matrix, silicone rubber is the most common matrix. Jolly developed MRE based on silicone rubber, result showed shear modulus of MRE in applied magnetic field increase about 40% more than the original [23]. Because the mechanical properties of silicone rubber are relatively poor, so a lot of researchers have used other rubber as the matrix. Hu fabricated the MRE by mixing polyurethane with silicon rubber, the test results showed that the MRE have a higher magnetic rheological effect [24].

In most of the experimental studies, the elastic surface layer of MRE sandwich beam is made of aluminum alloy material, as shown in Fig.3 [25]. Because the aluminum alloy has lower damping properties and higher stiffness compared with MRE. Moreover, the relative permeability of aluminum alloy is almost 1, so it will not affect the size and distribution of the applied magnetic field on the MRE. Free vibration, impact hammer and shaker excitation are main experiments have been conducted in studies. Experimental method uses experimental modal analysis technique, the main principle is using the real or mechanical structure model, product as the object to do dynamic experiments, data acquisition, signal analysis and processing, frequency response function estimation of response signal of test points of the exciting force and structure, and then dynamic modal parameter identification (curve fitting), can get quite a high accuracy of modal parameters (modal frequencies and modal shapes and modal damping, the typical experimental setup is shown in Fig.4 [26].

Fig.3 Sketch of a sandwich beam structure

Fig.4 Schematic typical experimental setup of MRE sandwich beam

5 Researches and findings

The theoretical analysis and experimental studies of MRE are mainly focused on the effects of various parameters on dynamic responses of MRE sandwich beam. Wei [27] tested and analyzed the vibration response characteristics of the MRE sandwich beam under the external uniform magnetic field, the results showed the amplitude of the vibration response of the laminated beam decreased and the first order natural frequency increased as the magnetic field increased, which is shown in Fig.5. Yeh [28] analyzed vibration characteristics of MRE sandwich beam, discussed the effects of the natural frequencies and the loss factors, the results showed that the modal loss factor and natural frequencies would increase with the increasing of the magnetic fields, as shown in Fig.6.

Fig.5 Amplitude-frequency response of the MRE beam under different magnetic field

Fig.6 Effects of magnetic field on the modal loss factor of the MRE sandwich beam

Hu analyzed the trends of the natural frequencies of the MRE sandwich beam under different magnetic field conditions, and simulation analysis was also studied to verify the correctness of the theoretical calculations [29]. The results agreed well with calculation, it indicated the natural frequencies and stiffness of the MRE sandwich beam increased with the magnetic field. Nayak et al investigated dynamic stability of rotating sandwich beam with MRE core[30]. They reported significant improvement in stability of the system in response to increasing magnetic field, rotational speed, and ratio of hub radius to beam length. Ramesh investigated the effects of magnetic field, taper angle of the top and bottom layers, aspect ratio, ply orientations and various end conditions on the various dynamic properties of tapered laminated composite MRE sandwich plate [31]. The results showed the loss factors and natural frequencies of the tapered composite sandwich plates increased with the magnetic field increased, the natural frequencies decreased with the taper angle of the face layers of the MRE sandwich plates increased.

Unlike most reseatchers that studing the vibration characteristics of MRE sandwich beam under the homogeneous magnetic field, Hu analyzed vibration characteristics under non-homogeneous magnetic field, the results showed the possibility of shift the natural frequency to lower frequency through applying non-homogenous magnetic field, which could be achieved by partially activated region of the sandwich beam, as shown in Fig.7 [32-33].

Fig.7 Vibration characteristics of MRE sandwich beam

There also have studies on the nonlinear behaviors of the MRE sandwich beams. Yildirim showed that increasing the imperfection led to higher natural frequencies, the first order natural frequency of the geometrically imperfect MRE sandwich increased with the external magnetic field increased [34]. Aguib analyzed the nonlinear static behavior of composite MRE sandwich beams, the results showed the modification of the magnetic field affected the stiffness and the damping characteristics of the beam, and the stresses and strains increased with the magnetic field increased, however, beam bending decreased with the magnetic field increased, as shown in Fig.8 [35].

Fig.8 Stiffness and the damping characteristics of the sandwich beams

6 Conclusions

This paper mainly summarized the studies on dynamic responses of MRE sandwich beams, a comprehensive review on constitutive models of materials, methods of theoretical modeling, experimental studies, and fabrication of MRE. There are many parameters influence the vibration characteristics, such as: magnetic field, boundary conditions, geometry. At present, most studies analyzed the vibration characteristics of the MRE sandwich beam under the uniform magnetic field, the results show that the natural frequency of MRE sandwich beams is increased with the magnetic field increased However, the studies on the vibration characteristics of MRE sandwich beams under the non-homogeneous magnetic fields reported rarely, the researches of it show that the natural frequency of MRE sandwich beams can be decreased with the location of magnetic field. In particular, the application of MRE sandwich beam structure is in the initial stage. This is not conducive to the full use of the characteristics of the MRE to develop the corresponding frequency damping device. Therefore, it is necessary to strengthen the researches on application of MRE sandwich beam in the future.