SCIENCE CHINA Technological Sciences RESEARCH PAPER April 013 Vol.56 No.4: 878 883 doi: 10.1007/s11431-013-5168-7 Nonlinear dynamic characteristics of magneto-rheological visco-elastomers YING ZuGuang 1* NI YiQing & SAJJADI Masoud 1 Department of Mechanics School of Aeronautics and Astronautics Zhejiang University Hangzhou 31007 China; Department of Civil and Structural Engineering The Hong Kong Polytechnic University Hung Hom Kowloon Hong Kong China Received June 5 01; accepted January 14 013; published online March 9 013 The smart magneto-rheological visco-elastomer (MRVE) has a promising application to vibration control. Its dynamic characteristics are described by complex moduli which are applicable to linear dynamics. However experimental results show remarkable nonlinear relations between force and deformation for certain large deformations and the nonlinear dynamic modeling needs to be developed. The present study focuses on the nonlinear dynamic characteristics of MRVE. The MRVE was fabricated and specimens were tested to show nonlinear mechanical properties and dynamic behaviors. The nonlinear effect induced by applied magnetic fields was investigated. A phenomenological model for the dynamic behaviors of MRVE was proposed to describe the nonlinear elasticity linear damping and hysteretic effect and the corresponding equivalent linear model in the frequency domain was also given for small deformations. The proposed model is applicable to the dynamics and control analysis of composite structures with MRVE. nonlinear hysteresis dynamic modeling magneto-rheological visco-elastomer complex stiffness Citation: Ying Z G Ni Y Q Sajjadi M. Nonlinear dynamic characteristics of magneto-rheological visco-elastomers. Sci China Tech Sci 013 56: 878 883 doi: 10.1007/s11431-013-5168-7 1 Introduction *Corresponding author (email: yingzg@zju.edu.cn) Magneto-rheological elastomer is a smart composite material which consists of magnetically polarizable particles and non-magnetic polymers for instance iron particles silicone oil and rubber [1 1]. The elastomer has time-invariant properties such as the naught settlement of magnetic particles in magneto-rheological fluids. Its dynamic characteristics including stiffness and damping can be changed reversibly under external magnetic fields applied in milliseconds. As the material with controllable viscousness and elasticity in dynamics it is called magneto-rheological viscoelastomer (MRVE). The MRVE has a promising application to vibration control in engineering and the study on its dynamic model is necessary [13 ]. Several MRVEs have been fabricated and tested for magnetic-mechanical properties and dynamic behaviors. A static modeling for the shear modulus of MRVE with applied magnetic field has been presented based on the magnetic dipole interaction and polymeric nonlinear elasticity [6 3 4]. A complex shear modulus has been presented to describe the dynamic properties of MRVE with magnetic field based on the polymeric dynamic characteristics in the frequency domain [19 5]. However the complex modulus is only applicable to the linear dynamics of MRVE in small deformation. Experimental results showed remarkable nonlinear relations between force and deformation for certain large deformations. Also a hysteretic model has been presented and used for describing the dynamic behaviors of magneto-rheological fluid dampers with applied magnetic field [6]. Therefore a non- Science China Press and Springer-Verlag erlin Heidelberg 013 tech.scichina.com www.springerlink.com
Ying Z G et al. Sci China Tech Sci April (013) Vol.56 No.4 879 linear dynamic model of MRVE needs to be developed for vibration control application. In this paper the nonlinear dynamic characteristics of MRVE were exhibited by experimental results and a phenomenological modeling was proposed for its nonlinear dynamic behaviors. An MRVE was firstly fabricated by using silicone rubber silicone oil and carbonyl iron particles. Tension and compression specimens were tested under applied magnetic fields for nonlinear mechanical properties and dynamic behaviors. The dynamic relations between force and deformation were described by backbone and loop curves. Then a phenomenological dynamic model was constructed which includes a nonlinear elasticity for the backbone a viscous damping and nonlinear hysteresis for the loop and coupling effect. The reduced nonlinear visco-elastic model in the time domain for low frequency and the equivalent linear model in the frequency domain for small deformation were also presented. The model parameters were identified based on test results by using the least square method and combining the characteristics of hysteretic curves and the indices of elastic potential and dissipative energies. The simulation results were compared with the test results to illustrate the capacity of the proposed model. The equivalent stiffness and loss factor were given to show the dependence on the frequency and magnetic field. Figure 1 MRVE section micrograph (500 optical magnification). Experiment on mechanical properties and dynamic behaviors of MRVE Figure Tensile force versus deformation. The MRVE specimens were fabricated by using silicone rubber silicone oil and carbonyl iron particles. The iron particles with an average diameter of 4 m and an average density of 3700 kg m 3 were used as magnetic fillers. The silicone oil with a viscosity of 5 Pa s was used for the performance regulator. The silicone rubber with two-components was used as the matrix material. According to certain ingredient percentage the iron particles were dispersed thoroughly in the silicone oil and the blended liquid was mixed with the silicone rubber further. Then the homogeneous mixture was poured into a mold and cured under certain temperature conditions. The MRVE specimens were obtained finally. ased on the micrograph of inner planar section (Figure 1) the MRVE has proportionally distributed iron particles and is a composite continuous medium without any inner air cavities. According to the test standard of polymeric materials the tension specimen had a gage size of 33 mm 6 mm 3.5 mm (weight percentage of iron particles silicone oil and silicone rubber is 6::). The tensile test was performed on a testing machine (Zwick Z.5) for mechanical properties. The magnetic field was applied perpendicularly to the tensile force. Numerical results on tensile forces and elongation deformations were obtained from the automatic control and dataprocessing system. Figure shows that the tensile force depends linearly on the elongation deformation without the magnetic field and the dependence relation becomes nonlinear under applied magnetic fields. The dependence relation can be regarded as linear only in small deformation and the elastic stiffness increases with magnetic field intensity. The compression specimen was a small cylinder with a diameter of 40 mm and a length of 14 mm (weight percentage of iron particles silicone oil and silicone rubber is 7:1:). The compression test was performed on the same testing machine. The magnetic field was applied in parallel with the compressive force. Figure 3 shows the relation between the compressive force and shortening deformation similar to the tension. The basic mechanical properties including the nonlinear elasticity and hardening rigidity of the MRVE under applied magnetic fields were obtained by the test results. The dynamic behaviors were tested by the tension and compression circulation under certain amplitude and frequency. The test specimen was a small cylinder with a diameter of 40 mm and a length of 14 mm (weight percentage of iron particles silicone oil and silicone rubber is 7:1:). The dynamic test was performed on a material testing machine (MTS-810). The magnetic field was applied in paral-
880 Ying Z G et al. Sci China Tech Sci April (013) Vol.56 No.4 modeled equivalently as Figure 3 Compressive force versus deformation. f ( u ) cu () v where f v denotes the viscous damping force c is a damping coefficient dependent on the magnetic field u is velocity and the over dot represents the derivative with respect to time. y combining eqs. (1) and () the dynamic force of the MRVE is described by f e +f v. As the velocity or frequency increases the nonlinear damping and its coupling with elasticity become outstanding which can be expressed by a nonlinear hysteresis. According to the hysteretic dynamic modeling for magneto-rheological fluids with applied magnetic field [6] the hysteretic force of the MRVE can be described by the typical ouc-wen model. The differential equation for the hysteretic force f h is lel with the tensile and compressive forces so that the basic mechanical properties of MRVE could be obtained. Figures 4 and 5 show the test results on dynamic forces and deformations of the specimen under different deformation frequencies (5 Hz 8 Hz) and magnetic field intensity (0.14 T). The dynamic behaviors including the nonlinear elasticity (curve slope) and damping (loop area) were observed. The non-elliptical curve without rigidity indicates the nonlinear damping and hysteresis characteristics as exhibited by magneto-rheological fluids with applied magnetic fields. 3 Nonlinear dynamic modeling and numerical results on MRVE 3.1 Nonlinear hysteretic model in time domain The dynamic relation between forces and deformations was described by backbone and loop curves. Then a phenomenological model for the dynamic characteristics of MRVE was constructed which includes nonlinear elasticity for the backbone viscous damping and nonlinear hysteresis for the loop and coupling effect. The elasticity and damping were expressed by elastic force and damping force respectively. ased on the experimental results the applied magnetic field would induce nonlinear elasticity. The nonlinear elastic force f e of the MRVE with magnetic field can be modeled as N i f ( u) au (1) e i 0 where u denotes the dynamic deformation a i are coefficients dependent on the magnetic field and N is an integer. f e is a linear elastic force for N=1 or nonlinear elastic force for N>1. In general the damping force has complicated relations with deformation. However in the case of small velocity or frequency the damping force of the MRVE can be i f u u f u f f (3) n n 1 h h h h where and n are coefficients dependent on the magnetic field. Therefore the nonlinear hysteretic model or the dynamic force of the MRVE is described by f ( u) f ( u) f ( u ) f ( u) (4) e v h where is a coefficient dependent on the magnetic field. The coefficients in model (4) can be determined by test results on dynamic forces and deformations under different frequencies and magnetic fields. The least square method can be used for identifying coefficients in eqs. (1) and (). The coefficients in eqs. (3) and (4) can be identified by combining the characteristics of hysteretic curves and the indices of elastic potential and dissipative energies. The test results on the relation between dynamic force and deformation of the MRVE specimen under a frequency of 8 Hz and a magnetic field of 0.14 T are shown in Figure 5. The nonlinear curve and loop area indicate a nonlinear elastic force and damping force respectively. The non-zero deformation under zero force and the non-elliptical curve without rigidity imply the hysteretic effect. According to the nonlinear hysteretic model (4) the coefficients for N=3 were obtained by using the least square method and combining the characteristics of hysteretic curve and the indices of elastic potential and dissipative energies as follows: a 0 =18.16 N a 1 =86.00 N mm 1 a = 13.70 N mm a 3 = 8.81 N mm 3 c=0.38 N s mm 1 =35.37 N =1.0 mm 1 =0.5 mm 1 =1.5 mm 1 and n=0.8. The modeling results are given in Figure 5 which agree well with the test results. oth the difference of potential energies and the difference of dissipative energies for experiment and simulation are less than 1%. The descending relation of a 1 > a > a 3 implies that the first several terms in eq. (1) can describe the nonlinear elasticity. As the frequency decreases the nonlinear damping and hysteretic effect can be left out. The test results on the rela-
Ying Z G et al. Sci China Tech Sci April (013) Vol.56 No.4 881 F( U) F ( U) F ( U) F ( U) e v h a E E U 0 R I [ ( ) j ( )] ( )[1 j ( )] (5) R a0 E U Figure 4 Dynamic force versus deformation for 5 Hz and 0.14 T. where F and U denote respectively the transformed force and deformation F e F v and F h are respectively the transformed linear elastic force damping force and hysteretic force is a frequency j 1 E R and E I are respectively the real and imaginary stiffnesses is the loss factor. a 0 =0 for the center of force according with deformation. In general the equivalent dynamic stiffness and loss factor of the MRVE increase with the frequency. When the frequency is less than a certain value the dynamic stiffness and loss factor depend linearly on the frequency and are expressed as R E 0 1 0 1 (6) where 0 1 0 and 1 are coefficients dependent on the magnetic field. efore the magnetic saturation the coefficients can be approximated for instance to 0 00 01 m 0 m 1 10 11 m 1 m 0 00 01 m 0 m 1 10 11 m 1 m (7) Figure 5 Dynamic force versus deformation for 8 Hz and 0.14 T. tion between dynamic force and deformation of the MRVE specimen under a frequency of 5 Hz and a magnetic field of 0.14 T are shown in Figure 4. According to the visco-elastic dynamic model combining nonlinear elasticity (1) and linear damping () the coefficients for N=3 were obtained by using the least square method as follows: a 0 =17.34 N a 1 =80.00 N mm 1 a = 14.56 N mm a 3 =6.53 N mm 3 and c=0.88 N s mm 1. The modeling results are given in Figure 4 which accord well with the test results. Also there exists the descending coefficient relation that is a 1 > a > a 3. 3. Equivalent complex stiffness in frequency domain As the deformation of the MRVE is small for instance in micro-vibration [19 1] the corresponding dynamic force can be regarded as linear and the coupling effect in hysteresis is slight. The dynamic force of the MRVE in the time domain (4) can be transformed equivalently into the following in the frequency domain: where m denotes the magnetic field intensity ij and ij (i=0 1; j=0 1 ) are constants. The constants in eq. (7) can be determined by the least square method based on test results on dynamic forces and deformations under different frequencies and magnetic fields. The damping coefficient c for the loss factor is expressed by the area of curve loop A r and the deformation amplitude u a as c A /(π u ). (8) r The dynamic forces dependent on deformations for the MRVE under certain frequency and magnetic field can be given to illustrate the dynamic behaviors and also used for calculating the equivalent stiffness and loss factor or damping in small deformation. In this case the corresponding dynamic forces are regarded as linearly dependent on deformations and modeled in the frequency domain by eq. (5). The constants in eq. (7) are obtained by using the least square method based on test results and then the equivalent stiffness and loss factor (6) are determined. Figures 6 and 7 show the test and simulation results on the equivalent stiffness and loss factor of the MRVE specimen varying with frequency under different magnetic fields (strain less than 4%) respectively. It is seen that the dynamic stiffness and loss factor depend linearly on the frequency almost in the given interval and depend nonlinearly on the magnetic field. According to the dynamic model (6) and (7) the coeffi- a
88 Ying Z G et al. Sci China Tech Sci April (013) Vol.56 No.4 cients were obtained as follows: 0 ={40.6 kn m 1 17.57 kn T m 1 074.31 kn T m 1 } 1 ={3.16 kn s m 1 33.87 kn s T m 1 37.78 kn s T m 1 } ={ 0.16 kn s m 1 3.19 kn s T m 1 5.86 kn s T m 1 } 0 ={0.130 0.843 1.69} 1 ={0.07 s 0.161 s 0.388 s} and ={ 0.001 s 0.014 s 0.041 s }. The coefficient j is much less than 1j or 0j and j is much less than 1j or 0j (j=0 1 ) which imply the linear dependences of the equivalent stiffness and loss factor on the frequency. Figure 6 illustrates the increase of the equivalent stiffness with the frequency and magnetic field intensity. Figure 7 illustrates that the loss factor increases with the frequency and has a maximal value for various magnetic fields. The stiffness and damping characteristics can be used for the fabrication of MRVE. 4 Conclusions The MRVE specimens have been fabricated by using sili- cone rubber silicone oil and carbonyl iron particles and tested under applied magnetic fields for nonlinear mechanical properties and dynamic behaviors. The phenomenological model for the nonlinear hysteretic dynamic behaviors of MRVEs has been proposed and reduced to the equivalent linear model in the frequency domain for small deformation. The model can describe the dynamics including the nonlinear elasticity linear damping and hysteretic effect of MRVE and accord well with the test results. It has been obtained that: 1) The applied magnetic fields can induce the nonlinear relation between forces and deformations; ) The dynamic behaviors at low frequency can be modeled by combining the nonlinear elasticity and linear damping; 3) The equivalent stiffness and loss factor depend linearly on the frequency in certain interval and depend nonlinearly on the magnetic field for small deformation. The proposed model is applicable to the dynamics and control analysis of composite structures with MRVE. This work was supported by the National Natural Science Foundation of China (Grant No. 110715) the Fundamental Research Funds for the Central Universities and the Hong Kong Polytechnic University through the Development of Niche Areas Programme (Grant No. 1-95). Figure 6 Equivalent stiffness versus frequency under different magnetic fields (dot for test; line for modeling). Figure 7 Loss factor versus frequency under different magnetic fields (dot for test; line for modeling). 1 Shiga T Okada A Kurauchi T. Magnetoviscoelastic behavior of composite gels. J Appl Polym Sci 1995 58: 787 79 Carlson J D Jolly M R. MR fluid foam and elastomer devices. Mechatronics 000 10: 555 569 3 Ginder J M Clark S M Schlotter W F et al. Magnetostrictive phenomena in magneto-rheological elastomers. Int J Mod Phys 00 16: 41 418 4 ellan C ossis G. Field dependence of viscoelastic properties of MR elastomers. Int J Mod Phys 00 16: 447 453 5 Demchuk S A Kuz min V A. Viscoelastic properties of magnetorheological elastomers in the regime of dynamic deformation. J Eng Phys Therm 00 75: 396 400 6 Shen Y Golnaraghi M F Heppler G R. Experimental research and modeling of magneto-rheological elastomers. J Intell Mat Syst Str 004 15: 7 35 7 Wang Y Hu Y Gong X et al. Preparation and properties of magnetorheological elastomers based on silicon rubber/polystyrene blend matrix. J Appl Polym Sci 007 103: 3143 3149 8 öse H. Viscoelastic properties of silicone-based magnetorheological elastomers. Int J Mod Phys 007 1: 4790 4797 9 Stepanov G V Abramchuk S S Grishin D A et al. Effect of a homogeneous magnetic field on the viscoelastic behavior of magnetic elastomers. Polymer 007 48: 488 495 10 Kallio M Lindroos T Aalto S et al. Dynamic compression testing of a tunable spring element consisting of a magnetorheological elastomer. Smart Mater Struct 007 16: 506 514 11 Koo J H Khan F Jang D D et al. Dynamic characterization and modeling of magneto-rheological elastomers under compressive loadings. Smart Mater Struct 010 19: 11700 1 Kaleta J Krolewicz M Lewandowski D. Magnetomechanical properties of anisotropic and isotropic magnetorheological composites with thermoplastic elastomer matrices. Smart Mater Struct 011 0: 085006 13 Tombul G S anks S P. Nonlinear optimal control of rotating flexible shaft in active magnetic bearings. Sci China Tech Sci 011 54: 1084 1094 14 Yuan Z C Shi J M. Research on EM pulse protection property of plasma-microwave absorptive material-plasma sandwich structure.
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