Nonlinear principal resonance of magneto-electro-elastic thin plate

XUE Chun-xia, REN Xiu-juan

(School of Science, North University of China, Taiyuan 030051, China)

Nonlinearprincipalresonanceofmagneto-electro-elasticthinplate

XUE Chun-xia, REN Xiu-juan

(SchoolofScience,NorthUniversityofChina,Taiyuan030051,China)

Abstract：Considering magneto-electro-elastic thin plate, the von Karman plate theory of large deflection and the geometric nonlinearity, the mathematical model of nonlinear undamped forced vibration is established.Making use of the improved Lindstedt-Poincare (L-P) method, the undamped forced vibration problem is solved, and the amplitude-frequency response equation of thin plate is obtained.Furthermore, the amplitude frequency response curves of system under different conditions are obtained by numerical simulation.The results show that the thickness of the plate, mechanical excitation, parameterε, pure piezoelectric material of BaTiO3, pure piezomagnetic material of CoFe2O4, different magneto-electro-elastic materials of BaTiO3/CoFe2O4and Terfenol-D/PZT will have an impact on the system frequency response.The main effects involve principal resonance interval, spring stiffness characteristic and amplitude jumping phenomena.

Key words：magneto-electro-elastic thin plate; improved Lindstedt-Poincare (L-P) method; principal resonance; amplitude-frequency response curve

CLDnumber： O322Documentcode： A

As a new composite intelligent material, piezoelectric and piezomagnetic composite material[1]possesses good magneto-electric coupling effect.It has the ability to convert energy from one form to the other among electric, magnetic and mechanical energies, so it has wide application in engineering.

Recently, with the large progress of study on magneto-electric material characteristics, more and more piezoelectric and piezomagnetic materials have attracted people’s attention.In order to get better application in industry, it has become particularly important to study the mechanical properties of magneto-electric materials.At present, there are many studies about piezoelectric and piezomagnetic materials done by domestic and foreign scholars: Rajesh and Ganesan[2]analyzed the free vibration problem of simply supported functionally graded and layered magneto-electro-elastic plates by the finite element method; CHEN et al.[3]gave out the state equation and buckling equation aiming at the stabilization problem of orthotropic magneto-electro-elastic rectangular plate; LIU and CHANG[4]studied the linear vibration problem of anisotropic magneto-electro-elastic rectangular plate; XUE et al.[5]studied the large deflection problem of magneto-electric rectangular thin plate.However, little attention has been paid to the nonlinear dynamic resonance of magneto-electric composite materials.For magneto-electric composite materials, the magneto-electric coupling effect can not be neglected.Therefore, nonlinear studies on magneto-electric composite materials have become particularly important, especially on those aspects involving its nonlinear vibration.

Taking the transversely isotropic magneto-electro-elastic thin plate as the research object, this paper combines the von Karman plate theory[6]of large deflection and geometric nonlinearity, and then analyzes the pertinent questions using improved Lindstedt-Poincare (L-P) method[7], finally establishes the mathematic model of nonlinear and undamped forced vibration.After selecting two kinds of magneto-electric composite materials, namely, BaTiO3/CoFe2O4[5]and Terfenol-D/PZT[8], this paper obtains the influence of different parameters on the nonlinear vibration characteristic of magneto-electro-elastic thin plate by analyzing its amplitude-frequency response curves, which makes the two magneto-electric materials better applied in the engineering practice.

1 Basic equations of magneto-electric-elastic thin plate

We consider a rectangular transversely isotropic magneto-electric-elastic thin plate in the Cartesian coordinate system (x,y,z), as shown in Fig.1.The length, width and thickness of the plate are,a,bandh, respectively.The coordinate planeOxyis attached to the middle plane of the plate with a simple harmonic mechanical loadqinz-direction, whereq=q0cosωt.

Fig.1 Schematic of a rectangular magneto-electric-elastic thin plate under mechanical load q

In terms of transversely isotropic magneto-electro-elastic rectangular thin plate, both the electric and magnetic fields in the plane can be ignored.According to the fundamental assumption of Kirchhoff thin plate theory[9], the following constitutive equations[10]can be obtained,

τxy=c66γxy,

(1)

(2)

(3)

whereσj,τj,DjandBjrepresent the normal stress, shearing stress, electric displacement and magnetic induction component, respectively;φandψare, respectively, the electric potential and magnetic potential;cij,εijandμijare the elastic, dielectric and

magnetic conductance constants, respectively;eij,qijandmijare, respectively, the piezoelectric, piezomagnetic and magneto-electric material constants; and for transversely isotropic material, the relationc11=c12+2c66holds.

The equations of dynamic equilibrium[11], including the balance of the electric and magnetic quantities, are given as

(4)

Di,i=0,

(5)

Bi,i=0,

(6)

whereuiis the elastic displacement component, andρis remembered as the mass of plate in unit area.

As the electric field and magnetic field of MEE thin plate in the surface can be ignored, we can get the following equation[12],

Bx=By=Dx=Dy=0.

Using von Karman’s theory of large deflection of plates, the following geometric equations are obtained as

(7)

whereu,vandware the elastic displacement components inx-,y- andz-direction, respectively; andtis the time variable.

Based on the assumption of Kirchhoff thin plate theory, we can assume the unknown elastic displacement vectoruas

(8)

whereuis the displacement vector of the plate.

Then, substituting Eqs.(2), (3) and (7) into Eqs.(5) and (6) gives

(9)

Δ1=e31μ33-m33q31,

Δ2=q31ε33-m33e31,

andφ0(x,y) andψ0(x,y) are functions independent of variablez.

We now consider a simply-supported plate.For this case, the mechanical boundary conditions on the sides of the plate are

The elastic displacementwcan be assumed as

(10)

The stress functionF(x,y) is introduced, thus Eq.(1) can be simplified as

(11)

By taking the second derivative of Eq.(7) and combing the resulting expressions, it can be shown as

(12)

Substituting Eqs.(9), (10) and (11) into Eq.(12), it can be obtained as

Recalling that the resultant force and moment are defined as

(14)

Integrating Eq.(4) with respect toz, equilibrium equation is given by

(15)

Substituting Eqs.(10) and (14) into Eq.(15), according to Bubnov-Galerkin method[13], the following dynamic equation about this system can be found

(16)

where

ω0is the natural frequency of the system.

To solve Eq.(16), we use the dimensionless method[14].In other words, we assume that

τ=ω0t,

The nondimensional equation about this system turns into as

?τ,

(17)

where

2 Solution of amplitude-frequency response equation

To solve Eq.(17), we use the improved L-P method[7].The order of nonlinear and extrinsic motivation terms are assumed asO(ε), whereεis a small parameter.Then Eq.(17) can be changed into

(18)

The initial conditions of the system are assumed as

whereAis the initial amplitude of this system.

On the assumption ofγ=τ, substituting this equation into Eq.(18) gives

2f″+f+εk3f3=εkcosγ.

(19)

(20)

(21)

Substituting Eqs.(20) and (21) into Eq.(19), the amplitude-frequency response equation can be obtained as

(22)

3 Numerical simulation

For magneto-electric-elastic thin plate, two kinds of new materials are selected, namely BaTiO3/CoFe2O4and Terfenol-D/PZT, with the plate sizes ofa=1 m andb=0.2 m.The material parameters are listed in the following Tables 1 and 2.

Table1MaterialcoefficientsofBaTiO3/CoFe2O4materialinthetableobtainedbythesimpleruleofmixturewith50%BaTiO3and50%CoFe2O4

c11(N/m2)c12(N/m2)e31(C/m2)q31(N·A·m-1)ε33(C2/N·m2)μ33(Ns2/C2)m33(Ns/VC)2.25×10111.25×1011-2.2290.26.35×10-98.35×10-52.334 1×10-9The density of BaTiO3/CoFe2O4 thin plates is 5 430 kg/m3.

Table2MaterialcoefficientsofTerfenol-D/PZTmaterialobtainedbythesimpleruleofmixturewith50%Terfenol-Dand50%PZT

c11(N/m2)c12(N/m2)e31(C/m2)q31(N·A·m-1)ε33(C2/N·m2)μ33(Ns2/C2)m33(Ns/VC)1.04×10116.205×1010-3.25-30.456.37×10-93.76×10-64.048 5×10-8The density of Terfenol-D/PZT thin plates is 8 350 kg/m3.

While the thickness of BaTiO3/CoFe2O4plate varies as 0.01 m, 0.015 m and 0.02 m,εandq0are fixed atε=1 andq0=0.01 MPa.It is obvious from Fig.2(a) that the width of the resonant region becomes narrow and the amplitude in the same frequency gradually decreases.While the mechanical loadq0varies as 0.01, 0.05 and 0.1 MPa,εandhare fixed atε=1 andh=0.01 m.The phenomenon can be shown in Fig.2(b), with the increase of extrinsic motivation amplitude, the width of the resonant region increases, and the amplitude in the same frequency gradually increases.

Fig.2 Amplitude-frequency response curves of the coupled BaTiO3/CoFe2O4 plate under different thickness and mechanical excitation

While the BaTiO3/CoFe2O4plate’s parameterεvaries as -1.2, -1, -0.8 and 0.8, 1, 1.2, respectively;handq0are fixed ath=0.01 m andq0=0.01 MPa.

According to Fig.3(a), whenεis less than zero, the system presents the nonlinear characteristics of “gradually soft” spring, the curve bending to the left, and the system’s principal resonance interval narrowing, the nonlinear soft effect gradually decreasing with the increase ofεvalue.Meanwhile, the amplitude-frequency curves show classic nonlinear phenomena, including multi-value, jump, delay, etc.The amplitude-frequency curve ofε=-1 is divided into two parts by its spine line.The spine line is the characteristic curves of materials without the influence of mechanical excitation.

According to Fig.3(b), whenεis greater than zero, the system takes on the nonlinear characteristics of “gradually hard” spring, the curve bending to the right, and its nonlinear hard effect gradually decreasing with the increase ofεvalue.

In a word, it can be seen that the response amplitude value of hard spring jumps to lower place while that of soft spring jumps to higher place.

Fig.3 Amplitude-frequency response curves of the coupled BaTiO3/CoFe2O4 plate while h=0.01 m and q0=0.01 MPa are fixed, and parameter ε varies as -1.2, -1, -0.8 and 0.8, 1, 1.2, respectively

While the material parametersε=1,h=0.01 m andq0=0.01 MPa are fixed, Fig.4 shows the influence of the plates made of BaTiO3/CoFe2O4, piezoelectric BaTiO3and piezomagnetic CoFe2O4materials on the amplitude-frequency response curves.It is observed that the principal resonance interval and the bending corresponding to the piezoelectric BaTiO3plate are the largest among them, while those corresponding to the piezomagnetic CoFe2O4is the smallest.

Fig.4 Amplitude-frequency response curves of the plate made of BaTiO3/CoFe2O4, piezoelectric BaTiO3 and piezomagnetic CoFe2O4 materials

While the materials’ parametersε=1,h=0.01 m andq0=0.01 MPa are fixed, Fig.5 shows amplitude-frequency response curves for the plate made of BaTiO3/CoFe2O4and Terfenol-D/PZT materials.It is observed that the principal resonance interval corresponding to the BaTiO3/CoFe2O4material plate is smaller than that corresponding to the Terfenol-D/PZT material plate.And the bending corresponding to the BaTiO3/CoFe2O4material plate is larger than the Terfenol-D/PZT material plate.

Fig.5 Amplitude-frequency response curves for the plate made of BaTiO3/CoFe2O4 and Terfenol-D/PZT materials

4 Conclusion

Nonlinear principal resonances of a magneto-electric-elastic thin plate with large deflection are investigated in this paper.The governing equation and its corresponding asymptotic solution based on the improved L-P method are derived.For a plate made of BaTiO3/CoFe2O4or Terfenol-D/PZT, the influence of the thickness, external force and parameterεare demonstrated.The significance of this material and its interesting features will be useful in the analysis and design of magneto-electric-elastic related structures.

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磁電彈薄板的非線性主共振

薛春霞， 任秀娟

(中北大學 理學院， 山西 太原 030051)

摘 要:針對磁電彈性薄板， 結(jié)合大撓度板理論， 考慮幾何非線性， 建立了非線性無阻尼強迫振動的數(shù)學模型， 應用改進的L-P法， 對非線性無阻尼強迫振動問題進行求解， 得到薄板穩(wěn)定狀態(tài)下的幅頻響應方程， 數(shù)值模擬了不同情況下系統(tǒng)的幅頻響應曲線圖。 通過比較分析得出： 板的厚度、 外激勵力、 參數(shù)ε的不同取值以及壓電材料 BaTiO3、 壓磁材料CoFe2O4及磁電彈材料BaTiO3/CoFe2O4與Terfenol-D /PZT均會對系統(tǒng)的幅頻響應曲線產(chǎn)生影響， 主要表現(xiàn)為對系統(tǒng)主共振區(qū)間， 彈簧軟硬特性， 幅值跳躍現(xiàn)象的影響。 這些結(jié)論在理論上可以更好地指導工程結(jié)構(gòu)的設計。

關鍵詞:磁電彈性薄板； 改進的L-P法； 主共振； 幅頻響應曲線

引用格式：XUE Chun-xia, REN Xiu-juan.Nonlinear principal resonance of magneto-electro-elastic thin plate.Journal of Measurement Science and Instrumentation, 2014, 5(4): 93-98.[doi: 10.3969/j.issn.1674-8042.2014.04.018]

Article ID：1674-8042(2014)04-0093-06

10.3969/j.issn.1674-8042.2014.04.018

Receiveddate： 2014-07-05

Foundation item:National Natural Science Foundation of China (No.11202190); Scientific Research Staring Foundation for the

Overseas Chinese Scholars, Ministry of Education, China; Research Project Supported by Shanxi Scholarship Council of China (No.2013-085)

Corresponding author：XUE Chun-xia (xuechunxia@nuc.edu.cn)