盛俊杰 李樹(shù)勇 張玉慶 陳花鈴

(1.中國(guó)工程物理研究院總體工程研究所, 綿陽(yáng) 621900) (2.西安交通大學(xué)機(jī)械工程學(xué)院, 西安 710049)(3.西安交通大學(xué)機(jī)械結(jié)構(gòu)強(qiáng)度與振動(dòng)國(guó)家重點(diǎn)實(shí)驗(yàn)室, 西安 710049)

介電彈性體材料致動(dòng)器的非線(xiàn)性動(dòng)態(tài)行為研究*

盛俊杰1?李樹(shù)勇1張玉慶1陳花鈴2,3

(1.中國(guó)工程物理研究院總體工程研究所, 綿陽(yáng) 621900) (2.西安交通大學(xué)機(jī)械工程學(xué)院, 西安 710049)(3.西安交通大學(xué)機(jī)械結(jié)構(gòu)強(qiáng)度與振動(dòng)國(guó)家重點(diǎn)實(shí)驗(yàn)室, 西安 710049)

介電彈性體材料(Dielectric Elastomer,簡(jiǎn)稱(chēng)DE),是制造柔性智能致動(dòng)器最有潛力的電活性聚合物(Electroactive polymer,縮寫(xiě)EAP)材料之一,可在電壓驅(qū)動(dòng)下產(chǎn)生大幅度的厚度與面積變形,最大面積應(yīng)變高達(dá)1600%.由于DE材料的固有阻尼特性,從而使其變形具有時(shí)間依賴(lài)性,因此動(dòng)態(tài)變形中,其能量轉(zhuǎn)換、宏觀變形等特性也必然受到阻尼的影響.借助熱力學(xué)自由能理論,考慮DE材料致動(dòng)器面內(nèi)動(dòng)態(tài)變形過(guò)程中的慣性力和阻尼力,構(gòu)建平面DE材料致動(dòng)器的非線(xiàn)性動(dòng)力學(xué)模型.研究了交變電載荷下DE系統(tǒng)的非線(xiàn)性動(dòng)態(tài)特性,包括其幅頻曲線(xiàn)、位移響應(yīng)和相平面圖.研究表明,存在阻尼的時(shí)候,DE系統(tǒng)的振動(dòng)會(huì)隨著時(shí)間的增加出現(xiàn)鎖頻現(xiàn)象,最后變成一種具有恒定振幅的振動(dòng).相平面圖和Poincaré映射圖研究表明,考慮阻尼后DE系統(tǒng)的穩(wěn)態(tài)相平面圖是一條閉合的曲線(xiàn),其Poincaré映射點(diǎn)集是有限的,代表其產(chǎn)生周期性運(yùn)動(dòng).研究成果為DE材料致動(dòng)器在各種動(dòng)態(tài)驅(qū)動(dòng)器或傳感器中的應(yīng)用提供理論依據(jù).

介電彈性體材料, 動(dòng)態(tài), 阻尼, 相平面, Poincaré映射

引言

介電彈性材料(Dielectric Elastomer,簡(jiǎn)稱(chēng)DE)是一種新型的智能電活性聚合物(Electroactive Polymer,簡(jiǎn)稱(chēng)EAP).相比于其他EAP材料,DE材料獨(dú)有的特點(diǎn)是變形大,最大面積應(yīng)變高達(dá)1600%[1],且彈性模量低,機(jī)電耦合效率高,工作溫度范圍寬和頻率范圍廣(0.1～20KHz).DE致動(dòng)器指的是在DE材料上下表面涂有柔性電極的三明治結(jié)構(gòu),在電極上施加電壓后,DE材料上產(chǎn)生的Maxwell電場(chǎng)力和電致伸縮電應(yīng)力的共同作用擠壓材料,結(jié)果使其面積增大,厚度減小,并逐漸成為近幾年國(guó)內(nèi)外的研究熱點(diǎn)之一[2-4].近年來(lái),研究學(xué)者設(shè)計(jì)出很多種基于DE材料的致動(dòng)器和換能器結(jié)構(gòu),并嘗試將DE應(yīng)用于智能機(jī)器人[5]、微機(jī)電系統(tǒng)[6]、揚(yáng)聲器[7]和能量回收[8-9]等領(lǐng)域.

在理論建模研究中,最具代表性的研究當(dāng)屬Harvard大學(xué)的Suo[10]小組,他們基于熱力學(xué)的自由能理論,構(gòu)建了一套非線(xiàn)性本構(gòu)方程,從而能夠從理論角度分析DE材料的力電耦合過(guò)程以及失穩(wěn)產(chǎn)生的臨界條件.由于該理論物理意義明確,擴(kuò)展性好,在國(guó)內(nèi)外得到了廣泛的應(yīng)用和推廣,基于該理論的研究也層出不窮,如:UT Austin大學(xué)的Huang[11]研究了力電耦合下的相變問(wèn)題;Harvard大學(xué)的Bertoldi[12]研究了多層DE材料中的不穩(wěn)定性的傳播問(wèn)題;新加坡A-star研究中心的Koh[13]等人研究了預(yù)拉伸下的穩(wěn)定性機(jī)理問(wèn)題;奧地利Johannes Kepler大學(xué)的Bauer[14]等人研究了變形失穩(wěn)的能量轉(zhuǎn)換效率問(wèn)題;哈爾濱工業(yè)大學(xué)的劉立武[15]應(yīng)用兩個(gè)材料常數(shù)的Mooney-Rivlin彈性應(yīng)變能函數(shù),分析了介電彈性體穩(wěn)定性行為,盛俊杰[16]在自由能模型基礎(chǔ)上研究了溫度對(duì)介電彈性體材料力電耦合變形的影響等.

在工程實(shí)際應(yīng)用中,對(duì)DE材料施加的載荷常常是周期性的電壓或者應(yīng)力,如微泵結(jié)構(gòu)、各種運(yùn)動(dòng)驅(qū)動(dòng)器、能量循環(huán)收集器等等[4].然而現(xiàn)有研究對(duì)DE材料在交變載荷下的動(dòng)力學(xué)變形行為以及力電耦合特性尚沒(méi)有得到足夠的重視.近兩年來(lái),學(xué)者們才開(kāi)始對(duì)DE材料動(dòng)態(tài)特性進(jìn)行研究.哈佛大學(xué)的Zhu從熱動(dòng)力學(xué)出發(fā),通過(guò)擾動(dòng)方法分別研究了DE氣球[17]的非線(xiàn)性振動(dòng)特性；蘭州大學(xué)的Yong等人[18]建立了一套DE動(dòng)態(tài)分析模型,研究了厚球殼的穩(wěn)定性；浙江大學(xué)的Li[19]研究了純剪切共振器的力電耦合性能及其動(dòng)態(tài)性能；Federal University of Goiás的Soares等[20]利用打靶法對(duì)預(yù)拉伸后的超彈性平面薄膜的動(dòng)態(tài)方程進(jìn)行了求解,并與有限元求解方法進(jìn)行了對(duì)比；最近TU Darmstadt的Xu[21]利用朗格朗日方程得到了平面DE的動(dòng)態(tài)運(yùn)動(dòng)方程,并研究了動(dòng)態(tài)載荷下的位移響應(yīng)和穩(wěn)定性,但是沒(méi)有考慮預(yù)應(yīng)力的影響;西安交通大學(xué)的Li[22]研究了DE材料共振器在純剪切工作模式下的動(dòng)態(tài)性能；西安交通大學(xué)的zhang[23]利用歐拉拉格朗日方法研究了三種外力邊界條件下DE致動(dòng)器的動(dòng)態(tài)性能.

研究發(fā)現(xiàn)阻尼是DE材料的固有特性[24],可以用來(lái)減振.最近兩年雖然一些研究機(jī)構(gòu)開(kāi)始關(guān)注DE材料的動(dòng)態(tài)性能,但是卻忽略了阻尼等的影響,幾乎沒(méi)有學(xué)者研究經(jīng)典的DE材料平面變形的振動(dòng)特性,不能全面反映DE材料的動(dòng)態(tài)力電耦合特征.由于平面致動(dòng)結(jié)構(gòu)是DE致動(dòng)器的最常用結(jié)構(gòu),通過(guò)熱力學(xué)自由能理論,在考慮DE材料致動(dòng)器面內(nèi)動(dòng)態(tài)變形過(guò)程中的慣性力和阻尼力的基礎(chǔ)上,構(gòu)建平面DE材料致動(dòng)器的非線(xiàn)性動(dòng)力學(xué)模型.基于所建立的模型,研究了交變電載荷下DE系統(tǒng)的非線(xiàn)性動(dòng)態(tài)特性.

1 DE致動(dòng)器的動(dòng)態(tài)運(yùn)動(dòng)方程

在靜力學(xué)問(wèn)題的研究中,DE材料的變形與時(shí)間無(wú)關(guān),問(wèn)題相對(duì)簡(jiǎn)單；在動(dòng)力學(xué)問(wèn)題研究中,DE材料的運(yùn)動(dòng)將會(huì)隨時(shí)間的變化而變化,由于需要考慮其慣性力及阻尼力的作用,且由于DE材料的變形是典型的力電耦合作用下的變形,屬于典型的非線(xiàn)性問(wèn)題.

通過(guò)自由能方法[17,19]來(lái)建立DE材料致動(dòng)器的面內(nèi)振動(dòng)模型能夠?yàn)镈E材料面內(nèi)應(yīng)用提供理論指導(dǎo).本文主要研究的是如圖1所示的DE材料致動(dòng)器的面內(nèi)變形動(dòng)態(tài)性能,認(rèn)為DE材料是一種理想的電介質(zhì),其變形λ1=l1/L1,λ2=l2/L2和λ3=l3/L3是時(shí)間的函數(shù).

圖1 平面DE致動(dòng)器的振動(dòng)示意圖Fig. 1 Vibration schematics of an in-plane dielectric elastomer actuator

由于研究的是在平衡位置附近的小幅振動(dòng)特性,忽略變形及溫度對(duì)極化率的影響.這種理想的DE材料的介電性能與聚合物熔融體一樣,其真實(shí)電位移D和真實(shí)電場(chǎng)強(qiáng)度E的關(guān)系可以表示為D=εE(ε是DE材料的介電常數(shù),典型代表值為ε=4×10-11F/m[10,25]),那么可以得到DE致動(dòng)器的電荷量和電壓的關(guān)系為:

(1)

在時(shí)間t時(shí)刻,當(dāng)有少量的電荷流過(guò)DE材料的兩側(cè)電極的時(shí)候,電壓做功為ΦδQ,當(dāng)DE的尺寸發(fā)生微小的δλ1和δλ2變化時(shí),外力做的功分別為P1L1δλ1和P2L2δλ2.此時(shí),x和y方向上微小單元的慣性力分別為ρL2L3x2(d2λ1/dt2)和ρL1L3y2(d2λ2/dt2)[19],阻尼力分別為cxdλ1/dt和cydλ2/dt.分別對(duì)微小單元上δλ1dx和δλ2dy的慣性力和阻尼力在x和y方向進(jìn)行積分,可求得慣性力和阻尼力所做的功.在任意熱力學(xué)系統(tǒng)中,自由能的改變量等于外力、電壓、慣性力和阻尼力所做功的總和,即:

L1L2L3δW=ΦδQ+P1L1δλ1+P2L2δλ2-

(2)

DE致動(dòng)器的自由能包括彈性應(yīng)變能和靜電能.應(yīng)變能仍用Gent模型描述[26],那么DE的自由能為：

(3)

式中Jm為DE材料的變形極限,Jm=100[26-28]；μ是DE材料的剪切模量,μ=1×106N/m2[10,25].

假設(shè)DE材料致動(dòng)器的初始尺寸相等L1=L2=L,由于我們主要研究的是等雙軸的變形,令λ1=λ2=λ,P1/(L2L3)=P2/(L1L3)=s,對(duì)式(1)～(3)求解進(jìn)行化簡(jiǎn)后就得到了DE材料致動(dòng)器的非線(xiàn)性動(dòng)力學(xué)方程:

(4)

(5)

2 考慮阻尼時(shí)致動(dòng)器的動(dòng)態(tài)特性

以正弦電壓作為激勵(lì)電壓,即

Φ(t)=Φdc+Φacsin(Ωt)

(6)

式中Φdc為直流電壓，單位:V；Φac為交流電壓幅值，單位:V；Ω為正弦電壓的頻率，單位:rad/s.

把公式(6)代入方程(4)中,化簡(jiǎn)后得到

(7)

圖2 阻尼對(duì)DE致動(dòng)器幅頻特性的影響Fig. 2 Effect of different damping on amplitude-frequency characteristic for DE actuator

雖然有阻尼的時(shí)候,DE致動(dòng)器系統(tǒng)的非線(xiàn)性振動(dòng)包含著阻尼引起的衰減振動(dòng),但由于在其振動(dòng)的過(guò)程中一直存在的外加交變電壓的力電耦合效果,其中電能和變形能會(huì)抵消振動(dòng)過(guò)程中的能量耗散,隨著時(shí)間的增加,其最后會(huì)達(dá)到一個(gè)恒定振幅的振動(dòng)狀態(tài).如圖3中的變形響應(yīng)曲線(xiàn)所示,這種非線(xiàn)性振動(dòng)出現(xiàn)了鎖頻現(xiàn)象,表明承受著周期性的阻尼振動(dòng),這和相關(guān)文獻(xiàn)報(bào)道類(lèi)似[21-23].

圖4給出了阻尼為0.05時(shí),DE系統(tǒng)響應(yīng)的相平面圖(a、b、c)和Poincaré映射圖(d、e、f).圖4(a)是激勵(lì)頻率為1.62時(shí)的相平面圖,此時(shí)穩(wěn)態(tài)的相平面圖上形成閉合的曲線(xiàn),代表著周期性的運(yùn)動(dòng)[29],這同樣反映在其Poincaré映射圖4(d)上,其點(diǎn)集是有限的,存在著周期吸引子.頻率為3.24和0.82的時(shí)候,也出現(xiàn)了類(lèi)似的情況.表明此條件下DE致動(dòng)器在固有頻率、2倍固有頻率和二分之一固有頻率下的振動(dòng)均是周期振動(dòng).

圖3 阻尼為0.05時(shí)DE致動(dòng)器在三種激勵(lì)頻率下的振動(dòng)響應(yīng)Fig. 3 Vibration response of DE actuator system for three excitation frequencies with a dimensionless damping effect of 0.05

圖4 阻尼為0.05時(shí)DE致動(dòng)器的相平面圖和Poincaré映射圖Fig. 4 Phase diagrams and poincaré of DE actuator system with a dimensionless damping effect of 0.05

3 結(jié)論

通過(guò)熱動(dòng)力學(xué)方程,考慮了慣性力和阻尼力的共同作用,建立了DE致動(dòng)器非線(xiàn)性力電耦合系統(tǒng)的動(dòng)態(tài)行為方程.研究發(fā)現(xiàn),存在阻尼的時(shí)候,DE系統(tǒng)的振動(dòng)會(huì)隨著時(shí)間的增加出現(xiàn)鎖頻現(xiàn)象,最后變成一種具有恒定振幅的振動(dòng).這是因?yàn)楸M管存在著阻尼的耗能作用,但是由于DE致動(dòng)器的振動(dòng)是由外加交變電場(chǎng)產(chǎn)生的Maxwell應(yīng)力引起的,靜電能和變形能補(bǔ)償了阻尼的能耗,這種力電耦合效果最后使得DE系統(tǒng)變成具有恒定振幅的振動(dòng).相平面圖和Poincaré映射圖研究表明,考慮阻尼后DE系統(tǒng)的穩(wěn)態(tài)相平面圖是一條閉合的曲線(xiàn),其Poincaré映射點(diǎn)集是有限的,代表其產(chǎn)生周期性運(yùn)動(dòng).

通過(guò)對(duì)平面運(yùn)動(dòng)DE致動(dòng)器系統(tǒng)非線(xiàn)性動(dòng)態(tài)力電耦合性能的研究,可以預(yù)測(cè)DE材料在動(dòng)態(tài)載荷下的非線(xiàn)性動(dòng)態(tài)變形規(guī)律,能為DE材料在各種動(dòng)態(tài)驅(qū)動(dòng)器或傳感器中的應(yīng)用提供理論依據(jù).

1 Keplinger C, Li T, Baumgartner R, et al. Harnessing snap-through instability in soft dielectrics to achieve giant voltage-triggered deformation.SoftMatter, 2011,8(2):285～288

2 Pelrine R, Kornbluh R, Pei Q, Joseph J. High-speed electrically actuated elastomers with strain greater than 100%.Science, 2000,287(5454):836～839

3 Carpi F, Rossi D. D, Kornbluh R, Pelrine R, Sommer-Lasen P. Dielectric Elastomers as Electromechanical Transducers. Amsterdam: Elsevier Press, 2008

4 Brochu P, Pei Q. Advances in dielectric elastomers for actuators and artificial muscle.MacromolecularRapidCommunication, 2010,31(1):10～36

5 Pei Q, Rosenthal M, Stanford S, Prahlad H and Pelrin R. Multiple-degrees-of-freedom electroelastomer roll actuators.SmartMaterialsandStructures, 2004,13(5): N86

6 Kovacs G, Düring L, Michel S, and Terrasi G. Stacked dielectric elastomer actuator for tensile force transmission.SensorsandActuatorsA:Physical, 2009,155(2):299～307.

7 Keplinger C, Sun J Y, Foo C C, Rothemund P, Whitesides G M, and Suo Z. Stretchable, Transparent, Ionic Conductors.Science, 2013,341(6149): 984～987.

8 Huang J, Shian S, Suo Z. and Clarke D R. Maximizing the energy density of dielectric elastomer generators using equi-biaxial loading.AdvancedFunctionalMaterials, 2013,23(40):5056～5061

9 陳寶鴻,周進(jìn)雄. 離子導(dǎo)體驅(qū)動(dòng)的介電彈性體軟機(jī)器研究進(jìn)展. 固體力學(xué)學(xué)報(bào), 2015,36(6):481～492 (Chen B H, Zhou J X. Dielectric elastomers based soft machines actuated by ionic conductors: progress and perspectives.ChineseJournalofSolidMechanics, 2015,36(6):481～492 (in Chinese))

10 Suo Z. Theory of dielectric elastomer.ActaMechanicaSolidaSinica, 2010,23(6):549～578

11 Huang R, Suo Z G. Electromechanical phase transition of dielectric elastomer.ProceedingsoftheRoyalSocietyAMathematicalPhysical&EngineeringSciences, 2012,468(2140):1014～1040

12 Bertoldi K, Gei M. Instability in multilayered soft dielectircs.JournaloftheMechanics&PhysicsofSolids, 2011,59(1):18～42

13 Koh SJA, Li T F, Zhou J X, et al. Mechanisms of large actuation strain in dielectric elastomer.JournalofPolymerSciencePartBPolymerPhysics, 2011,49(7):504～515

14 Kaltseis R, Keplinger C, Baumgartner R, et al. Method for measuring energy generation and efficiency of dielectric elastomer generators.AppliedPhysicsLetters, 2011,99(16):162904.

15 盛俊杰,張玉慶,李樹(shù)勇,劉磊,陳花玲. 溫度對(duì)介電彈性體材料力電耦合變形的影響. 固體力學(xué)學(xué)報(bào), 2015,36(2):129～136 (Sheng J J, Zhang Y Q, Li S Y, Liu L, Chen H L. Effect of temperature on the electromechanical deformation of dielectric elastomer.ChineseJournalofSolidMechanics, 2015,36(2):129～136 (in Chinese))

16 劉立武,孫壽華,劉彥菊,冷勁松. 具有線(xiàn)性介電常數(shù)的Ogden型介電彈性體的本構(gòu)關(guān)系和機(jī)電穩(wěn)定性. 固體力學(xué)學(xué)報(bào), 2010,31(2):181～192 (Liu L W, Sun S H, Liu Y J, Leng J S. Constibutive relation electromechanical stability analysis of Ogden type dielectirc elastomer with linear permittivity.ChineseJournalofSolidMechanics, 2010,31(2):181～192 (in Chinese))

17 Zhu J, Cai S, Suo Z. Nonlinear oscillation of a dielectric elastomer balloon.PolymerInternational, 2010,59(3):378～383

18 Yong H, He X, Zhou Y. Dynamics of a thick-walled dielectric elastomer spherical shell.InternationalJournalofEngineeringScience, 2011,49(8):792～800

19 Li T, Qu S, Yang W. Electromechanical and dynamic analyses of tunable dielectric elastomer resonator.InternationalJournalofSolidsandStructures, 2012,49(26):3754～3761

20 Soares R M, Gonc-alves P B. Nonlinear vibrations and instabilities of a stretched hyperelastic annular membrane.InternationalJournalofSolidsandStructures, 2012,49(3-4):514～526

21 Xu B, Mueller R, Theis A, Klassen M, Gross D. Dynamic analysis of dielectric elastomer actuators.AppliedPhysicsLetters, 2012,100(11):112903

22 Li B, Zhang J, Liu L, Chen H, Jia S, and Li D. Modeling of dielectric elastomer as electromechanical resonator.JournalofAppliedPhysics, 2014,116(12):124509

23 Zhang J S, Tang L L, Li B, Wang Y J, Chen H L. Modeling of the dynamic characteristic of viscoelastic dielectric elastomer actuators subject to different conditions of mechanical load.JournalofAppliedPhysics, 2015,117(8):084902

24 Wolf K, R?glin T, Haase F, Finnberg T, Steinhoff B. An electroactive polymer based concept for vibration reduction via adaptive supports. Proc SPIE, 2008, 6927: 69271F

25 Zhao X, Suo Z. Method to analyze electromechanical stability of dielectric elastomers.AppliedPhysicsLetters, 2007,91(6):061921-061921-3

26 Sheng J, Chen H, Li B, Wang Y. Nonlinear dynamic characteristics of a dielectric elastomer membrane undergoing in-plane deformation.SmartMaterialsandStructures, 2014,23(4):045010

27 Foo C C, Cai S, Koh S J A, et al. Model of dissipative dielectric elastomers.JournalofAppliedPhysics, 2012,111(3):034102-13

28 Sheng J, Chen H, Li B. Effect of temperature on the stability of dielectric elastomers.JournalofPhysicsDAppliedPhysics, 2011,44(36):365406-365414

29 聞邦椿,李以農(nóng),徐培民. 工程非線(xiàn)性振動(dòng). 北京: 科學(xué)出版社,2007 (Wen B C, Li Y N, Xu P M. Nonlinear Engineering Vibration. Beijing: Publication of Science Press, 2007(in Chinese))

*The project supported by the National Natural Science Foundation of China (11402246) and the Key Subject “Computational Solid Mechanics” of the China Academy of Engineering Physics

? Corresponding author E-mail: scu2005sjj@163.com

17 May 2016, revised 18 June 2016.

NONLINEAR DYNAMIC PERFORMANCE OF A DIELECTRIC ELASTOMER ACTUATOR*

Sheng Junjie1?Li Shuyong1Zhang Yuqing1Chen Hualing2,3

(1.InstituteofSystemsEngineering,ChinaAcademyofEngineeringPhysics,Mianyang621900,China)(2.SchoolofMechanicalEngineering,Xi′anJiaotongUniversity,Xi′an710049,China)(3.StateKeyLaboratoryforStrengthandVibrationofMechanicalStructures,Xi’anJiaotongUniversity,Xi′an710049,China)

Dielectric elastomers (DE) are one of the most potential electro-active polymer (EAP) used as high-performance actuators. A DE subjected to a voltage can generate large thickness and area expansion with a maximum area strain of 1600%. As DE has natural damping properties and its deformation is then dependent to time, its dynamic performance, energy conversion and large deformation are definitely influenced by damping. A free energy model is developed to study the dynamic characteristics of a dielectric elastomer actuator undergoing in-plane deformation subjected to the combined loadings of a mechanical press and an electriceld. The numerical results including the oscillation, phase diagrams and Poincaré maps are presented to show the inuence of the damping on the nonlinear dynamic characteristics of the dielectric elastomer. The numerical results indicate that the damping effect could cause the dynamic responses to constant vibration and decrease the amplitude. The phase paths are all presented in closed regions and the points of Poincaré map are finite, indicating that the DE system experiences a nonlinear periodic oscillation, and the dynamic oscillation of the DE system is stable. These conclusions provide the basis for the exploration of high-performance dielectric elastomers under dynamic mechanical and electrical loads.

dielectric elastomer, dynamic,damping, phase diagram, Poincaré map

*國(guó)家自然科學(xué)基金資助項(xiàng)目(11402246)、中國(guó)工程物理研究院重點(diǎn)學(xué)科項(xiàng)目“計(jì)算固體力學(xué)”資助

10.6052/1672-6553-2016-049

2016-05-17收到第1稿,2016-06-18收到修改稿.

? 通訊作者 E-mail: scu2005sjj@163.com