端部质量块对悬臂梁型挠曲电俘能器性能的影响
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摘要
端部质量块对悬臂梁型挠曲电俘能器的性能具有显著的影响.以电吉布斯自由能为基础,基于Euler-Bernoulli梁模型,利用挠曲电材料的变分方法,推导带端部质量块的挠曲电俘能器结构的机电耦合动力学控制方程和相应的力电边界条件;利用分离变量法推导结构在短路条件下的振型正交条件和归一化振型参数;结合模态叠加法推导出挠曲电俘能器在外加简谐激励条件下位移和电势响应的解析形式.数值分析结果表明,端部质量块质量的增加和尺寸的减小可以提高俘能器输出功率并降低其共振频率,并且具有调节挠曲电俘能器最优外加电阻负载的功能.此外,当端部质量块尺寸一定时,端部质量块的质量越大,采用质点模型分析所带来的误差越明显.
The tip mass has a significant influence on the performance of flexoelectric energy harvesters.In this paper,based on Euler-Bernoulli beam assumption and the electric Gibbs free energy,the dynamic electromechanical coupled governing equations and the corresponding boundary conditions of flexoelectric energy harvesters are derived using the variational principle method.The modal orthometric conditions and parameters for the normalized mode of vibration are obtained by the separation of variable method.Furthermore,the mode superposition method is used to derive the analytical expressions of the displacement and electric potential responses of the flexoelectric energy harvester with the extrinsic harmonic mechanical excitation.Numerical results indicate that the increase of the value of mass and decrease of the size of the tip mass not only can increase the output power density and decrease the resonant frequency of the energy harvester,but also can adjust the optimal resistance.In addition,when the size of mass is fixed,the value of mass of the tip mass is larger,and the error caused by considering the tip mass as a particle becomes more distinct.
The tip mass has a significant influence on the performance of flexoelectric energy harvesters.In this paper,based on Euler-Bernoulli beam assumption and the electric Gibbs free energy,the dynamic electromechanical coupled governing equations and the corresponding boundary conditions of flexoelectric energy harvesters are derived using the variational principle method.The modal orthometric conditions and parameters for the normalized mode of vibration are obtained by the separation of variable method.Furthermore,the mode superposition method is used to derive the analytical expressions of the displacement and electric potential responses of the flexoelectric energy harvester with the extrinsic harmonic mechanical excitation.Numerical results indicate that the increase of the value of mass and decrease of the size of the tip mass not only can increase the output power density and decrease the resonant frequency of the energy harvester,but also can adjust the optimal resistance.In addition,when the size of mass is fixed,the value of mass of the tip mass is larger,and the error caused by considering the tip mass as a particle becomes more distinct.
引文
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[26]YAN Z.Modeling of a piezoelectric/piezomagnetic nano energy harvester based on two dimensional theory[J].Smart Materials and Structures,2017,27(1):015016.
[27]ERTURK A,INMAN D J.压电能量收集[M].舒海生,赵丹,史肖娜,译.北京:国防工业出版社,2015:19-111.
[28]TANG L,WANG J.Size effect of tip mass on performance of cantilevered piezoelectric energy harvester with a dynamic magnifier[J].Acta Mechanica,2017,228(11):3997-4015.
[29]LIANG X,HU S,SHEN S.A new Bernoulli-Euler beam model based on a simplified strain gradient elasticity theory and its applications[J].Composite Structures,2014(111):317-323.
[30]CHU B,SALEM D R.Flexoelectricity in several thermoplastic and thermosetting polymers[J].Applied Physics Letters,2012,101(10):103905.
[31]BASKARAN S,HE X,WANG Y,et al.Strain gradient induced electric polarization inα-phase polyvinylidene fluoride films under bending conditions[J].Journal of Applied Physics,2012,111(1):014109.
[32]ZHOU Y,LIU J,HU X,et al.Flexoelectric effect in PVDF-based polymers[J].IEEE Transactions on Dielectrics and Electrical Insulation,2017,24(2):727-731.
[33]ALADWANI A,ALDRAIHEM O,BAZ A.A distributed parameter cantilevered piezoelectric energy harvester with a dynamic magnifier[J].Mechanics of Advanced Materials and Structures,2014,21(7):566-578.
[2]SODANO H A,INMAN D J,PARK G.A review of power harvesting from vibration using piezoelectric materials[J].Shock and Vibration Digest,2004,36(3):197-205.
[3]MITCHESON P D,MIAO P,STARK B H,et al.MEMSelectrostatic micropower generator for low frequency operation[J].Sensors and Actuators A Physical,2004,115(2):523-529.
[4]WILLIAMS C B,YATES R B.Analysis of a micro-electric generator for microsystems[J].Sensors and Actuators APhysical,1996,52(1):8-11.
[5]THOMANN H.Piezoelectric ceramics[J].Advanced Materials,1990,2(10):458-463.
[6]KOGAN S M.Piezoelectric effect during inhomogeneous deformation and acoustic scattering of carriers in crystals[J].Soviet Physics-Solid State,1964,5(10):2069-2070.
[7]INDENBOM V L.Flexoelectric effect and crystal-structure[J].Kristallografi,1981,26(6):1157-1162.
[8]NGUYEN T D,MAO S,YEH Y W,et al.Nanoscale flexoelectricity[J].Advanced Materials,2013,25(7):946-974.:[9]MAJDOUB M S,SHARMA P,AG∨IN T.Dramatic enhancement in energy harvesting for a narrow range of dimensions in piezoelectric nanostructures[J].Physical Review B,2008,78(12):121407.
[10]HUANG W,KWON S R,YUAN F G,et al.Aflexoelectric micro-accelerometer[C]∥2012International Mechanical Engineering Congress and Exposition.New York:ASME,2012:597-603.
[11]BHASKAR U K,BANERJEE N,ABDOLLAHI A,et al.A flexoelectric microelectromechanical system on silicon[J].Nature Nanotechnology,2015,11(3):263-266.
[12]CATALAN G,SINNAMON L J,GREGG J M.The effect of flexoelectricity on the dielectric properties of inhomogeneously strained ferroelectric thin films[J].Journal of Physics Condensed Matter,2004,16(13):2253-2264.
[13]MA W,CROSS L E.Large flexoelectric polarization in ceramic lead magnesium niobate[J].Applied Physics Letters,2001,79(26):4420-4422.
[14]MARANGANTI R,SHARMA P.Atomistic determination of flexoelectric properties of crystalline dielectrics[J].Physical Review B,2009,80(5):054109.
[15]SHARMA N D,LANDIS C M,SHARMA P.Piezoelectric thin-film superlattices without using piezoelectric materials[J].Journal of Applied Physics,2010,108(2):424304.
[16]ZUBKO P,CATALAN G,TAGANTSEV A K.Flexoelectric effect in solids[J].Annual Review of Materials Research,2013,43(1):387-421.
[17]SHEN S,HU S.A theory of flexoelectricity with surface effect for elastic dielectrics[J].Journal of the Mechanics and Physics of Solids,2010,58(5):665-677.
[18]MAJDOUB M S,SHARMA P,CAGIN T.Enhanced sizedependent piezoelectricity and elasticity in nanostructures due to the flexoelectric effect[J].Physical Review B,2008,77(12):125424.
[19]LIANG X,HU S,SHEN S.Size-dependent buckling and vibration behaviors of piezoelectric nanostructures due to flexoelectricity[J].Smart Materials and Structures,2015,24(10):105012.
[20]ZHANG R,LIANG X,SHEN S.A Timoshenko dielectric beam model with flexoelectric effect[J].Meccanica,2016,51(5):1181-1188.
[21]ZHOU Z D,YANG C P,SU Y X,et al.Electromechanical coupling in piezoelectric nanobeams due to flexoelectric effect[J].Smart Materials and Structures,2017,26(9):095025.
[22]杨昌平,苏雅璇,林晓辉,等.挠曲电效应对简支梁式压电传感器性能的影响[J].厦门大学学报(自然科学版),2017,56(6):823-830.
[23]LI X,LUO Y.Flexoelectric effect on vibration of piezoelectric microbeams based on a modified couple stress theory[J].Shock and Vibration,2017.doi:10.1155/2017/4157085.
[24]DENG Q,KAMMOUN M,ERTURK A,et al.Nanoscale flexoelectric energy harvesting[J].International Journal of Solids and Structures,2014,51(18):3218-3225.
[25]LIANG X,ZHANG R,HU S,et al.Flexoelectric energy harvesters based on Timoshenko laminated beam theory[J].Journal of Intelligent Material Systems and Structures,2017,28(15):2064-2073.
[26]YAN Z.Modeling of a piezoelectric/piezomagnetic nano energy harvester based on two dimensional theory[J].Smart Materials and Structures,2017,27(1):015016.
[27]ERTURK A,INMAN D J.压电能量收集[M].舒海生,赵丹,史肖娜,译.北京:国防工业出版社,2015:19-111.
[28]TANG L,WANG J.Size effect of tip mass on performance of cantilevered piezoelectric energy harvester with a dynamic magnifier[J].Acta Mechanica,2017,228(11):3997-4015.
[29]LIANG X,HU S,SHEN S.A new Bernoulli-Euler beam model based on a simplified strain gradient elasticity theory and its applications[J].Composite Structures,2014(111):317-323.
[30]CHU B,SALEM D R.Flexoelectricity in several thermoplastic and thermosetting polymers[J].Applied Physics Letters,2012,101(10):103905.
[31]BASKARAN S,HE X,WANG Y,et al.Strain gradient induced electric polarization inα-phase polyvinylidene fluoride films under bending conditions[J].Journal of Applied Physics,2012,111(1):014109.
[32]ZHOU Y,LIU J,HU X,et al.Flexoelectric effect in PVDF-based polymers[J].IEEE Transactions on Dielectrics and Electrical Insulation,2017,24(2):727-731.
[33]ALADWANI A,ALDRAIHEM O,BAZ A.A distributed parameter cantilevered piezoelectric energy harvester with a dynamic magnifier[J].Mechanics of Advanced Materials and Structures,2014,21(7):566-578.