摘要
层合压电结构具有较高的比强度和比模量,良好的接收性能和更宽的工作频率等诸多优势,广泛地应用于航空航天、动力测量、无损检测、生物医学等众多工程和技术领域。研究电弹性波在层合压电结构中的传播特性对于压电智能结构的设计制造及其工程中的应用有着重要理论意义和实际应用价值。由于压电材料和基体材料的各向异性,材料层的交错铺设,以及压电介质中机电耦合特性,使得层合压电结构是一个具有不连续分布、不连续材料属性以及机电耦合性质的复杂系统,目前对层合压电结构的动力响应分析在数值模型的建立、求解精度和效率等方面存在着一系列的技术难点尚待完善和解决。为此,针对层合压电板壳结构在机电耦合载荷作用下的瞬态响应问题,本文力求建立一套能够兼顾求解效率和精度,同时准确描述层合压电结构动力响应的分析体系。本文的主要工作如下：
(1)提出了适用于层合压电结构瞬态响应分析的压电层单元模型。针对层合压电结构中电弹性耦合波主要以面波形式传播的特性,将压电层单元中的位移场和电势场两个基本量离散为相应的节点面向量,利用层单元三个节点面内的位移和电势的二维分布函数来拟合层合压电结构中位移场和电势场的实际状态。基于压电层单元建立了层合压电板和层合压电圆柱壳的半解析数值模型,相比较有限元分析模型大幅减少了瞬态问题分析中所需单元数量和密度,只需较少的单元划分即可满足数值模拟的精度。
(2)提出了力、电载荷激励下的层合压电结构的瞬态位移响应和电势响应的高效求解算法。将层合压电结构半解析数值模型应用于层合压电板和层合压电圆柱壳的瞬态响应求解,基于Hamilton变分原理推导了结构在时空域内的运动控制偏微分方程。利用傅里叶变换的性质,将关于位移和电势完全耦合的运动控制偏微分方程,变换为波数域内的位移控制常微分方程,以及位移和电势之间的线性耦合方程,从而避免了数值计算中耗时低效的偏微分方程组直接求解。在波数域控制方程中,引入等效力载荷和等效耦合刚度矩阵的概念,降低了控制方程的阶次,减少了在线计算的存储空间。利用压电铺层材料参数矩阵的对称性,对复数域内的波数域运动控制方程和相应的特征值问题进行了实数化处理,进一步提高了算法的求解效率
(3)提出了层合压电圆柱壳中电弹性波传播的波动特性分析方法。基于层合压电圆柱壳半解析数值模型,建立了结构的自由振动控制方程。基于模态分析方法,推导了层合压电圆柱壳中电弹性波传播的相速度面、相慢度面、相波面、群速度面、群慢度面和群波面六个波动特征面的具体形式,可以直观的描述材料的各向异性和压电效应对结构波动特性的影响,观察不同波动传播方向上波动特性评价指标的变化。
(4)提出了层合压电板电势和位移瞬态响应对载荷参数和材料本构参数的敏感性分析方法。基于所提出的层合压电板瞬态响应求解算法,采用直接求导的方式在波数域内建立了瞬态响应的敏感性方程,并给出了两类设计参数下敏感性方程中虚载荷的计算方法。所推导的敏感性方程与瞬态响应求解中的运动控制方程对应着相同的广义特征值问题,因此本方法尤其适用于较多设计参数情况下的瞬态响应敏感性分析,并可在结构瞬态分析的基础上,增加很小的计算成本即实现瞬态响应和响应敏感性的同时输出。结合区间结构分析理论,用于计算层合压电板在不确定材料参数和载荷参数条件下的位移和电势瞬态响应边界。
Piezoelectric materials, as ideal materials of sensors and actuators in intelligent structure systems, have the characteristics of piezoelectric and inverse piezoelectric effects, which can perceive and respond to environment. These materials, forming the laminated piezoelectric smart structure with the function of self-test, self-adaption and quick response, are always pasted on the surface or embedded in the internal part of base layer material in the form of clad layer and chip. With so many advantages, such as better receptivity and wider working frequency, piezoelectric laminated structure has higher specific strength and specific modulus applied into the aerospace, dynamometry, nondestructive testing and biomedicine. There is a significantly important point doing research on the application of electric elastic wave to design and manufacture of laminated piezoelectric smart structure.
Because of the anisotropy matrices of composite material and piezoelectric materials, the interactive laying of materials and the coupling in piezoelectric media, laminated piezoelectric smart structure is dynamic system with discontinuous distribution, discontinuous material properties and electromechanical coupling. At present, a series of technical difficulties on numerical model of laminated piezoelectric structure's dynamic response analysis and the balance between accuracy and efficiency. Therefore, based on the transient response of electromechanical coupling from the structure of piezoelectric laminated plate and shell, this paper tries to build an analysis system which can both consider the balance between accuracy and efficiency and describes precisely dynamic response of laminated piezoelectric smart structure. The corresponding work is as follow:
(1) Based on the propagation characteristic of electric-elastic wave and three dimensional electric-elastic theory, a numerical model of laminated piezoelectric plate has been built based on piezoelectric layer element, which can deduce the deformation, transient distribution and response time of potential under the common action among force load, potential incentive and electromechanical coupling load. This method can solve structure transient response efficiently by associating several numerical methods such as Finite element method and overcome several difficulties in traditional method.
(2) The transient problem of electric elastic wave has been for piezoelectric laminated cylindrical shell has been analyzed. On the radial direction of cylindrical shell, displacement field and electric potential field will be dispersed in the form of three nodes isoparametric element. The coupling physical field of piezoelectric laminated cylindrical shell can be fitted by the vector function of cylindrical element's surface distribution and the scale and precision of calculation can be controlled by changing the density of cylindrical shell element. Based on Hamilton theory, the whole motion control equation of structure will be built after twice independent variational methods for node displacement and electric potential. In addition,, semi-analytical solutions of the wave number domain displacement and transient distribution of electric potential under several kinds of force-electric load are deduced. Bringing in equivalent load of the wave number domain and equivalent coupling stiffness matrix makes solving process more unified and it is convenient to analyze directly the influence of electrical coupling effect on laminated piezoelectric structure dynamic response. At the same time, this measurement will decrease the degree of freedom of control equation and workload of calculation. When the material properties of the piezoelectric layer and substrate layer meet certain conditions, motion control equation of the wave number domain and corresponding eigenvalue problem will be a real treatment increasing the calculation efficiency further.
(3) Based on the modal analysis theory, researches about problems of the propagation and scattering characteristics have been done. Free vibration control equation has been built according to piezoelectric laminated structure mechanics model of single layer cell. Moreover, six concepts of characteristic face are used for expressing the influence of material anisotropy and piezoelectric effect on structure fluctuation characteristics.
(4) Based on semi-analytical calculation model for laminated panels pressure dynamic response analysis, both transient responses of electric potential and displacement for loading parameters and the first-order sensitivity of the material constitutive parameters are deduced. Using properties of space domain-wave number domain transform, the first-order sensitivity of wave number domain response has been deduced from the control equation of wave number domain. The comparison between this method and finite difference method verifies advantages of efficiency and accuracy in this proposed method. Combing with interval structure analysis theory, a new method has been proposed for obtaining transient response border of displacement and electric potential quickly on the premise that there are many uncertain parameters and constraints.
引文
[1]张金升,尹衍升,李嘉,等.智能材料的结构和性能综述.中国陶瓷,2003,39(2)：43-47
[2]Rogers C A. Intelligent material systems-the dawn of a new materials age. Journal of Intelligent Material Systems and Structures,1993,4(1):4-12
[3]陶宝祺.智能材料结构.北京：国防工业出版社,1997
[4]刘祥建,陈仁文.压电振动能量收集装置研究现状及发展趋势.振动与冲击,2012,31(16)：169-176
[5]袁江波,谢涛,单小彪,等.压电俘能技术研究现状综述.振动与冲击,2009,28(10)：36-42
[6]Leontsev S O, Eitel R E. Progress in engineering high strain lead-free piezoelectric ceramics. Science and Technology of Advanced Materials,2010,11(4):44-62
[7]Bailey T, Ubbard J. Distributed piezoelectric-polymer active vibration control of a cantilever beam. Journal of Guidance Control and Dynamics,1985,8(5):605-611
[8]蒋建平,李东旭.压电层合结构力学模型评述.噪声与振动控制,2008(05)：11-20
[9]Crawley E F. Intelligent structures for aerospace-A technology overview and assessment. AIAA Journal,1994,32(8):1689-1699
[10]Morgan D P. History of SAW devices. In:Frequency Control Symposium,1998. Proceedings of the 1998 IEEE International. London,1998,439-460
[11]Gibson R F. A review of recent research on mechanics of multifunctional composite materials and structures. Composite Structures,2010,92(12): 2793-2810
[12]Egusa S, Wang Z, Chocat N, et al. Multimaterial piezoelectric fibres. Nature materials,2010,9(8):643-648
[13]Mason W P. Piezoelectricity, its history and applications. Journal of the Acoustical Society of America,1981,70(6):1561-1566
[14]Gallego J. Piezoelectric ceramics and ultrasonic transducers. Journal of Physics E: Scientific Instruments,1989,22(10):804-812
[15]Jaffe B, Roth R S, Marzullo S. Piezoelectric properties of lead zirconate-lead titanate solid-solution ceramics. Journal of Applied Physics,1954,25(6): 809-810
[16]Phelan J R J, Mahler R J, Cook A R. High D* pyroelectric polyvinylfluoride detectors. Applied Physics Letters,1971,19(9):337
[17]Shung K K, Cannata J M, Zhou Q F. Piezoelectric materials for high frequency medical imaging applications:A review. Journal of Electroceramics,2007,19(1): 141-147
[18]Trindade M, Benjeddou A, Ohayon R. Piezoelectric active vibration control of damped sandwich beams. Journal of Sound and Vibration,2001,246(4):653-677
[19]Baz A, Poh S. Performance of an active control system with piezoelectric actuators. Journal of Sound and Vibration,1988,126(2):327-343
[20]Hagood N W, Chung Walter H, Von F A. Modelling of piezoelectric actuator dynamics for active structural control. Journal of Intelligent Material Systems and Structures,1990,1(3):327-354
[21]Leleu S, Abou-Kandil H, Bonnassieux Y. Piezoelectric actuators and sensors location for active control of flexible structures. IEEE Transactions on Instrumentation and Measurement,2001,50(6):1577-1582
[22]Jalili N. Piezoelectric-based vibration control-From Macro to Micro/Nano Scale Systems. Newyork:Springer,2010
[23]Yan W, Chen W Q, Lim C W, et al. Application of EMI technique for crack detection in continuous beams adhesively bonded with multiple piezoelectric patches. Mechanics of Advanced Materials and Structures,2008,15(1):1-11
[24]Purekar A S, Pines D J. Damage detection in thin composite laminates using piezoelectric phased sensor arrays and guided lamb wave interrogation. Journal of Intelligent Material Systems and Structures,2010,21(10):995-1010
[25]Zhang K, Zhao L B, Guo S S, et al. A microfluidic system with surface modified piezoelectric sensor for trapping and detection of cancer cells. Biosensors and Bioelectronics,2010,26(2):935-939
[26]Wang X, Tang J. Damage detection using piezoelectric admittance approach with inductive circuitry. Journal of Intelligent Material Systems and Structures,2010, 21(7):667-676
[27]姚志刚.压电复合材料结构的复杂非线性动力学与控制的研究：[北京工业大学博士学位论文].北京：北京工业大学机电学院,2009
[28]Khot N S, Eastep F E, Koloney R M. Wing twist and camber for the rolling maneuver of a flexible wing without aileron. In:38th AIAA SDM Meeting. Florida,1997,759-776
[29]Khot N S, Eastep F E, Kolonay R M. Method for enhancement of the rolling maneuver of a flexible wing. Journal of Aircraft,1997,34(5):673-678
[30]Kessler S S, Spearing S M, Atalla M J, et al. Damage detection in composite materials using frequency response methods. Composites Part B:Engineering, 2002,33(1):87-95
[31]Yeum C M, Sohn H, Ihn J B, et al. Instantaneous delamination detection in a composite plate using a dual piezoelectric transducer network. Composite Structures,2012,94(12):3490-3499
[32]Yan Y J, Yam L H. Online detection of crack damage in composite plates using embedded piezoelectric actuators/sensors and wavelet analysis. Composite Structures,2002,58(1):29-38
[33]Fukunaga H, Hu N, Chang F K. Structural damage identification using piezoelectric sensors. International Journal of Solids and Structures,2002,39(2): 393-418
[34]Chen W Q, Cai J B,Ye G R, et al. Exact three-dimensional solutions of laminated orthotropic piezoelectric rectangular plates featuring interlaminar bonding imperfections modeled by a general spring layer. International Journal of Solids and Structures,2004,41(18):5247-5263
[35]Howells C A. Piezoelectric energy harvesting. Energy Conversion and Management,2009,50(7):1847-1850
[36]Starner T. Human-powered wearable computing. IBM Systems Journal,1996, 35(3):618-629
[37]Paradiso J A, Starner T. Energy scavenging for mobile and wireless electronics. Pervasive Computing, IEEE,2005,4(1):18-27
[38]Sodano H A, Inman D J, Park G. A review of power harvesting from vibration using piezoelectric materials. Shock and Vibration Digest,2004,36(3):197-206
[39]曾平,刘艳涛,吴博达,等.一种新型压电式无线发射装置.吉林大学学报工学版,2006,36(2)：78-82
[40]Lee M, Chen C Y,Wang S H, et al. A hybrid piezoelectric structure for wearable nanogenerators. Advanced Materials,2012,24(13):1759-1764
[41]Ottman G K, Hofmann H F, Bhatt A C, et al. Adaptive piezoelectric energy harvesting circuit for wireless remote power supply. IEEE Transactions on Power Electronics,2002,17(5):669-676
[42]Lee D G, Carman G P, Murphy D, et al. Novel micro vibration energy harvesting device using frequency up conversion. In:Solid-State Sensors, Actuators and Microsystems Conference. Lyon,2007,871-874
[43]Rodig T, Schonecker A, Gerlach G. A survey on piezoelectric ceramics for generator applications. Journal of the American Ceramic Society,2010,93(4): 901-912
[44]Leonov V, Vullers R J M. Wearable electronics self-powered by using human body heat:The state of the art and the perspective. Journal of Renewable and Sustainable Energy,2009,1(6):1-14
[45]舒小平.复合材料板壳理论及其应用.徐州：中国矿业大学出版社,2004
[46]Reddy J N, Wang C M. An overview of the relationships between solutions of the classical and shear deformation plate theories. Composites Science and Technology,2000,60(12):2327-2335
[47]Semedo G J, Soares M, Mota S C A, et al. Analysis of laminated adaptive plate structures using layerwise finite element models. Computers & structures,2004, 82(23):1939-1959
[48]Love A E H. A treatise on the mathematical theory of elasticity. Cambridge: Cambridge University Press,2013
[49]刘瑞霞,杨庆山,李作为,等.小垂度薄膜屋盖的气弹动力耦合作用方程.工程力学,2004,21(6)：41-44
[50]宋文晶,徐国彬.薄壁圆拱壳的半无矩理论分析.钢结构工程研究,1998,16(2)：181-188
[51]徐烈烜.圆柱壳的半无矩理论分析.武汉理工大学学报,2004,26(1)：72-75
[52]Guo Y H, Yu G Z, Xie X P. Uniform analysis of a stabilized hybrid finite element method for Reissner-Mindlin plates. Science China-Mathematics,2013(08): 1727-1742
[53]Feng M F, Yang R K, Xiong H X. Stablized finite element methods for the Ressner-Mindlin plate. Numerical Mathematics A Journal of Chinese Universities (English Series),1999(02):125-132
[54]Feng M F, Yang Y, Zhou T X. Nonconforming stabilized combined finite element method for Reissner-Mindlin plate. Applied Mathematics and Mechanics(English Edition),2009(02):197-207
[55]Reddy J N, Averill R C. Advances in the modeling of laminated plates. Computing Systems in Engineering,1991,2(5):541-555
[56]Gilbert F, Helmberger D V. Generalized ray theory for a layered sphere. Geophysical Journal of the Royal Astronomical Society,1972,27(1):57-80
[57]Pao Y H, Gajewski R R. The generalized ray theory and transient responses of layered elastic solids. Physical Acoustics,1977,13(6):183-265
[58]Pao Y H, Keh D C, Howard S M. Dynamic response and wave propagation in plane trusses and frames. AIAA Journal,1999,37(5):594-603
[59]Ganapathi M, Varadan T K, Sarma B S. Nonlinear flexural vibrations of laminated orthotropic plates. Computers & Structures,1991,39(6):685-688
[60]Cooper R M, Naghdi. Propagation of non-axially symmetric waves in elastic cylindrical shells. Journal of Acoustics of Society of America,1957,29(12): 1365-1373
[61]Greenspon J E. Vibrations of a thick-walled cylindrical shell comparison of the exact theory with approximate theories. The Journal of the Acoustical Society of America,1960,32:571-578
[62]丁皓江,梁剑,王耘.横观各向同性弹性层点力解.应用数学和力学,1996,17(4)：297-305
[63]Yang J S. Piezoelectric transformer structural modeling-A review. Ultrasonics, Ferroelectrics and Frequency Control,2007,54(6):1154-1170
[64]Chee C Y K, Tong L Y, Steven G P. A review on the modelling of piezoelectric sensors and actuators incorporated in intelligent structures. Journal of Intelligent Material Systems and Structures,1998,9(1):3-19
[65]Kapuria S, Kumari P, Nath J K. Efficient modeling of smart piezoelectric composite laminates:A review. Acta Mechanica,2010,214(1):31-48
[66]Gibson R F. A review of recent research on mechanics of multifunctional composite materials and structures. Composite structures,2010,92(12): 2793-2810
[67]Bailey T, Hubard J. Distributed piezoelectric-polymer active vibration control of a cantilever beam. Journal of Guidance, Control, and Dynamics,1985,8(5): 605-611
[68]Crawley E F, Luis D. Use of piezoelectric actuators as elements of intelligent structures. AIAA Journal,1987,25(10):1373-1385
[69]Crawley E F, Luis D. Use of piezo-ceramics as distributed actuators in large space structures. In:AIAA 26th Structures, Structural Dynamics, and Materials Conference. Orlando,1985,126-133
[70]Crawley E F, Luis D. Experimental verification of distributed piezoelectric actuators for use in precision space structures. In:AIAA 27th Structures, Structural Dynamics, and Materials Conference. San Antonio,1986,116-124
[71]Suleman A,Venkayya V B. A simple finite element formulation for a laminated composite plate with piezoelectric layers. Journal of Intelligent Material Systems and Structures,1995,6(6):776-782
[72]Lee C K. Theory of laminated piezoelectric plates for the design of distributed sensors/actuators. The Journal of the Acoustical Society of America,1990,87(3): 1144-1156
[73]Chandrashekhara K, Agarwal A N. Active vibration control of laminated composite plates using piezoelectric devices:a finite element approach. Journal of Intelligent Material Systems and Structures,1993,4(4):496-508
[74]Lee S J, Reddy J N, Rostam A F. Transient analysis of laminated composite plates with embedded smart-material layers. Finite Elements in Analysis and Design, 2004,40(5):463-483
[75]Lee S J, Reddy J N. Nonlinear deflection control of laminated plates using third-order shear deformation theory. Mechanics and Materials in Design,2004, 1(1):33-61
[76]Wang J, Yong Y K, Imai T. Finite element analysis of the piezoelectric vibrations of quartz plate resonators with higher-order plate theory. International Journal of Solids and Structures,1999,36(15):2303-2319
[77]Wang J, Yu J D, Yong Y K, et al. A new theory for electroded piezoelectric plates and its finite element application for the forced vibrations of quartz crystal resonators. International Journal of Solids and Structures,2000,37(40):5653-5673
[78]Wang J, Wang Y, Hu W K, et al. Parallel finite element analysis of high frequency vibrations of quartz crystal resonators on Linux cluster. Acta Mechanica Solida Sinica,2008,21(6):549-554
[79]Detwiler D T, Shen M H, Venkayya V B. Finite element analysis of laminated composite structures containing distributed piezoelectric actuators and sensors. Finite Elements in Analysis and Design,1995,20(2):87-100
[80]Simoes M J M, Correia I F P, Mota S C M, et al. Active control of adaptive laminated structures with bonded piezoelectric sensors and actuators. Computers & Structures,2004,82(17):1349-1358
[81]Correia V M F, Mota S C M, Mota S C A. Refined models for the optimal design of adaptive structures using simulated annealing. Composite Structures,2001, 54(2):161-167
[82]Franco C V M, Aguiar G M A, Suleman A, et al. Modelling and design of adaptive composite structures. Computer Methods in Applied Mechanics and Engineering, 2000,185(2):325-346
[83]Wang J, Yang J S. Higher-order theories of piezoelectric plates and applications. Applied Mechanics Reviews,2000,53(4):87-99
[84]Reddy J N. A generalization of two-dimensional theories of laminated composite plates. Communications in Applied Numerical Methods,1987,3(3):173-180
[85]Robbins D H, Reddy J N. Analysis of piezoelectrically actuated beams using a layer-wise displacement theory. Computers & structures,1991,41(2):265-279
[86]Rao S, Sunar M. Analysis of distributed thermopiezoelectric sensors and actuators in advanced intelligent structures. AIAA Journal,1993,31(7):1280-1286
[87]Sunar M, Rao S S. Thermopiezoelectric Control Design. AIAA Journal,1997, 35(3):534-539
[88]Saravanos D A, Heyliger P R. Coupled layerwise analysis of composite beams with embedded piezoelectric sensors and actuators. Journal of Intelligent Material Systems and Structures,1995,6(3):350-363
[89]Saravanos D A, Heyliger P R,Hopkins D A. Layerwise mechanics and finite element for the dynamic analysis of piezoelectric composite plates. International Journal of Solids and Structures,1997,34(3):359-378
[90]Tauchert T R. Piezothermoelastic behavior of a laminated plate. Journal of Thermal Stresses,1992,15(1):25-37
[91]Tzou H S. A new distributed sensor and actuator theory for "intelligent" shells. Journal of Sound and Vibration,1992,153(2):335-349
[92]王洪军,温杰,聂辉.四边简支压电层合厚板的一阶理论解析解.国防交通工程与技术,2006,4(3)：22-25
[93]姚林泉,沈纪苹.考虑几何与压电非线性行为压电层合轴对称圆板的精确解.应用力学学报,2007,24(4)：633-636
[94]Chen C Q, Shen Y P, Wang X M. Exact solution of orthotropic cylindrical shell with piezoelectric layers under cylindrical bending. International Journal of Solids and Structures,1996,33(30):4481-4494
[95]Wang J G, Qu L, Qian F. State vector approach of free-vibration analysis of magneto-electro-elastic hybrid laminated plates. Composite Structures,2010, 92(6):1318-1324
[96]Aldraihem O J, Khdeir A A. Exact deflection solutions of beams with shear piezoelectric actuators. International Journal of Solids and structures,2003,40(1): 1-12
[97]Aldraihem O J, Khdeir A A. Smart beams with extension and thickness-shear piezoelectric actuators. Smart Materials and Structures,2000,9(1):1-9
[98]Benjeddou A. Advances in piezoelectric finite element modeling of adaptive structural elements:A survey. Computers & Structures,2000,76(1):347-363
[99]Gopinathan S V, Varadan V V, Varadan V K. A review and critique of theories for piezoelectric laminates. Smart Materials and Structures,2000,9(1):24-28
[100]Irschik H. A review on static and dynamic shape control of structures by piezoelectric actuation. Engineering Structures,2002,24(1):5-11
[101]Balamurugan V, Narayanan S. Finite element modeling of stiffened piezolaminated plates and shells with piezoelectric layers for active vibration control. Smart Materials and Structures,2010,19(10):105-113
[102]Allik H, Hughes T J. Finite element method for piezoelectric vibration. International Journal for Numerical Methods in Engineering,1970,2(2):151-157
[103]Tzou H S, Tseng C I. Distributed piezoelectric sensor/actuator design for dynamic measurement/control of distributed parameter systems:a piezoelectric finite element approach. Journal of Sound and Vibration,1990,138(1):17-34
[104]Samanta B, Ray M C, Bhattacharyya R. Finite element model for active control of intelligent structures. AIAA Journal,1996,34(9):1885-1893
[105]Hwang W S, Park H C. Finite element modeling of piezoelectric sensors and actuators. AIAA Journal,1993,31(5):930-937
[106]尹林,王晓明.用压电元件实现复合材料层合板振动控制的数值分析.振动工程学报,1996,9(2)：107-115
[107]Chandrashekhara K, Tenneti R. Thermally induced vibration suppression of laminated plates with piezoelectric sensors and actuators. Smart Materials and Structures,1995,4(4):281-290
[108]Auricchio F,Sacco E, Vairo G. A mixed FSDT finite element for monoclinic laminated plates. Computers & structures,2006,84(8):624-639
[109]Sze K Y, Yao L Q. Modelling smart structures with segmented piezoelectric sensors and actuators. Journal of Sound and Vibration,2000,235(3):495-520
[110]Carrera E. An improved Reissner-Mindlin-type model for the electromechanical analysis of multilayered plates including piezo-layers. Journal of Intelligent Material Systems and Structures,1997,8(3):232-248
[111]Sheikh A H, Topdar P, Halder S. An appropriate FE model for through-thickness variation of displacement and potential in thin/moderately thick smart laminates. Composite Structures,2001,51(4):401-409
[112]王锋,李道奎,唐国金.基于高阶位移模式的压电层合板动力有限元模型.计算力学学报,2007,24(6)：891-898
[113]王锋,唐国金,李道奎.含压电片复合材料层合板的高阶计算模型.复合材 料学报,2005,22(1)：164-170
[114]Hajianmaleki M, Qatu M S. Vibrations of straight and curved composite beams: A review. Composite structures,2013,100(6):218-232
[115]张福学.现代压电学.北京：科学出版社,2002
[116]王矜奉,物理学,姜祖桐,等.压电振动.北京：科学出版社,1989
[117]Rogacheva N N. The theory of piezoelectric shells and plates. Boca Raton:CRC Press,1994
[118]Liu Gr, Tani J, Ohyoshi T, et al. Transient waves in anisotropic laminated plates. Journal of vibration, acoustics, stress, and reliability in design,1991,113(2): 230-234
[119]Vel S S, Batra R C. Three-dimensional analytical solution for hybrid multilayered piezoelectric plates. Journal of Applied Mechanics,2000,67(3): 558-567
[120]Qiu J H, Tani J. Vibration control of a cylindrical shell using distributed piezoelectric sensors and actuators. Journal of Intelligent Material Systems and Structures,1995,6(4):474-481
[121]Tzou H S, Gadre M L. Theoretical analysis of a multi-layered thin shell coupled with piezoelectric shell actuators for distributed vibration controls. Journal of Sound and Vibration,1989,132(3):433-450
[122]Lester Harold C,Lefebvre Sylvie.Piezoelectric actuator models for active sound and vibration control of cylinders.Journal of Intelligent Material Systems and Structures,1993,4(3):295-306
[123]Sonti V R, Jones J D. Curved piezoactuator model for active vibration control of cylindrical shells. AIAA Journal,1996,34(5):1034-1040
[124]Perry M A, Bates R A, Atherton M, et al. A finite-element-based formulation for sensitivity studies of piezoelectric systems. Smart Materials and Structures,2008,17(1):1-7
[125]Leugering G, Novotny A A, Menzala G P, et al. Shape sensitivity analysis of a quasi-electrostatic piezoelectric system in multilayered media. Mathematical Methods in the Applied Sciences,2010,33(17):2118-2131
[126]Kim S J, Jones J D. Influence of piezo-actuator thickness on the active vibration control of a cantilever beam. Journal of Intelligent Material Systems and Structures,1995,6(5):610-623
[127]Han X, Jiang C, Gong S, et al. Transient waves in composite-laminated plates with uncertain load and material property. International Journal for Numerical Methods in Engineering,2008,75(3):253-274