Free Vibration of Functionally Graded Material Beams With Surface-Bonded Piezoelectric Layers in Thermal Environment
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摘要: 研究了上下表面粘贴压电层的功能梯度材料Euler-Bernoulli梁在升温及电场作用下的屈曲和自由振动行为.在精确考虑轴线伸长基础上,建立了压电功能梯度材料层合梁在热-电-机载荷作用下的几何非线性动力学控制方程.其中,假设功能梯度材料性质沿厚度方向按照幂函数连续变化,上下压电层为各向同性均匀材料.在小振幅和谐振动假设下,上述非线性偏微分方程组被转化为两套相互耦合的常微分方程组,即过屈曲问题的控制方程和过屈曲构形附近的线性振动控制方程.采用打靶法数值求解上述两个耦合的常微分方程边值问题,获得了在均匀电场和横向非均匀升温场作用下两端固定压电-功能梯度材料层合梁在屈曲前和过屈曲构型附近的自由振动响应.绘出了梁的过屈曲平衡路径以及前3阶固有频率随热、电载荷及材料梯度参数变化的特性曲线.结果表明,梁的前3阶频率在屈曲前随着温度升高而减小,在进入过屈曲后它们却随着温度升高而增加.通过施加电压在压电层产生拉应力可有效地提高粱的热屈曲临界载荷,从而提高其固有频率.Abstract: Free vibration of statically thermal post-buckled functionally graded material beams with surface-bonded piezoelectric layers subjected to both temperature rise and voltage is studied. By accurately considering the axial extension and based on Euler-Bernoulli beam theory, geometrically nonlinear dynamic governing equations for FGM beams with surfacebonded piezoelectric layers subjected to thermo-electro-mechanical loadings were formulated. It was assumed that the material properties of the middle FGM layer vary continuously as a power law function of the thickness coordinate and that the piezoelectric layers are isotropic and homogenous. By assuming that the amplitude of beam. s vibration is small and its response harmonic, the above mentioned non-linear partial differential equations were reduced to two sets of coupled ordinary differential equations; the one for the postbuckling, and the other for linear vibration of the beam superimposed upon the post buckled configuration. By using a shooting method to solve the two sets of ordinary differential equations with fixed-fixed boundary conditions numerically, response of postbuckling and free vibration in the vicinity of the postbuckled configuration of the beam with fixed-fixed ends and subjected to transversely non-uniform heating and uniform electric field were obtained. Thermo-electric postbuckling equilibrium paths and characteristic curves of the first three natural frequencies versus the temperature, the electricity and the material gradient parameters were plotted. It is found that the three lowest frequencies of the prebuckled beam decrease with an increase in the temperature, but those of a buckled beam increase monotonically with the temperature rise. The results also show that the tensional force produced in the piezoelectric layers by the voltage can efficiently increase the critical buckling temperature and the natural frequency.
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[1] Coffin D W,Bloom F.Elastica solution for the hygrothermal buckling of a beam[J].International Journal of Non-Linear Mechanics,1999,34(5):935-947. doi: 10.1016/S0020-7462(98)00067-5 [2] Vaz M A,Solano R F.Post-buckling analysis of slender elastic rods subjected to uniform thermal loads[J].Journal of Thermal Stresses,2003,26(9):847-860. doi: 10.1080/01495730306293 [3] Vaz M A,Solano R F.Thermal post-buckling of slender elastic rods with hinged ends constrained by a linear spring[J].Journal of Thermal Stresses,2004,27(4):367-380. doi: 10.1080/01495730490427591 [4] 李世荣,程昌钧.加热弹性杆的热过屈曲分析[J].应用数学和力学,2000,21(2):119-125. [5] Li S-R,Zhou Y-H,Zheng X-J.Thermal post-buckling of a heated elastic rod with pinned-fixed ends[J].Journal of Thermal Stresses,2002,25(1):45-56. doi: 10.1080/014957302753305862 [6] Li S-R,Batra R C.Thermal buckling and post-buckling of Euler-Bernoulli beams supported on nonlinear elastic foundations[J].AIAA Journal,2007,45(3):711-720. [7] Li S-R,Zhou Y-H.Geometrically nonlinear analysis of Timoshenko beams under thermomechanical loadings[J].Journal of Thermal Stresses,2003,26(9):867-872. [8] Li S-R,Teng Z-C,Zhou Y-H.Free vibration of heated Euler-Bernoulli beams with thermal post-buckling deformations[J].Journal of Thermal Stresses,2004,27(9):843-856. doi: 10.1080/01495730490486352 [9] Sankar B V.An elasticity solution for functionally graded beams[J].Composites Science and Technology,2001,61(5):689-696. doi: 10.1016/S0266-3538(01)00007-0 [10] Chakraborty A,Gopalakrishnan S,Reddy J N.A new beam finite element for the analysis of functionally graded materials[J].International Journal of Mechanical Sciences,2003,45(3):519-539. doi: 10.1016/S0020-7403(03)00058-4 [11] Bhangale R K,Ganesan N.Thermoelastic buckling and vibration behavior of a functionally graded sandwich beam with constrained viscoelastic core[J]. Journal of Sound and Vibration, 2006, 295(1/2):294-316. doi: 10.1016/j.jsv.2006.01.026 [12] Xia X-K,Shen H-S.Vibration of postbuckled FGM hybrid laminated plates in thermal environment[J].Engineering Structures,2008,30(9):2420-2435. doi: 10.1016/j.engstruct.2008.01.022 [13] 夏贤坤,沈惠申.功能梯度材料剪切板热屈曲后的非线性振动[J].振动工程学报,2008,21(2):120-125. [14] 李世荣,张靖华,赵永刚.功能梯度材料Timoshenko梁的热过屈曲分析[J].应用数学和力学,2006,27(6):709-715. [15] Crawley E F,de Luis J.Use of piezoelectric actuators as elements of intelligent structures[J].AIAA Journal,1987,25(10):1373-1385. doi: 10.2514/3.9792 [16] Zhou Y-H,Wang J-Z.Vibration control of piezoelectric beam-type-plates with geometrical nonlinear deformation[J].Int J Non-Linear Mech,2004,39(6):909-920. doi: 10.1016/S0020-7462(03)00074-X [17] Zhou Y-H,Wang J-Z,Zheng X-J,et al.Vibration control of variable thickness plates with piezoelectric sensors and actuators based on wavelet theory[J].Journal of Sound and Vibration,2000,237(3):395-410. doi: 10.1006/jsvi.2000.3031 [18] 林启荣,刘正兴,王宗利.电场作用下压电层合梁的分析[J].应用数学和力学,2001,22(9):969-975. [19] Fridman Y,Abramovich H.Enhanced structural behavior of flexible laminated composite beams[J].Composite Structures,2008,82(1):140-154. doi: 10.1016/j.compstruct.2007.05.007 [20] 于涛,仲政.均布荷载作用下功能梯度悬臂梁弯曲问题的解析解[J].固体力学学报,2006,27(1):15-20. [21] Huang X-L,Shen H-S.Vibration and dynamic response of functionally graded plates with piezoelectric actuators in thermal environments[J].Journal of Sound and Vibration,2006,289(1/2):25-53. doi: 10.1016/j.jsv.2005.01.033 [22] LI Shi-rong,Batra Romesh C,MA Lan-sheng.Vibration of thermally post-buckled orthotropic circular plate[J].Journal of Thermal Stresses,2007,30(1):43-57. doi: 10.1080/01495730600897161 [23] William H P,Brain P F,San A T,et al.Numerical Recipes—The Art of Scientific Computing[M].London:Cambridge University Press,1986.
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