贴压电层的功能梯度层合梁的振动与稳定性分析
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摘要
多载荷作用下单层或多层智能材料结构元件的静动态力学行为已成为固体力学研究的热点内容,同时也是结构智能控制的关键性问题之一。本论文讨论和分析了压电-功能梯度材料单层及层合梁结构的静动态力学行为。主要研究了贴压电层的功能梯度梁的过屈曲行为和过屈曲构型附近的自由振动,包括以下三个方面的内容:
1.在精确考虑轴线伸长基础上,建立了功能梯度Euler梁在热过屈曲构型附近小振幅线性自由振动的控制方程。采用打靶法同时求解热过屈曲和振动问题的控制方程,得到了随温度载荷变化的热过屈曲平衡路径以及前几阶固有频率的数值解。分析和讨论了梁的材料梯度参数、温度场分布参数等对过屈曲变形和振动响应的影响。
2.基于精确Timoshenko理论,建立了压电-功能梯度Timoshenko层合梁在热-电-机载荷作用下的几何非线性控制方程。打靶法数值求解了相应的强非线性边值问题,获得了在均匀电场和横向非均匀升温场内两端固定Timoshenko梁的静态非线性屈曲和过屈曲数值解。绘出了梁的变形随热载荷、逆压电效应及材料梯度参数变化的特性曲线。结果表明,通过施加电压在压电层产生拉应力可以有效地提高梁的热屈曲临界载荷,延缓热过屈曲发生。
3.定量分析了上下表面粘贴压电层的FGM层合梁,在横向非均匀升温以及对压电层施加驱动电压载荷下的几何非线性大变形以及大变形构形附近的小振幅自由振动。对于屈曲前的微幅振动,得到了两端固定和简支压电-FGM层合梁在横向均匀电场和非均匀升温作用下线性振动的解析解。对于过屈曲梁振动问题,采用打靶法数值求解了两端固支梁的非线性边界值问题,给出了与热、电载荷参数有关的过屈曲平衡路径及频率-载荷曲线,详细分析和讨论逆压电效应、热载荷和材料梯度性质对变形和振动特性的影响。
In recent years, the problem of static deformation and dynamic response of single-layer or multi-layer intelligent structures subjected to multiple loads have been extensively investigated and become one of the most important and key issues in solid mechanics and intelligent control field. In this thesis, responses of post-buckling and linear free vibration of FGM beams with surface-bonded piezoelectric layers are studied. The main content andresents are as following:
1. By accurately considering the axial extension, governing equations for the small amplitude free vibration of functionally graded material Euler beams in the vicinity of thermal post-buckling configuration under thermal loadings, were formulated. By using shooting method to solve simultaneously the governing equations both for thermal post-buckling and vibration, numerical solutions of thermal post-buckling equilibrium paths and the first three natural frequencies of the FGM beam depending on the temperature rise are obtained. The effect of the parameters of material gradients, temperature field on the responses of post-buckling and vibration of the beam are examined.
2. Based on geometrically nonlinear theory for axial extensive Timoshenko beam, nonlinear governing equations, for functionally graded beams with surface-bonded piezoelectric layers subjected to thermo-electro-mechanical loadings were formulated. By using a shooting method, the obtained nonlinear boundary value problem was numerically solved and buckling and post-buckling response of the beams with the both ends fixed, subjected to transversely non-uniform heating and uniform electric field, were presented. Characteristic curves of the post-buckling deformation of the beam varying with thermal load, the electrical load, and the power law index are plotted. Numerical results show that the tensional force produced in the piezoelectric layers by the voltage can efficiently increase the critical buckling temperature and defer the thermal post-buckling to occur.
3. Geometrically nonlinear deformation and free vibration for functionally graded beams with surface-bonded piezoelectric layers in the vicinity of the post-buckling configuration, subjected to thermo-electro-mechanical loadings were analysised quantitatively. For the small amplitude free vibration befor the pre-buckling, analytical solutions of the laminated beam with surface-bonded piezoelectric layers subjected to transversely non-uniform heating and uniform electric field were arrived at. By using a shooting method, the boundary value problems for the post-buckling and linear vibration of the beam with the ends fixed was numerically solved. Characteristic curves of the post-buckling deformation and the lower order frequencies versus thermal load, the electrical load, and the power law index are plotted. Influence of the inverse piezoelectric effect, the temperature rise and the properties of the material gradients on the deformation and vibration are discussed.
1.在精确考虑轴线伸长基础上,建立了功能梯度Euler梁在热过屈曲构型附近小振幅线性自由振动的控制方程。采用打靶法同时求解热过屈曲和振动问题的控制方程,得到了随温度载荷变化的热过屈曲平衡路径以及前几阶固有频率的数值解。分析和讨论了梁的材料梯度参数、温度场分布参数等对过屈曲变形和振动响应的影响。
2.基于精确Timoshenko理论,建立了压电-功能梯度Timoshenko层合梁在热-电-机载荷作用下的几何非线性控制方程。打靶法数值求解了相应的强非线性边值问题,获得了在均匀电场和横向非均匀升温场内两端固定Timoshenko梁的静态非线性屈曲和过屈曲数值解。绘出了梁的变形随热载荷、逆压电效应及材料梯度参数变化的特性曲线。结果表明,通过施加电压在压电层产生拉应力可以有效地提高梁的热屈曲临界载荷,延缓热过屈曲发生。
3.定量分析了上下表面粘贴压电层的FGM层合梁,在横向非均匀升温以及对压电层施加驱动电压载荷下的几何非线性大变形以及大变形构形附近的小振幅自由振动。对于屈曲前的微幅振动,得到了两端固定和简支压电-FGM层合梁在横向均匀电场和非均匀升温作用下线性振动的解析解。对于过屈曲梁振动问题,采用打靶法数值求解了两端固支梁的非线性边界值问题,给出了与热、电载荷参数有关的过屈曲平衡路径及频率-载荷曲线,详细分析和讨论逆压电效应、热载荷和材料梯度性质对变形和振动特性的影响。
In recent years, the problem of static deformation and dynamic response of single-layer or multi-layer intelligent structures subjected to multiple loads have been extensively investigated and become one of the most important and key issues in solid mechanics and intelligent control field. In this thesis, responses of post-buckling and linear free vibration of FGM beams with surface-bonded piezoelectric layers are studied. The main content andresents are as following:
1. By accurately considering the axial extension, governing equations for the small amplitude free vibration of functionally graded material Euler beams in the vicinity of thermal post-buckling configuration under thermal loadings, were formulated. By using shooting method to solve simultaneously the governing equations both for thermal post-buckling and vibration, numerical solutions of thermal post-buckling equilibrium paths and the first three natural frequencies of the FGM beam depending on the temperature rise are obtained. The effect of the parameters of material gradients, temperature field on the responses of post-buckling and vibration of the beam are examined.
2. Based on geometrically nonlinear theory for axial extensive Timoshenko beam, nonlinear governing equations, for functionally graded beams with surface-bonded piezoelectric layers subjected to thermo-electro-mechanical loadings were formulated. By using a shooting method, the obtained nonlinear boundary value problem was numerically solved and buckling and post-buckling response of the beams with the both ends fixed, subjected to transversely non-uniform heating and uniform electric field, were presented. Characteristic curves of the post-buckling deformation of the beam varying with thermal load, the electrical load, and the power law index are plotted. Numerical results show that the tensional force produced in the piezoelectric layers by the voltage can efficiently increase the critical buckling temperature and defer the thermal post-buckling to occur.
3. Geometrically nonlinear deformation and free vibration for functionally graded beams with surface-bonded piezoelectric layers in the vicinity of the post-buckling configuration, subjected to thermo-electro-mechanical loadings were analysised quantitatively. For the small amplitude free vibration befor the pre-buckling, analytical solutions of the laminated beam with surface-bonded piezoelectric layers subjected to transversely non-uniform heating and uniform electric field were arrived at. By using a shooting method, the boundary value problems for the post-buckling and linear vibration of the beam with the ends fixed was numerically solved. Characteristic curves of the post-buckling deformation and the lower order frequencies versus thermal load, the electrical load, and the power law index are plotted. Influence of the inverse piezoelectric effect, the temperature rise and the properties of the material gradients on the deformation and vibration are discussed.
引文
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