热环境中功能梯度材料Euler梁的自由振动

Free vibration of Euler beams with functionally graded material in thermal environment

研究功能梯度材料Euler梁在温度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,建立功能梯度Euler梁在热载荷作用下的几何非线性控制方程.将控制方程的响应分解为热过屈曲静态解和振动解两部分,得到功能梯度材料梁在热过屈曲构型附近小振幅线性自由振动的微分方程.其中,假设功能梯度的材料性质沿厚度方向按照幂函数连续变化,采用打靶法数值求解所得强非线性边值问题,获得在横向升温场内两端固定Euler梁的热过屈曲平衡路径以及前三阶固有频率的数值解.分析和讨论梁的材料梯度参数、温度场分布参数等因素对过屈曲变形和振动响应的影响.

The behavior of free vibration and thermal post-buckling of Euler beams with functionally graded material in non-statior, ary thermal environment was studied. By accurately considering the axial extension and based on Euler beam theory, geometrically nonlinear governing equations of functionally graded beams subjected to thermo-mechanical loadings were formulated. By decomposing the response of the governing equations into two parts such as static thermal post-bucking solution and dynamic vibration solution, the linear differential equations were obtained for beam small vibration in the vicinity of the thermal post-buckling condition. By assuming that the material properties of the beam would vary continuously as a power function across the thickness, and using a shooting method, the strongly nonlinear boundary value problem obtained was numerically solved and the thermal post-buckling equilibrium path of the Euler beam with both ends fixed and subjected to transverse heating as well as the numeric solutions of its first thirdorder natural frequencies were achieved. The effect of the parameters of material gradients, and temperature field distribution on the responses of post-buckling and vibration of the beam was analyzed and discussed, also.