热冲击下轴向运动FGM梁的自由振动分析

基于Timoshenko梁理论研究两端夹紧、一端夹紧一端简支、两端简支三种不同边界条件下的轴向运动功能梯度材料(FGM)梁在热冲击载荷作用下的自由振动响应。利用Hamilton原理推导热冲击下轴向运动FGM梁的自由振动控制微分方程,并采用分离变量法求解一维热传导方程。通过微分求积法(DQM)在梁的长度方向进行离散,将原方程转化为四阶广义特征值问题,求解FGM梁自由振动的无量纲固有频率并进行特性分析。考虑了不同热冲击载荷,不同梯度指数和不同轴向运动无量纲速度对FGM梁自振频率的影响。结果表明:热冲击载荷越大,对降低FGM梁的固有频率的效果越明显;在轴向运动速度和热流输入不改变的情况下,逐渐增大材料梯度指数会使FGM梁的固有频率随之减小;FGM梁对热冲击短时间内有减缓作用,相对于均匀材料一阶失稳所需时间更长,受到热冲击的FGM梁在轴向运动时也更快达到失稳状态。

Free vibration analysis of axially moving FGM beams under thermal shock

Based on the Timoshenko beam theory, free vibrations of axially moving functionally graded material (FGM) beams under thermal shock were investigated for three different conditions with clamped-clamped, clamped-simply supported and simply supported-simply supported boundary. Using the Hamilton principle, the governing differential equation of free vibration of an axially moving FGM beam under thermal shock was derived and the one-dimensional heat conduction equation was solved by variable separation method. The beam was discretized along the length direction by using differential quadrature method (DQM) and the original equation was turned into fourth order generalized characteristic value problem, then the dimensionless natural frequencies of free vibration of the FGM beam were solved and analyzed. The influences of the different heat flow input, graded index and dimensionless axially moving speed on natural frequencies of free vibration of the FGM beam were presented. The results show that the larger the thermal shock load is, the more obvious the effect is to reduce the natural frequency of the FGM beam. Under the condition of certain axially moving speeds and heat flow input, increasing gradient indexes of functionally graded material gradually will reduce the dimensionless natural frequencies of the beams. The FGM beam will slow down the thermal shock in a short period of time and the first order instability will take long time relative to homogeneous material. Under thermal shock, the FGM beam easier reach an instability state in axially moving.