复杂边界条件轴向功能梯度梁动力学分析

为分析复杂边界条件及截面属性对轴向功能梯度梁动力学特性的影响,采用Gauss-Lobatto节点与Chebyshev多项式对变截面轴向功能梯度梁变形场进行离散,利用Chebyshev谱方法和Lagrange方程推导了系统的离散控制方程,通过投影矩阵法施加经典及弹性连接边界条件,分析了材料梯度指数、截面变化率、弹性支撑边界条件、末端集中质量等因素对系统固有频率的影响.结果表明:截面变化率和材料梯度指数对固有频率的影响因边界条件不同而不同;悬臂梁的前一阶量纲一的固有频率随着截面变化率的增大而增大,其他情况下则随截面变化率的增大而减小;悬臂梁的一阶固有频率随材料梯度指数先增大后逐渐减小,而其余各阶频率均显示出增大的趋势,而两端固支梁的前两阶频率呈现减小趋势,后两阶频率则显示出增大的趋势;两端简支梁各阶固有频率皆随材料梯度指数增大而增大;随着弹性支承刚度的增大各阶固有频率均呈阶梯状增大,但当弹性支撑刚度较小时,旋转弹簧相较线性弹簧对系统固有频率影响更大;末端附加质量将使固有频率减小且对高阶固有频率的影响更大.

Dynamic analysis of axially functionally graded beams with complex boundary conditions

To investigate the effects of boundary conditions and geometric properties on dynamic characteristics of axially functionally graded beams, Gauss-Lobatto sampling and Chebyshev polynomials are used to discretize deformation fields of the beams, and the discrete governing equations are obtained by utilizing Chebyshev spectral method and Lagrange''s equation. After employing projection matrices, classical as well as elastic boundary conditions are incorporated in the governing equations. The effects of various parameters, such as material gradient index, cross-sectional area, and attached tip mass on the vibration of the beams are analyzed. The results show that these effects differ for different boundary conditions. As the taper ratio increases, the first natural frequency of the cantilever beam increases simultaneously. While for the beams with other boundary conditions, their natural frequencies decrease. With the raising of the material gradient index, the first natural frequency of the cantilever beam increases firstly and then decreases, but other frequencies all increase. But for the fixed-fixed beam, its first two natural frequencies decrease, the third and the fourth increase. For the pinned-pinned beam, all natural frequencies increase with increasing material index. When the elastic support becomes stiffer, all natural frequencies increase with a step. The effect of the rotational spring is more pronounced than the translational spring when the elastic supports are of low stiffness. The attached tip mass makes the natural frequencies smaller and this effect appears more pronounced for higher modes.