功能梯度梁临界荷载与固有频率的微分求积法计算  

Calculation of Critical Load and Natural Frequency of Functionally Graded Beam by Differential Quadrature Method

作　　者：熊海超 葛仁余 马国强 刘小双 余本源 XIONG Haichao;GE Renyu;MA Guoqiang;LIU Xiaoshuang;YU Benyuan(School of Architecture and Civil Engineering,Anhui Polytechnic University,Wuhu 241000,China)

机构地区：[1]安徽工程大学建筑工程学院,安徽芜湖241000

出　　处：《安徽工程大学学报》2021年第2期49-55,共7页Journal of Anhui Polytechnic University

基　　金：安徽省自然科学基金资助项目(1808085ME147);国家级大学生创新创业训练计划基金资助项目(202010363121)。

摘　　要：运用微分求积法(DQM)研究了轴向功能梯度变截面Euler-Bernoulli梁的屈曲临界荷载和固有频率,以及轴向荷载对功能梯度变截面梁固有频率的影响。首先基于Euler-Bernoulli梁理论,建立了求解功能梯度材料Euler-Bernoulli变截面梁屈曲临界荷载和固有频率的变系数常微分方程;然后基于微分求积法原理将梁的变系数常微分方程的特征值问题转化为一组线性代数方程组的特征值问题;再由QR法计算获得功能梯度变截面梁的屈曲临界荷载和固有频率。数值计算结果表明,采用等步长均匀网格时,微分求积法计算数值不稳定甚至失真,而用变步长非均匀网格获得计算值精度较高,如切比雪夫多项式的根作为离散节点分布形式;研究还表明,轴向拉力使梁的固有频率增大,压力使梁的固有频率减小,当第1阶固有频率为零时,对应的轴向压力即为梁的屈曲临界荷载。The differential quadrature method is used to study the critical load and natural frequency of an axial functionally graded Euler Bernoulli beam with variable cross-section.Based on Euler Bernoulli beam theory,an ordinary differential equation for solving the critical load and natural frequency of a functionally graded material Euler Bernoulli beam with variable cross section is established.Then,based on the principle of differential quadrature,the eigenvalue problem of ordinary differential equation with variable coefficients is transformed into the eigenvalue problem of a set of linear algebraic equations,and the critical load and natural frequency are obtained by QR method.The numerical results show that the DQM method is not stable or even distorted when the uniform grid with equal step size is used,while the accuracy of the value obtained by the variable step size non-uniform grid is higher,such as the root of Chebyshev polynomial as the distribution form of discrete nodes.The study also shows that the axial tension increases the natural frequency of the beam,while the pressure reduces the natural frequency of the beam.When the first natural frequency is zero,the corresponding axis The axial pressure is the critical load of the beam.

关 键 词：轴向功能梯度梁 固有频率 屈曲临界荷载 微分求积法 

分 类 号：O326[理学—一般力学与力学基础]