微分求积法求解悬臂L梁固有振动特性研究

Study on Natural Vibration Characteristics of L⁃Shaped Cantilever Beams With the Differential Quadrature Method

悬臂L梁结构由于具有柔性大、可设计性强、空间利用充分,振动过程中变形方式多样等独特优势而受到了广泛的关注与研究.该文提出了一种基于微分求积法求解末端附加质量块的矩形等截面均质悬臂细长L梁的各阶固有频率和模态的方法.在双坐标系下,基于Euler-Bernoulli梁理论建立了悬臂L梁的动力学方程,然后通过选取Chebyshev多项式的根作为节点坐标、选取Lagrange插值基函数、求解各阶权系数、处理边界条件等步骤,最终利用求解矩阵广义特征值问题的方法求得结构各阶固有频率及模态.在边界条件的处理上,直接将边界条件施加于边界点上,通过对比研究验证了该文固有频率理论解的正确性.最后分析了末端质量、内外梁的长度比、宽度、厚度对各阶固有振动特性的影响.该方法可以进一步应用推广到相关结构振动的研究中.

The L⁃shaped cantilever beam structure has many unique advantages such as large flexibility,strong designability,full utilization of space and various deformation modes during vibration,and is widely regarded and studied.A differential quadrature method was proposed to solve the natural frequencies and modes of rec⁃tangular⁃section homogeneous slender L⁃shaped cantilever beams with additional end masses.In the double co⁃ordinate systems,the dynamic equations for the L⁃shaped cantilever beam based on the Euler⁃Bernoulli beam theory were established.With selected roots of the Chebyshev polynomial as the node coordinates,the La⁃grange interpolation basis function was employed,the weight coefficients of each order were solved,and the boundary conditions were considered,to obtain the natural frequencies and modes of all orders of the structure through resolution of the generalized matrix eigenvalue problem.The theoretical solution of the natural frequen⁃cies was verified in comparison with the previous theoretical results and the finite element results.Finally,the effects of the end mass,the length ratio,the width and the thickness of the inner and outer beams on the natu⁃ral vibration characteristics of all orders were discussed.This method can be further applied to the study of re⁃lated structural vibrations.