考虑运动效应的直梁振动的解析解与传递矩阵解  

Analytical Solution and Transfer Matrix Solution to Straight Beam Vibration Considering Motion Effects

作　　者：王馨悦 随岁寒 李成 WANG Xinyue;SUI Suihan;LI Cheng(School of Rail Transportation,Soochow University,Suzhou 215131;School of Mechanical Engineering,Shangqiu Institute of Technology,Shangqiu 476000)

机构地区：[1]苏州大学轨道交通学院,江苏苏州215131 [2]商丘工学院机械工程学院,河南商丘476000

出　　处：《常州工学院学报》2023年第3期1-8,共8页Journal of Changzhou Institute of Technology

基　　金：国家自然科学基金资助项目(11972240);河南省教育厅高等学校重点科研项目(23B130002)。

摘　　要：运动直梁的自由振动常见于机械系统中,以往研究多给出数值解,在此分别通过解析法和传递矩阵法求解运动直梁的自由振动问题,并与有限元法的计算结果进行比较,讨论了两种方法的适用情况。解析法首先推导积分方程,再根据变分原理求解出近似解析解。传递矩阵法则是先由Hamilton原理得到控制方程,同时建立剪力与弯矩方程,然后通过分离变量法求解,再代入边界条件即可得到矩阵形式的方程,令矩阵方程的系数行列式为零即可推得梁的各阶固有频率。通过数值算例,检验解析解与传递矩阵解的可靠性。结果表明运动梁自由振动的前三阶固有频率与运动速度成反比,与梁的厚度成正比。将解析法和传递矩阵法的结果与有限元法的结果进行比较发现,解析法适用于尺寸较大但速度较小甚至静止无运动的梁,而传递矩阵法适用于小尺寸梁。The free vibration of a moving straight beam is commonly seen in mechanical systems.Unlike most previous studies where numerical solutions were provided,the analytical method and the transfer matrix method are used to solve the free vibration of a moving straight beam.The results from the two me\|thods are compared with those of the finite element method and the applicability of both the methods is discussed.In the analytical method,the integral equation is derived firstly and then the approximate analytical solution is solved via the variational principle.In the transfer matrix method,the governing equation is obtained using Hamilton principle,the shear force and bending moment equations are established,and then solved by the method of separation of variables.The equation in matrix form can be obtained by substituting the boundary conditions,and the natural frequency of each order of the beam can be deduced by making the determinant of the coefficients of the matrix equation zero.The reliability of analytical and transfer matrix solutions is proved through numerical examples.The results show that the first three natural frequencies are inversely proportional to the moving speed and directly proportional to the thickness of the beam.Comparing the results of the analytical and transfer matrix methods with those of the finite element method,one can find that the analytical method is suitable for beams with larger size and smaller speed such as a stationary beam without motion,while the transfer matrix method is suitable for beams with smaller size.

关 键 词：运动直梁 自由振动 解析法 传递矩阵法 固有频率 

分 类 号：O327[理学—一般力学与力学基础]