基于改进AMDM的变截面Timoshenko梁的自由振动分析  

Free vibration analysis of non-uniform Timoshenko beams using improved AMDM

作　　者：姚东辉[1] 任勇生[1] YAO Donghui;REN Yongsheng(College of Mechanical and Electronic Engineering,Shandong University of Science and Technology,Qingdao 266590,China)

机构地区：[1]山东科技大学机械电子工程学院,山东青岛266590

出　　处：《振动与冲击》2025年第8期89-100,共12页Journal of Vibration and Shock

基　　金：国家自然科学基金(11672166)。

摘　　要：梁的自由振动问题通常可以归结为对运动偏微分方程的求解。修正的Admoian分解法(Adomian modified decomposition method,AMDM)是一种精度较高的求解偏微分方程近似解析解的数学方法,但其计算精度取决于收敛域,在求解横截面尺寸随位置线性或非线性变化的非均匀Timoshenko梁的振动问题时收敛性较差。为了改进收敛性,将振动方程中与截面形状相关的项分离出来,根据形状相关项Taylor展开式的收敛情况将梁分割成若干小段,分别对每个小段应用AMDM,最后通过给定的边界条件求解固有频率。上述改进AMDM为求解截面尺寸非线性变化的Timoshenko梁的振动问题提供了一种通用方法。该研究分析了悬臂梁边界条件下梁的锥度、形状函数的幂次等参数对收敛性和固有频率的影响。通过与有限元仿真的对比,验证了方法的有效性和准确性。The free vibration of beams can usually be modeled by differential equations.The Adomian modified decomposition method(AMDM)is a mathematical method for solving approximate analytical solutions of differential equations,which has high accuracy.However,its accuracy depends on the convergence domain and cannot converge when solving vibration problem of Timoshenko beams with cross section varying along axial coordinate in linear or non-linear form.In order to improve convergence,firstly,the shape dependent terms in the differential equations of vibration was extracted in this paper.According the convergence of the Taylor expansion of these terms,the beam was divided into several segments.AMDM was applied to each segment of the beam.Then,boundary conditions were substituted to solve natural frequencies and modal functions.This improved AMDM provides a universal method for solving the vibration problem of Timoshenko beams with cross section varying in non-linear form.The influence of factors such as taper and power of shape functions on convergence and natural frequency was investigated.Numerical results show that the method proposed in this paper is in good agreement with FEM simulation.

关 键 词：变截面 TIMOSHENKO梁 改进修正的Admoian分解法(AMDM) 形状函数 固有频率 

分 类 号：TH113[机械工程—机械设计及理论]