张伟杰
    (河南理工大学土木工程学院，焦作454000)

[摘要] 应用Hamilton变分原理推导出变截面Timoshenko悬臂梁在谐波位移激励下强迫振动的控制微分方程及相应的边界条件，并用常微分方程求解器( Ordinary Differential Equation Solver)进行求解，同时求得等截面及变截面Timoshenko悬臂梁自由振动的前三阶振动频率和相应的振型，计算结果通过解析解以及SAP2000有限元程序进行校核。数值算例表明：该方法具有收敛速度快、精度高等特点，从而为工程中塔式结构的动力特性分析提供理论参考。
[关键词] 变截面悬臂梁；Timoshenko梁；常微分方程求解器；自由振动；共振响应

ODE solution of dynamic characteristics of Timoshenko cantilever beams with variable cross-section
Zhang Weijie
( School of Civil Engineering, Henan Polytechnic University, Jiaozu0 454000, China)

Abstract: The differential equations and corresponding boundary conditions of Timoshenko cantilever beams with variable cross-section which impelled by harmonic displacement were developed by Hamilton variational principle and solved by Ordinary Differential Equation Solver. At the same time, the first three natural frequencies and corresponding mode shapes of the Timoshenko cantilever beams with same or variable cross-section were obtained. In order to illustrate the accuracy and practical usefulness of this method, numerical solutions by this study were presented and compared with exact solutions and solutions by SAP2000. The presented method is simple in convergent and sufficiently accurate, so as to provide theoretical reference in the analysis of dynamic characteristics of tower structures in civil engmeenng.
Keywords: cantilever with variable cross-section, Timoshenko beam; Ordinary Differential Equation Solver, free vibration; the resonant response