基于无网格法的Timoshenko曲梁动力分析

曲梁具有外形美观、受力性能良好的优点，故在工程中得到广泛应用。本文基于移动最小二乘近似和一阶剪切变形理论，提出一种对Timoshenko曲梁自由振动和受迫振动进行分析的无网格方法。通过一系列离散点离散曲梁，建立曲梁无网格模型，然后推导曲梁势能和动能方程，通过哈密顿原理给出曲梁自由振动和受迫振动的控制方程，因为本文方法不能直接施加边界条件，所以使用完全转换法处理本质边界条件，最后求解方程得到频率和振动模态。文末通过算例验证了本文方法的有效性，且通过收敛性分析表明本文方法具有较好的收敛性，并进一步分析了不同边界条件、跨高比和变截面变曲率对曲梁自由振动和受迫振动的影响，将计算结果与文献解或ABAQUS解进行对比分析，表明本文方法具有较高的精度，且适用于实际工程情况。

Analysis of free vibration and forced vibration of Timoshenko curved beam based on meshless method

Curved beam has the advantages of beautiful appearance and good mechanical performance, so it has been widely used in engineering. A moving-least squares (MLS) meshless method is proposed in this paper to solve the free vibration and forced vibration problems of Timoshenko curved beams, and the first-order shear deformation theory (FSDT) is adopted. Firstly, a series of discrete points is used to establish the meshless model of the curved beam. Then, the potential energy and kinetic energy equations of the curved beam are derived. The equations governing the free vibration and forced vibration of the curved beam are given according to the Hamiltonian principle, and the full transformation method is employed to deal with the essential boundary conditions because the boundary conditions cannot be directly applied by the meshless method. Lastly, the natural frequencies and vibration modes are obtained by solving the corresponding equations. At the end of this paper, the effectiveness of the meshless method is verified by numerical examples, and the results show that the proposed method has good convergence characteristics. In addition, the effects of various boundary conditions, span height ratios, variable sections and curvatures on the free vibration and forced vibration of curved beams are discussed. The calculated results are compared with the literature solution or ABAQUS solution, which shows that the meshless method has high accuracy and is suitable for actual engineering projects.