无单元法求解欧拉梁及梁系的自由振动问题

Solution to free vibration of Euler beam and girder system with Element-Free Galerkin method

双变量无单元法以广义移动最小二乘法为理论基础,同时考虑挠度和转角双变量。采用双变量无单元法建立了欧拉梁的质量矩阵和刚度矩阵,并进行自由振动的计算。不同边界条件欧拉梁动力特性的算例表明:双变量无单元法比与只考虑挠度的单变量无单元法具有更高的插值精度,并能在高阶振型计算中获得明显优于有限元的计算精度。通过试算法对影响半径中的scale乘子进行了讨论,认为在动力计算中scale取3.5较合理。最后在欧拉梁的基础上,将无单元法应用于梁系模型的自由振动计算,显示了该法在复杂模型中的精确性。

Generalized moving least square method is the theoretical basis of a new Element-Free Galerkin （EFG） double-variable approximation. Deflection and angle of rotation are both considered in the new method. Mass matrix and stiffness matrix of Euler beam are established with EFG and free vibration is analyzed. Dynamic characteristics of three Euler beams with different boundary conditions are calculated. It indicated that double-variable approximation has smaller interpolation error than single-variable approximation, and it is more accurate than FEM of higher-order modes. With trial method, scale of influence radius is discussed and then 3.5 is regarded as its reasonable values. Based on Euler beam, free vibration of girder system is calculated with EFG and the accuracy in complex model is shown.