平面曲梁面内自由振动分析的自适应有限元法

A SELF-ADAPTIVE FEM FOR FREE VIBRATION ANALYSIS OF PLANAR CURVED BEAMS WITH VARIABLE CROSS-SECTIONS

该文将杆系结构自由振动精确分析的Wittrick-Williams算法、导护型Newton法和基于单元能量投影(EEP)超收敛计算的自适应有限元法有机结合,应用于平面变截面曲梁面内自由振动的分析,可以得到数值精确解,即频率和振型的精度均可满足用户事先给定的误差限。通过对无限细密网格上的有限元模型作自由度的凝聚可转化为精确动力刚度模型的分析对比,为自适应有限元法建立了与精确动力刚度法之间的等价关系、等价公式和等价算法。并对精确动力刚度法中两阶段算法给出了自适应有限元的实施方案。该文给出了有代表性的数值算例,计算结果表明:该方法是一种精确、可靠、高效的自由振动分析方法。

This paper presents a self-adaptive Finite Element Method （FEM） for the free vibration analysis of planar curved beams. The method integrates several techniques such as the Wittrick-Williams algorithm and the guided and guarded Newton method in the exact Dynamic Stiffness Method （DSM） for the vibration analysis of skeletal structures, and the self-adaptive FEM for linear BVP based on the Element Energy Projection （EEP） super-convergence calculation. The method can yield exact numerical results, i.e. the accuracy of the frequencies and the modes can satisfy the user-preset error tolerances. The finite element model on the infinitely dense mesh can be reduced to the exact dynamic stiffness model by condensation, thus it can produce exact results theoretically. Based on this comparison, the equivalence between the self-adaptive FEM and the exact DSM is set up. As a result, the corresponding equivalent formulae and equivalent algorithm are established and the two-phase algorithm for the exact DSM is extended to the self-adaptive FEM. The representative numerical examples show that this method is accurate, reliable and efficient.