In this paper, the finite free-form beam element is formulated by the isogeometric approach based on the Timoshenko beam theory to investigate the free vibration behavior of the beams. The non-uniform rational B-splines (NURBS) functions which define the geometry of the beam are used as the basis functions for the finite element analysis. In order to enrich the basis functions and to increase the accuracy of the solution fields, the h-, p-, and k-refinement techniques are implemented. The geometry and curvature of the beams are modelled in a unique way based on NURBS. All the effects of the the shear deformation, and the rotary inertia are taken into consideration by the present isogeometric model. Results of the beams for non-dimensional frequencies are compared with other available results in order to show the accuracy and efficiency of the present isogeometric approach. From numerical results, the present element can produce very accurate values of natural frequencies and the mode shapes due to exact definition of the geometry. With higher order basis functions, there is no shear locking phenomenon in very thin beam situations. Finally, the benchmark tests described in this study are provided as future reference solutions for Timoshenko beam vibration problem.

翻译：本文基于Timoshenko梁理论，采用等几何方法构建了有限自由形式梁单元，用于研究梁的自由振动行为。用于定义梁几何的非均匀有理B样条（NURBS）函数被用作有限元分析的基函数。为丰富基函数并提高解场的精度，实施了h细化、p细化和k细化技术。基于NURBS以独特方式对梁的几何形状和曲率进行建模。本文提出的等几何模型考虑了剪切变形和转动惯量的所有影响。为展示所提等几何方法的精度和效率，将梁的无量纲频率计算结果与其他现有结果进行了对比。数值结果表明，由于几何定义精确，本文单元能提供非常精确的固有频率值和振型。采用高阶基函数时，极薄梁情况下不会出现剪切闭锁现象。最后，本研究提供的基准测试可作为未来Timoshenko梁振动问题的参考解。