A recursive rotational-coordinate-based formulation of a planar Euler–Bernoulli beam is developed, where large displacements, deformations, and rotations are considered. Different from the traditional rotational-coordinate-based formulations, relative rotational angles rather than absolute ones are used as generalized coordinates. The number of generalized coordinates is minimized, which is inherited from traditional rotational-coordinate-based formulations. A recursive scheme is used for adjacent elements of the beam that is considered as a chain-like structure. Integrals in the mass matrix and generalized force vector of each element of the beam can be analytically derived, and a numerical trick based on Taylor polynomial approximations is adopted to avoid numerical singularity. The current formulation entirely avoids the evaluation of integrals in each time step, which greatly improves the computational efficiency. Three widely-used examples are studied to illustrate the performance of the proposed method. Results indicate that the present formulation can achieve the same accuracy, as well as much higher efficiency, compared to some traditional formulations. In addition, calculation time of the current formulation almost linearly increases with the increasing number of elements of the beam, indicating that the computational complexity of the current formulation is \(O\left ( N \right )\).

开发了基于递归旋转坐标的平面欧拉-伯努利梁公式，其中考虑了大的位移，变形和旋转。与传统的基于旋转坐标的公式不同，相对旋转角度而不是绝对旋转角度被用作广义坐标。通用坐标的数量已减到最少，这是从传统的基于旋转坐标的公式继承而来的。递归方案用于被认为是链状结构的梁的相邻元素。可以解析得出梁中每个单元的质量矩阵和广义力矢量的积分，并采用基于泰勒多项式逼近的数值技巧来避免数值奇异。当前的公式完全避免了在每个时间步中对积分的求值，从而大大提高了计算效率。研究了三个广泛使用的示例，以说明该方法的性能。结果表明，与某些传统配方相比，本配方可以实现相同的精度以及更高的效率。此外，当前公式的计算时间几乎随着光束元素数量的增加而线性增加，这表明当前公式的计算复杂度为 与某些传统配方相比，效率更高。此外，当前公式的计算时间几乎随着光束元素数量的增加而线性增加，这表明当前公式的计算复杂度为 与某些传统配方相比，效率更高。此外，当前公式的计算时间几乎随着光束元素数量的增加而线性增加，这表明当前公式的计算复杂度为\（O \ left（N \ right）\）。