A new recursive geometrically exact formulation for three-dimensional Euler–Bernoulli beams with large deformations

A new recursive geometrically exact formulation (RGEF) for three-dimensional Euler–Bernoulli beams with large deformations is proposed in this work. The proposed RGEF introduces three major innovations compared to the existing beam formulations. First, each element only has three degrees of freedom, which can significantly reduce the dimension of the equations of motion. Second, the computational complexity is only \({\text{O}}(n)\) $\text{O}(n)$ due to the recursive scheme, meaning the calculation time increases linearly with the number of elements. Third, the inertia force and mass matrix can be explicitly integrated in advance by adopting the velocity approximation approach, which can significantly reduce computational costs. Moreover, the proposed RGEF is singularity-free due to the interpolation of relative rotation vectors. The proposed RGEF can be used for both open-loop and closed-loop multibody systems, and several static and dynamic benchmark examples are presented to demonstrate the effectiveness of the formulation.