A versatile strategy to compute nonlinear normal modes of flexible beams

Flexible beams are usually modeled under the assumption of large displacement, finite rotation, but with small strains. Such hypothesis allows the equation of motion to be built using co-rotational finite elements. The co-rotational formulation decomposes the total motion of a structural element into two parts: a rigid body and an elastic (small) deformation. This way, a geometric nonlinearity caused by the large displacements and rotations of the beam's cross sections can be efficiently modeled. The novelty of this paper consists in incorporating this modeling technique inside a standard method to compute nonlinear normal modes (NNMs). The resulting method becomes a dedicated one to the analysis of complex flexible beams, including those with nonuniform cross sections and with pre-deformations. Those cases are not easily incorporated by other methods in the literature. The harmonic balance method (HBM) is used here to approximate the periodic solutions of the system. The arc-length parametrization is used to perform the continuation with respect to the energy level. The alternating frequency-time (AFT) method is used to compute the Fourier coefficients of the nonlinear elastic forces computed from the co-rotational finite elements. Two examples are used to illustrate the performance of the proposed method: bi-clamped flexible beams with nonuniform cross sections and a flexible riser (offshore oil pipes) in catenary configuration.