Coupled vortex-induced modeling for spatially large-curved beam with elastic support

Abstract

The existing studies on mathematical modeling for the forced vibration response of naturally curved beams under elastic boundary conditions mainly focus on slightly curved structures. This work proposes the mathematical model for naturally curved beam with a large curvature under the elastic boundary. Under the assumption of small deformations, the six degrees of freedom are reduced to four degrees of freedom through spatial bending beam theory and differential geometry. The dynamics governing equations of the naturally curved beam are derived using Hamilton’s principle and Frenet–Serret formula. An improved Van der Pol model is introduced to investigate the vortex induced vibration (VIV) or vortex induced divergence (VID) response. Hammer test, wind tunnel experiment and finite element method validate the proposed model for spatially curved helical beam. It is found that under the fixed boundary condition, transposition occurs in the first two coupled modes in the normal direction of the helical beam. The errors of the first two nature frequencies are 0.016% and 0.088%, respectively. The modal analysis under the elastic boundary shows that the first-order decoupled modes in each direction will “disappear” with the variation of stiffness. Under the boundary condition of small stiffness, the first two modes of the normal direction change abruptly at the subtended angle of 76∘∼90∘. This abrupt change produces negative damping and leads to structural unstability. When the first two frequencies are close to each other, the water velocity region of VIV or VID response will intersect and the orthogonal modes in normal direction should be calculated coupled. In this case, the amplitude of the first order mode in the normal direction will always be greater than that of the second order mode. The maximum amplitude in u direction can reach 0.014 m. The obtained results provide useful information for further research on the fluid-induced vibration response of natural large-curved beam .