Eigencharacteristics and nonlinear response of partially submerged flexible L-shaped beam with eccentric tip mass

Abstract

In this work, free and forced vibration analysis of a partially submerged flexible L-shaped offshore structure carrying concentrated eccentric tip mass is presented. The column is modelled as interconnected Euler–Bernoulli beam elements having fixed condition at proximal end and carrying a concentrated mass at the terminal end. The governing equations of motion describing system’s behaviour in transverse and longitudinal direction are  derived using multi-body system approach. The variable separable method is used to obtain the eigenfrequency equation of fluid–structure system, and exact solutions are graphically presented. The influence of depth of immersion, added tip mass and its eccentricity on the eigenfrequency of the column, is reported. The results obtained are verified with the existing literature as well as FEA simulations for the limiting cases of depth of immersion and varying tip mass. Further, the mode shape functions and eigenfrequency parameters are used in conjunction with Galerkin’s method to obtain nonlinear model of the system under bi-directional time-dependent base excitations. The large deformation of the beams is under consideration, and inextensibility condition is used to incorporate the longitudinal deformation effects in transverse direction. The steady-state solutions of the system under combined external and inherent 1:1 internal resonance conditions are obtained using perturbation method. The existence of multiple jump phenomena and multi-valued solutions are explored through frequency response curves. The system exhibits combination of saddle-node and pitchfork bifurcations leading to instability in the system responses due to sudden change in amplitudes at critical points. The obtained results are compared with the numerical solutions to achieve the close agreement. The system is found to undergo large-amplitude vibrations at critical values axial and transverse base excitation frequencies. The fluid interaction with the structure has a stabilizing influence on the frequency response, while increase in the tip mass leads to increase in vibration amplitude of the system. The first and second beam exhibit frequency response of different features due to the nature of geometric nonlinearities. The effect of essential system parameters on the column’s responses has been analysed graphically, and vulnerability of system’s stability is also discussed graphically.