Periodic response to fluid loading of single-degree of freedom systems with stiffness nonlinearity

Abstract

The paper considers single-degree of freedom systems with a nonlinear restoring force and subjected to periodic fluid loading due to waves and currents. The desired number of displacement harmonics are found in the frequency domain using two alternative iterative schemes both of which are based on the Newton–Raphson method and involve an explicit description of both the nonlinear restoring force and the relative velocity-squared drag loading in the time domain. The first option, called the multi-diagonal Jacobian method (MDJM), uses essentially the full Jacobian, whereas the second option, the single-diagonal Jacobian method (SDJM), uses only the main diagonal of the Jacobian. It is shown that the SDJM involves adding an artificial stiffness and an artificial damping to the system. The methods have been implemented for the particular case of cubic nonlinearity, which corresponds to the Duffing equation with linear viscous damping, and a nonlinear fluid loading which may have a non-zero mean. The results have been validated primarily by noting that the residual load at convergence is negligibly small and, for a selected number of cases, also by comparison with the results of the Runge–Kutta–Nystrom time integration. For the system parameters considered here, both the schemes showed good convergence properties. However, as the SDJM needs much less computation per iteration than the MDJM, it works out to be the method of choice.