Nonlinear Multimodal Model for Tuned Sloshing Dampers With Nonflat Bottoms

Abstract

A third-order nonlinear multimodal model is developed for tanks with non-flat bottoms. A 6-node finite element model is used to determine the mode shapes of the velocity potential and the associated natural sloshing frequencies. Using this modal information, equations of motion are developed that accommodate the nonlinear coupling among the first three sloshing modes. Damping arising from screens is incorporated into the model using the principle of virtual work. The equations of motion are ordinary differential equations that are solved using the Runge-Kutta-Gill numerical time-stepping method. The model is evaluated with an existing third-order nonlinear multimodal model for a flat-bottom tank and is found to be in excellent agreement. Demonstrative simulations are conducted for tanks with sloped-, boxed-, and ramped-bottoms. The resulting sloshing forces and wave heights at the tank end wall are calculated and presented using time series plots and frequency response plots. The excitation of higher order sloshing modes through modal coupling results in larger wave heights, and shallower wave troughs. The sloshing forces are less impacted by the responses of higher modes. Secondary resonances are clearly visible in several frequency response plots at frequencies that correspond to the natural sloshing frequency of a higher order mode divided by an integer. The model is applicable to tanks that are of intermediate water depth with moderate excitation amplitudes, where the response of the second and third order sloshing modes are less than the fundamental mode.