Analytical solution to a coupled system including tuned liquid damper and single degree of freedom under free vibration with modal decomposition method

This research focuses on employing a linear analytical approach to transform free surface waves and velocities into mode coordinates, with the aim of investigating the free vibration behavior of a coupled system consisting of a Single Degree of Freedom and a sloshing tank. Through a series of manipulations and simplifications of the coupled equations, a fourth-order ordinary differential equation is derived to showcase the overall response of the system, highlighting the contribution of each odd mode. Key concepts explored include system stability, mode-specific natural periods, establishment of initial boundary conditions, and formulation of the complete system response. The analytical method applied to study Tuned Liquid Dampers, a type of elevated sloshing tank, reveals that in higher modes, the lower frequency aligns with the structural natural frequency, while the higher frequency is approximately n times the structural natural frequency (where n is the odd mode number). This approach also elucidates why the system's response does not exhibit a higher-frequency component in higher modes. The study further investigates concepts such as employing an initial perturbation to excite higher frequencies and the potential for approximating the system through the first mode. Additionally, a numerical model was developed using variable separation and modal decomposition methods to complement and validate the analytical approach. Finally, further verification of the model was performed using the Preismann scheme applied to the relevant equations and the central upwind applied to nonlinear equations.