Non-linear multimodal model for tuned liquid dampers of arbitrary tank geometry

Tuned liquid dampers (TLDs) utilize sloshing fluid to absorb and dissipate structural vibrational energy. Simple TLD tank geometries may not always be feasible due to space limitations. While the non-linear modelling of sloshing fluid is currently limited to tanks of simple geometries, this paper develops a non-linear multimodal model which describes the sloshing behaviour of a fluid in a flat-bottom tank of arbitrary geometry. The mode shapes of the sloshing fluid are found by solving the Helmholtz equation over the tank domain using the finite element method. The Bateman–Luke variational principle is used to develop a system of ordinary differential equations which account for the coupling of the sloshing modes through the non-linear free surface boundary conditions. Damping is incorporated into the model by considering the drag produced on a set of damping screens inserted in the fluid. The system of ordinary differential equations is solved using the Runge–Kutta–Gill Method to predict the wave heights and sloshing forces. In general, the mode shapes in an arbitrary tank will have components in two orthogonal (x- and y-) directions. This out-of-plane behaviour is an important consideration for TLD design. The model is validated with existing models for the special cases of rectangular and circular tanks. Lastly, new shake table tests are conducted on a tank of complex geometry.