A smooth polynomial shaped command for sloshing suppression of a suspended liquid container

A smooth polynomial shaped command with an adjustable command time length is proposed for eliminating the residual vibrations of a multi-mode system. The ability of eliminating jerks and vibrational modes, regardless of their number, offers the most advantage of the proposed command. A numerical simulation is conducted to test the command’s effectiveness by eliminating the residual sloshing oscillations of a liquid-filled container conveyed by an overhead crane in a rest-to-rest manoeuvre. The governing equations of the liquid free-surface level are derived by modelling the sloshing dynamics by a series of mass–spring–damper harmonics. The proposed model accounts for the coupling between the pendulum dynamics and the sloshing equivalent mechanical model. The command’s robustness to the system parameters’ uncertainties, liquid depth and cable length, are investigated as well. The ability of adjusting the command length and retaining higher sloshing modes in command-designing are also outlined.