Most of the studies on multimode vibration reduction of floating raft systems remain at the stage of theoretical analysis, while relevant experimental studies are rare. In this paper, an experimental setup for a scaled-down floating raft system is constructed. The vertical track nonlinear energy sink (TNES) cells are experimentally applied for the first time to suppress the multimode vibration of the floating raft system. To ensure the damping effect is not affected by additional mass, the TNES cells are always installed on the floating raft system. The TNES is able to achieve vibration suppression by allowing unrestricted movement of its vibrator. Two damping systems with different TNESs placement are proposed. Experimental modal analysis is performed on the floating raft structure to determine its response characteristics. Based on the results, different magnitudes of excitation force are applied to each order mode to effectively suppress the multimode vibration of the system. Meanwhile, the nonlinear system coupled with TNES cells is analyzed using the harmonic balance method (HBM) to obtain the approximate analytical solution. The system responses obtained from the analytical solution are compared with the experimental data results. The particle swarm optimization (PSO) algorithm is used to obtain the optimal parameters of the TNESs for vibration suppression. The results show that the multimodal vibration of the floating raft structure is effectively suppressed by the TNES cells in both experimental and theoretical analyses. The vibration suppression effect for all modes is significantly improved by the PSO algorithm. In conclusion, the research content of this paper provides a practical and feasible vibration suppression solution for the design of floating raft systems.