In power systems with high penetration of distributed generations (DGs), multi-modal broadband harmonic oscillations pose a serious threat to secure and stable operation. Existing active and passive damping methods are limited in their capability to effectively suppress resonance peaks. Considering background harmonics and system impedance characteristics, this paper proposes a novel direct-mounted damping device to suppress broadband harmonic oscillations in the distribution network. In accordance with IEC specified electromagnetic compatibility standards for low-voltage grids, a unified quantitative indicator for broadband oscillation analysis is established. Additionally, a general procedure for damping device parameter design is developed based on modal analysis and the particle swarm optimization (PSO) algorithm. Simulation results show that the proposed topology achieves superior broadband oscillation suppression. Experiments on a multi-inverter grid-connected system further demonstrate the generality and effectiveness of the proposed method in suppressing multi-modal broadband harmonic oscillations within high-DG-penetration distribution networks.