Spacecraft micro-vibrations, characterized by multiple excitation sources, broadband frequency, and complex transmission, challenge conventional suppression methods and threaten mission reliability. To address the trade-off between lightweight design and vibration mitigation in passive damping structures, this paper proposes a topology optimization design methodology for maximizing the modal damping ratio of a free-layer damping structure. Utilizing the variable density method, the optimization model is established with sensitivity and density filtering techniques to achieve optimal material distribution under multi-modal targets. Case studies on two-end fixed and cantilever plates demonstrate that the optimized layouts enhance target modal damping ratios and reduce vibration transmissibility peaks.