Multi-modal shroud damping vibration reduction design based on the improved whale optimization algorithm and the energy method

Traditional shroud damping designs are effective for vibration reduction and structural performance but lack precision and cannot simultaneously optimize multiple critical modes. To address these issues, this study proposes a novel multi-modal shroud damping design integrating an improved Whale optimization algorithm (IWOA), modal weighting coefficients, and the energy method. The objective is to maximize the weighted damping ratio to determine optimal normal pressure. The IWOA enhances global search through an adaptive dual-leader strategy, circle map, and Cauchy inverse cumulative function. CEC2017 benchmark tests confirm its ability to escape local optima. A tailored modal weighting strategy enables accurate multi-mode optimization. Four quantitative weighting coefficients, namely damping, excitation, resonance margin, and Campbell intersection, are grouped into response amplitude and resonance probability factors, incorporating TRL, MRL, complexity, and accuracy. Validation results show only 0.8 % deviation from traditional methods, while computation time is reduced to 0.5116 s – a 20-fold efficiency gain. In the comprehensive weighting scheme, the average damping ratio increases by 15 % for the first mode and by 31 % for the second mode. These results demonstrate strong engineering value and practical feasibility of the proposed approach.