Diophantine Spectral Control for Nonlinear Internal Resonance Suppression

Abstract

This paper presents Diophantine spectral control to suppress nonlinear internal resonance in multi-modal systems by optimizing small stiffness perturbations to maximize the minimum Diophantine resonance distance. Internal resonance enables efficient energy transfer between modes, resulting in amplified vibrations and complex dynamic responses. Conventional suppression techniques, such as damping enhancement, feedback stabilization, and auxiliary absorbers, typically require significant energy input or substantially alter the inherent system response. Applied to a forced two-mode oscillator near 2:1 internal resonance, Diophantine spectral control increases the resonance distance from 0.20 to 0.8711 with frequency shifts below 3.2%. Steady-state analysis of the nominal case shows approximately 69% reduction in primary-mode RMS amplitude, over 90% reduction in the secondary mode, 71% decrease in the energy transfer index, and 73% attenuation of peak spectral amplitude at the forcing frequency, while preserving the dominant response frequency. Comprehensive robustness analysis across variations in frequency ratio, forcing amplitude, nonlinear coefficients, damping ratios, and optimal-gain perturbations confirms consistent suppression exceeding 65% in the primary mode and 88% in the secondary mode. These results from numerical simulations demonstrate that the proposed method achieves effective resonance avoidance with minimal control effort.