Optimal configuration for vibration frequencies in a ring of harmonic oscillators: The nonidentical mass effect

The parameter diversity effect in coupled nonidentical elements has attracted persistent interest in nonlinear dynamics. Of fundamental importance is the so-called optimal configuration problem for how the spatial position of elements with different parameters precisely determines the dynamics of the whole system. In this work, we study the optimal configuration problem for the vibration spectra in the classical mass–spring model with a ring configuration, paying particular attention to how the configuration of different masses affects the second smallest vibration frequency (ω₂) and the largest one (ω_N). For the extreme values of ω₂ and ω_N, namely, (ω₂)min, (ω₂)max, (ω_N)min, and (ω_N)max, we find some explicit organization rules for the optimal configurations and some approximation rules when the explicit organization rules are not available. The different distributions of ω₂ and ω_Nare compared. These findings are interesting and valuable for uncovering the underlying mechanism of the parameter diversity effect in more general cases.