Theoretical investigation of the simultaneous resonance of a cylindrical bubble under triple-frequency acoustic excitation

In the present paper, the simultaneous resonance of a cylindrical bubble under triple-frequency acoustic excitation is investigated theoretically. Specifically, based on the multi-scale method, the dimensionless oscillation equations and the second-order analytical solutions of the primary-subharmonic-subharmonic (PRI-SUB-SUB), primary-superharmonic-superharmonic (PRI-SUPER-SUPER) simultaneous resonances are obtained. Based on the analysis of the frequency response, the nonlinear dynamic behavior of the cylindrical bubble and its influencing factors are analyzed. The primary conclusions include: (1) Under triple-frequency acoustic excitation, the frequency response of PRI-SUB-SUB presents a single peak, and that of PRI-SUPER-SUPER presents two peaks. (2) The polytropic exponent affects both the peak value and position of the resonance peak in the frequency response. (3) The unstable region in frequency response curve of the simultaneous resonance is significantly affected by the total amplitude and equilibrium radius, presenting a positive correlation.