Multiple scales analysis of the nonlinear dynamics of coupled acoustic modes in a quasi 1-D duct

We derive novel solutions based on the method of multiple scales (MMS) to the nonlinear equations governing the time-dependent amplitudes of coupled acoustic modes in a quasi one-dimensional duct with non-uniform cross-section and axially inhomogeneous mean velocity, temperature and pressure. The modal amplitude equations constitute a multi-degree-of-freedom system of linearly and nonlinearly coupled ordinary differential equations with quadratic and cubic nonlinearities. Due to the presence of both quadratic and cubic terms, the perturbation expansion for the MMS solution necessary includes terms of three orders: O ( ϵ ) , O ( ϵ 2 ) and O ( ϵ 3 ) , where ϵ is the small parameter. In a recent study Swarnalatha et al. (J Sound Vib 553, 2023), we developed a novel modification of the Krylov–Bogoliubov method of averaging (KBMA) to analytically solve the modal-amplitude equations. The KBMA solutions are included in the present study to compare the MMS and KBMA approaches for the nonlinear equations with and without linear coupling. In general, the MMS and KBMA solutions are in good agreement with each other and with the numerical solutions to the modal amplitude equations. Two representative internal resonance cases that arise in a two-mode system are considered, i.e., ω 2 ≈ 2 ω 1 and ω 2 ≈ 3 ω 1 , where ω 1 and ω 2 are the linear natural frequencies of the first and second modes. For the ω 2 ≈ 2 ω 1 case, both the numerical and KBMA solutions contain low-frequency oscillations in the outer envelope of the limit-cycle oscillations, but the method of multiple scales does not capture these oscillations. It is seen that when the amplitude equations are linearly uncoupled, the low-frequency oscillations in the outer envelope disappear. The criteria for the stability of the limit cycles are analyzed using the MMS and the KBMA solutions and the stability boundaries illustrated.