Multiple Scales Analysis of the Nonlinear Dynamics of Coupled Acoustic Modes in a Quasi 1-D Duct

Abstract We present 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(𝜖2) and 𝑂(𝜖3). In addition to the MMS solutions, those obtained using the Krylov-Bogoliubov method of averaging (KBMA) are presented. The MMS and KBMA solutions are in good agreement with each other and with the numerical solutions to the governing equations. Two representative internal resonance cases that arise in a two-mode system are considered, i.e., 𝜔2 ≈ 2𝜔1 and 𝜔2 ≈ 3𝜔1, where 𝜔1 and 𝜔2 are the linear natural frequencies of the modes. For the 𝜔2 ≈ 2𝜔1 case, both the numerical and KBMA solutions contain low- frequency oscillations in the outer envelope of the limit-cycle like oscillations, but the method of multiple scales does not capture these oscillations. It was observed 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.