Existence and location of internal resonance of two-mode nonlinear conservative oscillators

Abstract

Internal resonances can be widely observed in nonlinear systems; even a simple nonlinear system can exhibit intricate internal resonances when vibrating at large amplitudes. In this study, the existence and locations of internal resonances of a general two-mode system with an arbitrary eigenfrequency ratio are considered. This is achieved by first considering the symmetric case, where the internal resonances are found to be approximately captured by the Mathieu equation. It is shown that the bifurcations can exist in pairs; and, for each pair, the bifurcated solution branches capture modal interactions with the same commensurate frequency relationship but different phase relationships. To determine the existence and locations of internal resonances, the divergence and convergence for correlated bifurcation pairs are then considered. Lastly, the internal resonances in asymmetric cases are analytically derived, where the asymmetry induced bifurcation splitting is captured by a non-homogeneous extended Mathieu equation. This work explores the mechanism underpinning internal resonances, and explains their topological features, such as which internal resonances are observed as amplitude increases. A graphical method is also proposed for efficient determination of the existence and locations of internal resonances.