A New Incremental Harmonic Balance Method With Two Time Scales for Quasi-Periodic Motions of an Axially Moving Beam With Internal Resonance Under Single-Tone External Excitation

Quasi-periodic motion is an oscillation of a dynamic system characterized by $m$ frequencies that are incommensurable with one another. In this work, a new incremental harmonic balance (IHB) method with only two time scales, where one is one of the $m$ frequencies, referred to as a fundamental frequency, and the other is an interval distance of two adjacent frequencies, is proposed for quasi-periodic motions of an axially moving beam with three-to-one internal resonance under single-tone external excitation. It is found that the interval frequency of every two adjacent frequencies, located around the fundamental frequency or one of its integer multiples, is fixed due to nonlinear coupling among resonant vibration modes. Consequently, only two time scales are used in the IHB method to obtain all incommensurable frequencies of quasi-periodic motions of the axially moving beam. The present IHB method can accurately trace from periodic responses of the beam to its quasi-periodic motions. For periodic responses of the axially moving beam, the single fundamental frequency is used in the IHB method to obtain solutions. For quasi-periodic motions of the beam, the present IHB method with two time scales is used, along with an amplitude increment approach that includes a large number of harmonics, to determine all the frequency components. All the frequency components and their corresponding amplitudes, obtained from the present IHB method, are in excellent agreement with those from numerical integration using the fourth-order Runge–Kutta method.