Multi-frequency harmonic balance method for nonlinear vibration of pipe conveying fluid under arbitrary dual-frequency excitation

The studies on nonlinear forced vibrations of pipes conveying fluid typically focus on single-frequency excitation, although most pipes are subject to multi-frequency excitation. Few studies on nonlinear vibration responses under multi-frequency excitation consider the relationship between frequencies as integer multiples. In this paper, a new approach called the multi-frequency harmonic balance method (MFHBM) is proposed to address nonlinear vibration responses under multiple excitations of arbitrary frequencies. A dynamic model of an engineering pipe, excited at one end by two arbitrary frequencies under elastic constraints, is established. The two excitation frequencies are represented by discrete frequency values, and the solution to the equation is expanded into a form of harmonic superposition. Furthermore, based on the characteristics of trigonometric functions, approximate solutions in harmonic superposition forms are derived. The correctness and high computational accuracy of the proposed method are validated through comparisons with numerical results. Furthermore, by comparing the multi-scale method (MSM) with the superimposed responses of two single-frequency excitations, the necessity of the proposed method for addressing nonlinear vibration responses to multiple arbitrary-frequency excitations is demonstrated. The findings indicate that when the two excitation frequencies approach the resonance frequency, the dual-frequency excitation responses no longer satisfy the superposition of the single-frequency excitation responses.