Multi-scale analysis on nonlinear gyroscopic systems with multi-degree-of-freedoms

A method of multiple scales is developed for n-degree-of-freedom weakly nonlinear gyroscopic systems. A general procedure is proposed to establish solvability conditions. The conditions have n different versions whose equivalence cannot be mathematically demonstrated. The procedure is applied to a 4-degree-of-freedom nonlinear gyroscopic system that is the 4-term Galerkin truncation of the governing equation of a pipe conveying fluid flowing in the supercritical speed. The investigation focuses on the primary external resonance in the first frequency ω1 and the two-to-one internal resonance of the first two frequencies ω1 and ω2. The multi-scale analysis shows that the amplitude–frequency response curve in each of the first two modes has a peak bending to the left when ω2>2ω1, two peaks with the same height and the opposite bending directions when ω2=2ω1, and a peak bending to the right when ω2