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<2ω₁. In all those cases, the 4 different versions of the solvability conditions yield same outcomes. The responses in the last two modes uninvolved in the resonances decay to zero exponentially. The numerical integration results are qualitative agreement with the analytical ones.