Abstract: Firstly, based on the relationship between the Average Lagrangian Function(A.L.F) and amplitude phase modulations, the system average equations were derived in another way which is different from the classical average method. And then the amplitude frequency equation was derived from the system amplitude phase modulations, the Newton numerical iteration was used to resolve the amplitude frequency equation, and the complicated nonlinear interaction between the gruops of models with closely spaced natural frequencies connected with 2∶1 internal resonance was discussed. Secondly, by investigating the relationship between 1∶1 internal resonance and 2∶1 internal resonance, the basic rule of the nonlinear interaction and its exhibiting as a whole in the groups of models with closely spaced natural frequencies was put forward. The conclusions reached here are very important to the active vibration control and modal analysis to the large flexible aeronautics and astronautics structures with closely spaced natural frequencies.

Key words: nonlinear vibr ation, 1∶ 1 internal r esonance, 2∶1 int ernal r esonance, gr oups of models withclosely spaced fr equencies