Structural modal interactions with internal resonances and external excitation

In (his paper the non-linear interaction of a three-degrees-of-freedom structural model subJected to external excitation is examined. The non-linearity of the system results in different critical regions of internal resonance and this has a significant effect on the response. We use the method of multiple scales to determine a first order uniform asymptotic expansion of the solution. including the modulation equations governing the dynamics of the amplitudes and phases of the interacting modes. Then. we investigate behavior and stability of the equilibrium and dynamic solutions as the forcing amplitude and resonance detunings are slowly varied. The solutions are stable when sigma(3) = 0.01 and the response amplitude alpha(3) has a saddle node bifurcation for minimum value of the left branch for all other variations. The first and second modes reach minimum values at about sigma(3) = 0 while the third mode reaches the maximum value at the same point. Each mode is separated into two branches for decreasing and increasim, mu(2) and beta(4) respectively.