Parametrically Excited Nonlinear Multi-Degree-of-Freedom Systems.
ARMY MILITARY PERSONNEL CENTER ALEXANDRIA VA
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An analysis of parametrically excited nonlinear multi-degree-of-freedom systems is presented. The nonlinearity considered is cubic and small so that the system of equations is weakly nonlinear. Modal damping is included and the parametric excitation is harmonic. The systems examined include those with distinct natural frequencies as well as those with a single repeated frequency. The significant role played by the existence of an internal resonance is explored in depth. The derivative-expansion version of the method of multiple scales, a perturbation technique, is used to develop solvability conditions for the various combinations of internal and parametric resonances considered. Regions where trivial and nontrivial solutions exist are defined and the stability of the solutions within each region is discussed. Nontrivial, unstable solutions have been shown to exist in regions where nontrivial stable solutions are known. Numerical solutions do not hint at the existence of these solutions. The role of internal resonance in parametrically excited systems is explored. Strong modal interaction is demonstrated as a consequence of the presence of the cubic nonlinearity and the internal resonance. Because of this modal coupling, modes other than the one excited can dominate the response. A multiplicity of jumps is shown to exist.