| The extensive applications of linear normal modes analysis and the existingnonlinear factors in vibration systems make it necessary that nonlinear normal modes(NNMs) analysis is developed. NNMs analysis is needed for structural dynamicdesigns and is also one of main approaches for one to obtain approximate solution tononlinear system with multi degrees of freedom. In this paper, we focus on thenonlinear vibration systems with two degrees of freedom, and have done someresearch on analytical solution to its nonlinear vibration by constructing its nonlinearnormal modes. The main work is as follow:1) The differences and connections between linear normal modes and NNMs areprobed by comparing their definitions and basic characteristics. The linear normalmodes vary with certain laws while NNMs have no rules to follow and are alsoinfluenced greatly by the variations of initial conditions. Under free disturbance, theenergy also influences NNMs; the greater the energy, more significantly the NNMschanges; the energy up to a certain extent will cause bifurcation between the modes.2) Using the method of multiple scales, the NNMs are constructed respectivelyfor the non-internal resonance and internal resonance situations of nonlinear systemswith two degrees of freedom. When non-internal resonance is not involved, theapproximate analytical solution and the numerical solution to the first-order NNM iscompared by changing the initial conditions, showing that their anastomosis isinfluenced by initial conditions and that the less the initial conditions, the better theanastomosis is.3) When internal resonance is involved, we compared the approximate analyticalsolutions and the numerical solutions to the1:1,1:2and1:3internal resonancesituations. The results suggest that NNMs are greatly affected by the strength ofnonlinearities, and that the weaker the nonlinearities, the better the anastomosisbetween the analytical solution and the numerical solution is. |
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