Nonlinear Free Vibration of a Beam Based on Nonlinear Modal Analysis and Harmonic Balance Methods

Nonlinear free vibration of an Euler-Bernoulli beam withlarge deflection is presented. It is convenient to assumethat the nonlinear beam vibrates in its linear mode; hence,the mode shape of the linearized system is used to spatiallydiscretize the governing dynamic equation. Consideringonly one degree of freedom, the equation takes the formof Duffing equation and can be analyzed using methodssuch as harmonic balance. However, this method is notaccurate especially at higher energy level that the modeshapes may be considerably different from linear modes.Nonlinear normal modes (NNMs) analysis has beenproposed to extend the concept of normal modes innonlinear oscillators. The continuous beam is discretizedusing Rayleigh-Ritz method which allows modelingvarious boundary conditions. Nonlinear free vibrationbehavior of a simply-supported and cantilever beam isexamined using harmonic balance method with singleterm and NNMs analysis considering two or more termsin Rayleigh-Ritz method. A comparison between theresults of the two cases illustrates that they are in goodagreement at low energy level. The results obtained byNNMs calculation are smaller than the results of harmonicbalance method with only one term. It is shown that thenumerical method implemented for nonlinear modalanalysis can predict the nonlinear mode shape and modalinteraction contrary to harmonic balance method withsingle term which is capable to calculate the frequenciesonly at low energy level. Nonlinear modal analysis for thesimply-supported beam shows that the nonlinearityflattens the modal peaks.

کلیدواژه ها:

Nonlinear normal modes ، Harmonic balancemethod ، Euler-Bernoulli beam ، Rayleigh-Ritz method