The Steady-State Response Of Forced Vibration For A Buckling Beam Under Axial Press

The beam exists widely in machinery, construction and other projects as a kindof slim bar. When the pressure exceeds a critical value, the axial buckling will occuron the beam. Nowadays the applications of the buckled beam have an increasingproportion in engineer. The study of the vibration response of the buckled beam is ofgreat significance.First, the governing equation of a simply-supported viscoelastic buckled beamto a harmonic base excitation and a couple of constant axial force is established.Based on Euler-Bernoulli beam theory, the partial differential governing equations ofthe buckled beam is got by infinitesimal method. In the paper, the disturbanceequation of transverse vibration of the buckled beam is derived from the freegoverning equation via a coordinate transform. The Galerkin method is applied totruncate the systems to a multi degree-of-freedom of nonlinear vibration system.Second, a nonlinear analysis of the response of a simply-supported viscoelasticbuckled beam to a couple of constant axial force and harmonic excitation ispresented. In addition, the harmonic excitation has two types: weak excitation andstrong excitation. The method of multiple scales is developed to present thesolvability condition of approximate solutions. In the presence of the2:1internalresonance and the primary external resonance, various jumping phenomena arerevealed in the amplitude-frequency characteristic curves, and the effects of relatedparameters, such as the coefficient of viscoelasticity, the external excitationamplitude and the axial force, on the phenomena are examined. At the same time, theexistence of the phenomenon of saturation is proved.Third, the effect of the application of the method of multiple scales in different orders to the amplitude-frequency characteristic curves is examined. Base on theexist of the cubic nonlinearities in the a multi degree-of-freedom of nonlinearvibration system, the two-order multi-scale and the three-order multi-scale aredeveloped to present the solvability condition of approximate solutions. In thepresence of the1:1internal resonance and the weak harmonic excitation, differentphenomena are revealed in the amplitude-frequency characteristic curves.Finally, the results of the thesis are summarized and the further work issuggested.