Abstract
This study presents the nonlinear vibration and chaotic response of a beam subjected to harmonic excitation. The multi-level residue harmonic balance method is applied to solve the geometrically cubic nonlinear vibration of the simply supported beam. The obtained results agree well with those of the numerical integration method. The amplitude frequency response curves are presented to illustrate the nonlinear dynamic system response both for a damping and without damping model. Also, the chaotic response is examined for a simply supported beam with a nonlinear dynamic system.
Highlights
Dynamic responses such as vibration behavior are an important part of structural analysis
Extensive research has been carried out, for example, on geometrically nonlinear beams,[1] nonlinear vibration of a curved beam with quadratic and cubic nonlinearities,[2] large deflection of a supported beam due to pure bending moment,[3] nonlinear dynamics of an axially moving viscoelastic beam,[4] nonlinear dynamics of a buckled beam subjected to primary resonance,[5] theoretical study of and experiments on a buckle beam,[6] and nonlinear vibrations and stability of an axially moving Timoshenko beam.[7]
As mentioned above, the nonlinear vibrations of beams are formulated by nonlinear partial differential equation in space and time with the different boundary conditions
Summary
Dynamic responses such as vibration behavior are an important part of structural analysis. Extensive research has been carried out, for example, on geometrically nonlinear beams,[1] nonlinear vibration of a curved beam with quadratic and cubic nonlinearities,[2] large deflection of a supported beam due to pure bending moment,[3] nonlinear dynamics of an axially moving viscoelastic beam,[4] nonlinear dynamics of a buckled beam subjected to primary resonance,[5] theoretical study of and experiments on a buckle beam,[6] and nonlinear vibrations and stability of an axially moving Timoshenko beam.[7] Nonlinear vibration and stability of duffing oscillators have been enormously investigated.[8,9,10] Lin et al.[11] studied the nonlinear dynamic of a cantilever beam excited by periodic force. The partial differential equations are discretized into ordinary differential equations as a mathematical model containing cubic or quadratic and cubic nonlinearity. Analytical solutions of such kind of nonlinear forced vibrations are highly complicated. Owing to the presence of nonlinearity in the mathematical models formulated from the governing equation, explicit solutions are rarely obtained
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