Nonlinear Dynamics and Bifurcations of an Axially Moving Beam

Abstract

The present paper analyzes the dynamic behavior of a simply supported beam subjected to an axial transport of mass. The Galerkin method is used to discretize the problem; a high dimensional system of ordinary differential equations with linear gyroscopic part and cubic nonlinearities is obtained. The system is studied in the sub and super-critical speed ranges with emphasis on the stability and the global dynamics that exhibits special features after the first bifurcation. A sample case of a physical beam is developed and numerical results are presented concerning the convergence of the series expansion, linear subcritical behavior, bifurcation analysis and stability, and direct simulation of global postcritical dynamics. A homoclinic orbit is found in a high dimensional phase space and its stability and collapse are studied.

On the transversal vibrations of a conveyor belt with a low and time-varying velocity. Part I: the string-like case

“…Knowledge about linear models is important in order to understand results found in non-linear models, especially for those cases which are weakly non-linear. For non-linear models describing the dynamic behavior of belts, we refer the readers to [4], [6], and [7]. In [7] the role played by the external frequency of the nonconstant belt velocity and the bending stiffness is studied.…”

“…Investigating transverse vibrations of a belt system is a challenging subject which has been studied for many years (see [1][2][3][4] for an overview) and is still of interest today.…”

Dynamic analysis of an axially translating viscoelastic beam with an arbitrarily varying length

“…Pelicano and Vestroni [26] and Pelicano et al [27] also investigated bifurcations and parametric resonances of a moving beam; several different viscoelastic models were proposed for different applications, and the key results were verified by experimental measurements. A spectral element model was introduced by Lee and Oh [28] to study the dynamics and stability of an axially moving viscoelastic beam subject to axial tension.…”

On the general theory of chaotic dynamics of flexible curvilinear Euler–Bernoulli beams

“…Pitchfork bifurcations, the stability of equilibria, as well as post-bifurcation velocity range were detected, illustrated and discussed. This study was extended by Pellicano and Vestroni [5], where the rub and super-critical speed intervals and their influence on the stability, bifurcation and global nonlinear dynamics of the Euler-Bernoulli axially moving beam were extensively analyzed. Exact solutions in closed form of forced vibrations of prismatic damped Euler-Bernoulli beams as well as Green functions for the beams with different homogeneous and elastic boundary conditions were reported by Abu-Hilal [6].…”