Nonlinear dynamics of traveling beam with longitudinally varying axial tension and variable velocity under parametric and internal resonances

Abstract

In this investigation, the nonlinear dynamics of an axially accelerating viscoelastic beam on a pully mounting system have been analyzed. The axial tension of the beam is modeled as a function of the traveling velocity, support stiffness parameter as well as spatial coordinate. Geometric cubic nonlinearity appeared in the equation of motion due to elongation set in the neutral axis of the beam. The integropartial differential equation of motion of the axially accelerating beam associated with the simplysupported end conditions is solved analytically by adopting the direct perturbation method of multiple time scales (MMS). As a result, a set of complex variable modulation equations are generated, which governs the modulation of amplitude and phase. This set of modulated equations is numerically solved to explore the influence of the support stiffness parameter upon the stability and bifurcation of the beam which has not been addressed in the existing literature. Apart from this, the impact of fluctuating velocity component, viscoelastic coefficient, longitudinal stiffness parameter, internal and parametric detuning parameters on the stability and bifurcation analysis is studied, revealing significant dynamic characteristics of the traveling system. The Fourth-order Runge-Kutta method is applied is to find the dynamic solution of the system. The system displays stable periodic, quasi-periodic, and mixed-mode dynamic responses along with the unstable chaotic behavior for a specific set of system parameters. The results obtained through an analytical-numerical approach may help the design and operation of an axially moving beam.