Parametrically excited axially accelerating viscoelastic beam subjected to time-varying axial speed, longitudinally varying axial tension and internal resonance

Abstract

The present work investigates the nonlinear characteristics of an axially moving viscoelastic beam of the Kelvin–Voigt model on a pulley mounting system. The equation of motion is developed by including the terms like external and internal damping, geometric cubic nonlinearity, Coriolis, time-dependent axial speed, and longitudinally varying axial tension. The direct perturbation method of multiple time scales (MMS) is adopted to get an approximate solution for the nonlinear integro-partial differential equation of motion which leads to a set of complex variable modulation equations. These modulated equations are solved numerically to study the trivial state stability plot considering the effect of the support stiffness parameter for the first time. In addition to this, the impact of material, and viscous damping parameters on stability boundary is also studied. Continuation algorithm is implemented to analyse the influence of the support stiffness parameter, internal and parametric detuning parameters, fluctuating velocity component, and longitudinal stiffness parameter on the stability and bifurcation of the steady state solutions. Variation in internal frequency detuning parameter and variable axial speed component reveals different kinds of equilibrium solution curves in the nonlinear travelling system. The stability and bifurcation features observed in the present nonlinear system with change in the above control parameters are interesting and also not available in the existing literature.