Multi-scale analysis of a moving beam under parametric and auto-parametric resonances

The present investigation analyses the impact of auto-parametric resonance on the dynamic behaviour of a viscoelastic beam. The perturbation technique-direct method of multiples scales (MMS) is applied to convert the higher-order nonlinear equation of motion to a set of first-order partial differential equations. In developing an appropriate model for a traveling system, attention is focused to include different damping terms, geometric nonlinearity, Coriolis term and time dependent speed in the dynamic equation. The system is investigated in the sub-critical speed ranges with an importance on the stability and bifurcation evaluation for steady-state solutions under parametric excitations. The system behaves sensitively by displaying variations in stability and bifurcation features as a impact of variation in auto-parametric frequency, fluctuating velocity, and axial stiffness. This is the prime outcome of this investigation. Apart from this, the present work is also compared with the previous study and huge dissimilarities are highlighted. The investigation reveals different interesting outcomes, which may help for future references for the analysis and design of the traveling continua.