Nonlinear dynamics of a viscoelastic traveling beam with time-dependent axial velocity and variable axial tension

In this paper, stability, various bifurcations and chaotic behavior of a traveling Euler beam are studied. Kelvin-Voigt viscoelastic model is adopted for the beam material. The beam has time dependency in velocity as well as tension in axial direction which is considered for the first time for traveling viscoelastic beam. The multi-frequency parametric resonance is assumed to be comprised of simultaneous principal parametric resonance of first mode due to the time dependency of speed, while the principal parametric resonance of second mode is due to the variable tension in the axial direction. The two-frequency parametric resonance is considered along with 3:1 internal resonance. Such multi-resonant case has not been considered so far in any available literature. The higher-order integro-partial differential equation of motion is attacked with direct method of multiple scales. Solvability condition is incorporated to get two complex variable modulation equations for amplitude and phase. From these two equations, a set of normalized reduced equations are derived through Cartesian transformation which are subsequently utilized to get numerical solutions for the nonlinear system. Continuation algorithm is used to depict the effect of system parameters such as fluctuating tension component, fluctuating velocity component, internal frequency detuning parameter, different damping on the frequency response curves, their stability and bifurcation analysis. This research work reveals interesting results which are not found so far in the existent literature.