Stability, Bifurcation and Chaos of a Traveling Viscoelastic Beam Tuned to 3:1 Internal Resonance and Subjected to Parametric Excitation

Abstract

Analytical-numerical approach has been adopted to investigate the stability, bifurcation and dynamic behavior (including chaotic behavior) of axially moving viscoelastic beam subjected to parametric excitation resulting from speed variation in the presence of 3:1 internal resonance between the first two modes of vibration. The governing equation of transverse vibration is a nonlinear integro-partial-differential equation with time-dependent coefficients. The direct method of multiple scales is employed to analyze the joint influence of the combination of parametric resonance and internal resonance with the focus on steady state responses. Equilibrium solutions along with their stability and bifurcations are determined by continuation algorithm while direct time integration is used for dynamic behavior for various system parameters. The results are compared with the previous works depicting the principal parametric resonances of the first and second modes. Significant comparative analysis results are reported in the stability and bifurcation of frequency response analysis. The dynamic responses show a range of behavior viz. stable periodic, mixed mode, quasiperiodic and unstable chaotic motion of the system. Numerical results illustrate various typical and interesting nonlinear phenomena of the traveling system which are not found in the existent literature.