Nonlinear Dynamics of Traveling Continua with Low Flexural Stiffness under Parametric and Internal Resonances

This study focuses on the trivial state stability boundary, steady state periodic response and chaotic behavior in the transverse motion of an axially moving viscoelastic continuum subject to parametric excitation and internal resonance. This parametric excitation comes from harmonic fluctuations of the traveling speed. The motion is restricted by viscous damping as well as material damping. The derived nonlinear integro-partial-differential equation of motion is solved through analytic-numerical approach. Direct method of multiple scales is used to solve the nonlinear equation of motion. A continuation algorithm is used to find the steady state response. Furthermore, Runge-Kutta method is applied to find the dynamic behavior of the system. The stable periodic response and the unstable chaotic motions are identified using different tools including the time traces, Phase portraits, Poincare map, fast Fourier transforms. Evolution of maximum Lyapunov exponent is used to locate the range of system parameters for which chaotic behavior exists.