Combination, principal parametric and internal resonances of an accelerating beam under two frequency parametric excitation

• Simultaneous principal parametric, combination parametric and internal resonances for a traveling viscoelastic beam subject to two frequency excitation are considered.
• Direct method of multiple scales is applied to solve the fourth order integro-partial differential equation.
• The system shows multiple equilibrium solutions curves with various bifurcations like sub-critical and super critical pitch fork, Hopf and saddle node bifurcations.
• In dynamic solution, system displays stable periodic, quasiperiodic, mixed mode and unstable chaotic behavior with variation of various system parameters.

This study analyses the nonlinear transverse vibration of an axially moving beam subject to two frequency excitation. Focus has been made on simultaneous resonant cases i.e. principal parametric resonance of first mode and combination parametric resonance of additive type involving first two modes in presence of internal resonance. By adopting the direct method of multiple scales, the governing nonlinear integro-partial differential equation for transverse motion is reduced to a set of nonlinear first order ordinary partial differential equations which are solved either by means of continuation algorithm or via direct time integration. Specifically, the frequency response plots and amplitude curves, their stability and bifurcation are obtained using continuation algorithm. Numerical results reveal the rich and interesting nonlinear phenomena that have not been presented in the existent literature on the nonlinear dynamics of axially moving systems.