Nonlinear parametric vibration of axially moving beams: Asymptotic analysis and differential quadrature verification

Nonlinear parametric vibration is investigated in principal parametric resonance of an axially accelerating viscoelastic beam. The axial speed is characterized as a simple harmonic variation about the constant mean speed. The material time derivative is used in the viscoelastic constitutive relation. The transverse motion can be governed a nonlinear integro-partial-differential equation. The asymptotic analysis is performed to determine steady-state responses. It is proved that the mode uninvolved in the resonance has no effect on the steady-state response. The differential quadrature scheme is developed to verify results via the method of multiple scales. It is demonstrated that the straight equilibrium configuration becomes unstable and a stable steady-state emerges when the axial speed variation frequency is close to 2 times of any linear natural frequency. The numerical calculations validate the analytical results in the principal parametric resonance.