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**Steady-state response of axially moving viscoelastic beams with pulsating speed: comparison of two nonlinear models.**
*(English)*
Zbl 1093.74526

Summary: Principal parametric resonance in transverse vibration is investigated for viscoelastic beams moving with axial pulsating speed. A nonlinear partial differential equation governing the transverse vibration is derived from the dynamical, constitutive, and geometrical relations. Under certain assumption, the partial differential equation reduces to an integro-partial differential equation for transverse vibration of axially accelerating viscoelastic nonlinear beams. The method of multiple scales is applied to two equations to calculate the steady-state response. Closed-form solutions for the vibration amplitude are derived from the solvability condition by eliminating secular terms. The stability of straight equilibrium and nontrivial steady-state response are analyzed by use of the Lyapunov linearized stability theory. Numerical examples are presented to highlight the effects of speed pulsation, viscoelasticity and nonlinearity. Finally, we compare the results obtained from two equations.

### MSC:

74H45 | Vibrations in dynamical problems in solid mechanics |

74K10 | Rods (beams, columns, shafts, arches, rings, etc.) |

74D05 | Linear constitutive equations for materials with memory |

74H10 | Analytic approximation of solutions (perturbation methods, asymptotic methods, series, etc.) of dynamical problems in solid mechanics |