Analysis And Simulation For An Axially Traveling Viscoelastic Beam Under The Internal Resonance

The transverse nonlinear forced vibration of axially moving viscoelastic beams with a three-to-one internal resonance is analytically and numerically studied in this paper.The material obeys the Kelvin model and the standard linear solid model in which the material derivative is taken part in the viscoelastic constitution relation instead of the simple partial time derivative.The method of multiple scales is first developed to present the governing partial differential equations of motion for the continuous system.The solvability condition of nonlinear forced vibration in the transverse motion is derived under the internal resonance.The steady-state response and stable boundary are determined.The effects of system parameters on the steady-state response are examined.And the effects of the transverse nonlinear vibration on the stress distribution of the axially moving beam are studied for the first time.Approximate analytical outcomes are qualitatively and quantitatively supported by numerical simulations.Under the 3:1 internal resonance condition,the steady-state periodic response of the forced vibration of a traveling viscoelastic beam is studied.The viscoelastic behaviors of the traveling beam are described by the standard linear solid model and the kelvin mode.The direct multi-scales method is used to derive the relationships between the excitation frequency and the response amplitudes.For the first time,the real modal functions are employed to analytically investigate the periodic response of the axially traveling beam.The undetermined coefficient method is used to approximately establish the real modal functions.The approximate analytical results are confirmed by the Galerkin truncation.Numerical examples are presented to highlight the effects of viscoelastic behaviors on the steady-state periodic responses.To illustrate the effect of the internal resonance,the energy transfer between the internal resonance modes and the saturation-like phenomenon in the steady-state responses is presented.The effects of the transverse nonlinear vibration on the stress distribution of the axially moving beam are studied for the first time.The parametric excitation of the moving beam is constituted by the pulsating axial speed.Three-to-one internal resonance condition is satisfied by selecting appropriate geometric parameters of the moving beam.The standard linear solid model is adopted in the viscoelastic constitutive relation.The nonlinear vibration of the axially moving viscoelastic beam with parametric and internal resonances is studied by using the direct multiple scales method with numerical confirmation.The convergent real modal functions are constructed for applying the multiple scales method.The stability of the steady-state solutions is judged by using the Routh-Hurwitz theory.Based on the approximate analytical solution,the distribution of tensile stress and bending stress on the axially moving beam is presented.Furthermore,based on the maximum stable cyclic stress,the limit cycle number of the axially moving beam is utilized to evaluate the fatigue life.The influences of the internal resonance on the steady-state responses and the fatigue life of the axially moving beam are revealed.Numerical examples illustrate that large unwanted resonances occur and the second-order mode receive vibration energy from to the first-order mode.Furthermore,the numerical results demonstrate that the nonlinear vibration significantly reduces the fatigue life of the axially moving beam.