Nonlinear combination parametric resonance of axially accelerating viscoelastic strings constituted by the standard linear solid model

Nonlinear combination parametric resonance is investigated for an axially accelerating viscoelastic string.The governing equation of in-planar motion of the string is established by introducing a coordinate transform in the Eulerian equation of a string with moving boundaries.The string under investigation is constituted by the standard linear solid model in which the material,not partial,time derivative was used.The governing equation leads to the Mote model for transverse vibration by omitting the longitudinal component and higher order terms.The Kirchhoff model is derived from the Mote model by replacing the tension with the averaged tension over the string.The two models are respectively analyzed via the method of multiple scales for principal parametric resonance.The amplitudes and the existence conditions of steady-state response and its stability can be numerically determined.Numerical calculations demonstrate the effects of the string material parameters,the initial tension,and the axial speed fluctuation amplitude.The outcomes of the two models are qualitatively and quantitatively compared.