Parametric Vibration Of Axially Moving Beams:Speed Dependent Tension And Tension Dependent Speed

Axial motion structure widely exists in daily life and production.A variety of engineering devices,such as elevator rope,saw blade,conveyor belt,tape,power transmission tape and recording tape,can be simplified to the mechanical model of axial motion structure.Large lateral vibrations will occur when they are subjected to external or intrinsic excitation during movement in a particular direction.The study of lateral vibration is of great engineering significance in social production and development.When the perturbed velocity and the perturbation tension are combined,the dynamic characteristics of the axial moving structure will be greatly affected.Therefore,it is very important to study its nonlinear dynamics.At the same time,they can also lay a certain foundation for engineering application and research of more complex models under related conditions.In this paper,the research status of axial motion structure is briefly introduced.Firstly,the influence of the relation between perturbation tension and perturbation axial velocity on the system is analyzed when the parameter vibration of axially moving viscoelastic beam is carried out.Previous studies on parametric vibration of axially moving beams subjected to both time-varying tension and time-varying velocity are rare.And their common feature is that axial tension and axial velocity are independent of each other.In this paper,the Kelvin viscoelastic constitutive relation is considered,the dynamic model of the axial moving viscoelastic beam is established by using the generalized Hamilton principle,and the relationship between the perturbation velocity and the perturbation tension is analyzed,and a new model is established for the study of the parameter vibration of the axial motion system.Secondly,the direct multi-scale method is applied to approximate the system,and the stability boundary conditions of the system are obtained according to the solvability condition and the Routh-Hurwitz criterion.When the non-homogeneous boundary conditions appear in the system,the existing solvability conditions will be invalidated.In this paper,an improved solvability condition,modified coefficient,is used to solve this problem.The dynamic characteristics of the axial beam subjected to 1:3 resonance in parametric vibration are studied in this paper.The effects of viscoelastic coefficient,fluctuating tension and pulsating velocity on the dynamic stability and steady-state response of the system are proved by some numerical examples.Finally,the approximate analytical results are verified numerically by differential quadrature method(DQM),and the consistency between approximate analytical solutions and numerical solutions is intuitively expressed by the graphical solution.