Nonlinear Dynamics Of A Rotating Inextensional Ring With The In-plane And Out-of-plane Motions

In engineering applications,the partial nonlinear dynamics of the toroidal system have a significant effect on the stable operation of the mechanical structure.In this paper,the nonlinear dynamics of the ring system caused by the in-plane and out-of-plane coupled vibration of the inextensional ring are studied.The mathematical model is established by using the Hamiltonian principle.The inextensional characteristics and the rotating gyro effect in the circular plane and the external harmonic excitation are comprehensively considered in the model.Through the numerical analysis of the equation of motion of the ring system,the nonlinear dynamic characteristics and interaction mechanism of the in-plane motion and out-of-plane motion of the inextensional ring are revealed.Firstly,the supported thin ring rotating at constant speed is taken as the research object.Based on the Hamiltonian principle,the potential energy function and kinetic energy function including shear deformation and rotational inertia effect are obtained.The effects of elastic support potential energy and external load potential energy and damping potential energy are considered in the equation of motion of the ring system.And the equation of motion of the ring is established by variational method.Then,the equation of motion of the system is linearized to analyze the natural frequency of the ring.The partial differential equation is transformed into an ordinary differential equation with variable coefficients.And the nth-order modal motion equation of the ring is further derived.Finally,the obtained ordinary differential equations of motion are solved by numerical methods.The global bifurcation diagram,time history diagram,phase diagram and Poincare diagram are given.This paper analyzes and discusses the vibration characteristics of the inextensional ring system,and reveals the nonlinear behavior of the ring,such as period and chaos.This paper also discusses the effects of changes in parameters such as excitation size,support stiffness and damping on system behavior.