Hopf Bifurcation At The Critical Point Of Nonlinear Rotors With Viscous Internal Damping

The objective of the study is to discuss the stability and Hopf bifurcation of a nonlinear rotor system with viscous internal damping and nonlinear elasticity.Hysteresis damping model and viscous internal damping are used to illustrate the instability caused by the damping of the rotor shaft material and to get the boundary condition pc of the stability of the rotor system respectively.The eigenvalues of the linear approximation of a multi-degree-of-freedom nonlinear rotor system have conjugate complex roots with zero real parts in pairs.In order to analyze Hopf bifurcation characteristics of the rotor system,this paper discusses the effects of analyzing changes in control parameter p on the eigenvalues and eigenvectors by introducing matrix perturbation of complex modes.It can be seen from the analysis that the corresponding eigenvector forms a two-dimensional center subspace with a pair of unstable modes,and Hopf bifurcation occurs at the critical point pc of the rotor system.Then,we apply the multi-scale method and the central manifold theorem to deal with the feedback control problem of instable modes in the rotor subsystem with viscous damping.Here we use the pole allocation method of modal control to find the gain vector.At the end of the article,we give a numerical example of the rotor system to illustrate the effectiveness of this paper.