Bifurcation Of Non-semi-simple Zero Eigenvalues At The Critical Point Of The Statical Bifurcation Of Nonlinear Rotor System

The objective of the study is to discuss the eigenvalue bifurcation and instability of the center subspace of a nonlinear rotor system with gyroscopic,inertial and potential forces,and nonlinear forces of the shaft,whose linear approximation has a m-multiple non-semi-simple zero eigenvalues,Re(λ)= 0,Im(λ)= 0.That is to discuss how the parameter changes affect the variations of non-semi-simple zero eigenvalues of the center subspace.The eigenvectors of the non-semi-simple system forms a center subspace.It will occurs statical bifurcation as the control parameter passes through a critical value.We get a group of eigenvectors by using generalized modal theory.In order to analyze the nature of the statical bifurcation we needs to compute the derivative of the eigenvalues respect to the control parameters of the critical point.However,for the complex case with multiple non-semi-simple zero eigenvalues it is difficult to get the derivative of the eigenvalues.According to the analysis,original m-multiple eigenvalues can perturbe into m distinct independent eigenvalues.It is the basis we judge the statical bifurcation as the control parameter passes through the critical value.In order to analyze the eigenvalue bifurcation,the Puiseux expansion is used to develop the expressions of variations of non-semi-simple eigenvalues.The method for computing the generalized modes of the center subspace and expression of variations of m-multiple non-semi-simple zero eigenvalues are given.A rotor model as an application example is given to show the validity of the present method.