Application of Krein’s theorem and bifurcation theory for stability analysis of a bladed rotor

This paper considers the stability and eigenvalue analyses for a bladed rotor which goes under cylindrical and conical whirling. The model consists of a group of flexible blades which are modeled by beams and rigid disk on the elastic bearings. The model is a Hamiltonian system which is perturbed by small dissipative forces. Krein’s theorem reveals that the forward whirling mode and the blade collective motion may cause instability when their frequencies cut themselves in the Campbell diagram. An unstable interaction between the blades and the conical whirling is discovered. The eigenmode and eigenvalue evolutions are determined on the stability boundary. The bifurcation analysis is performed by applying multiple scales method around the stability boundary. It is shown that the damping distribution between the blades and the bearings may shift the unstable mode.     

Stability and Hamiltonian Hopf bifurcation for a nonlinear symmetric bladed rotor

The modal interaction which leads to Hamiltonian Hopf bifurcation is studied for a nonlinear rotating bladed-disk system. The model, which is discussed in the paper, is a Jeffcott rotor carrying a number of planar blades which bend in the plane of the motion. The rigid rotating disk is supported on nonlinear bearings. It is supposed that this dynamical system is a Hamiltonian system which is perturbed by small dissipative and nonlinear forces. Krein’s theorem is employed for obtaining a stability criterion. The nonlinear eigenvalue equations on the stability boundary are turned into ordinary differential equations (ODEs) by differentiating them over the rotating speed. By solving these ODEs, the eigenmodes and the eigenvalues on the stability boundary are obtained. The bifurcation analysis is performed by applying multiple scales method around the boundary. The rotor nonlinear behavior and damping effects are studied for different conditions on the rotating speed and nonlinearity type by the bifurcation equation. It is shown that the damping distribution between the blades and bearings may shift the unstable mode. Depending on the nonlinearity type, subcritical and supercritical Hopf bifurcation are possible.     

Predictive modeling of creep in polymer/layered silicate nanocomposites

A predictive creep model is developed which uses the properties of matrix and reinforcement to predict the creep of polymer/layered silicate nanocomposites. Up to this point, primarily empirical creep models such as Findley and Burgers models have been used for creep of polymer/clay nanocomposites. The proposed creep model is based on the elastic-viscoelastic correspondence principle and a stiffness model of these nanocomposites. Also, the added stiffness of polymeric matrix due to the constraining effect of layered silicates on polymer chains in the nanocomposite is considered by a parameter termed constraint factor. The results of the proposed model show good agreement with experimental creep data for different clay contents, stresses and temperatures. Comparing the model predictions with experimental data, a logical relationship between the method of processing and the constraint factor is discovered which shows that in-situ polymerization can be more efficient for improving creep resistance of polymer/layered silicate nanocomposites relative to melt processing.