Emmanuelle Sarrouy

Non-Linear Periodic and Quasi-Periodic Vibrations in Mechanical Systems - On the use of the Harmonic Balance Methods

In this chapter, the general formulation and extensions of the harmonic balance method will be presented. The chapter is divided into four parts. Firstly we propose to present the general formulation and the basic concept of the harmonic balance method to find periodic oscillations of non-linear systems. Secondly a generalization of the method is exposed to treat quasi-periodic solutions. Thirdly, a condensation procedure that keeps only the non-linear degrees of freedom of the mechanical system is described. This technique may be of great interest to reduce the original non-linear system and to calculate the dynamical behaviour of non-linear systems with many degrees of freedom. The last part presents the classical continuation procedures that let us follow the evolution of a solution as a system parameter varies.

Stochastic Analysis of the Eigenvalue Problem for Mechanical Systems Using Polynomial Chaos Expansion— Application to a Finite Element Rotor

This paper proposes to use a polynomial chaos expansion approach to compute stochastic complex eigenvalues and eigenvectors of structures including damping or gyroscopic effects. Its application to a finite element rotor model is compared to Monte Carlo simulations. This lets us validate the method and emphasize its advantages. Three different uncertain configurations are studied. For each, a stochastic Campbell diagram is proposed and interpreted and critical speeds dispersion is evaluated. Furthermore, an adaptation of the Modal Accordance Criterion is proposed in order to monitor the eigenvectors dispersion.

Periodic Solution Analysis of a Rotor-Stator Contact Nonlinear Problem

A typical problem encountered when studying turbo-machineries is studied: contact between a rotor and his stator. The contact is supposed to be permanent and frictionless; both rotor and stator are linked to the carter in an elastic and dissipative way. Under these modelling assumptions, nonlinear vibrations appear through the geometry of the problem when the system is excited by an out-of-balance within the rotor. Equations are written in the rotating frame in order to simplify the resultant frequency content of the dynamic quantities followed along the excitationnal frequency range. The stability and bifurcation analysis of a particular equilibrium is carried on. This first stage exhibits two Hopf bifurcation points. The associated periodic solutions are constructed and followed, using a shooting method. A specific bifurcation diagram is then established, where the incommensurate period of the solution is shown.Copyright