The extraction of Campbell diagrams from the dynamical system representation of a foil-air bearing rotor model. (15th August 2019) - Bonello, Philip
Highlights: For the first time, a method is established for deriving Campbell diagrams for foil-air bearing (FAB) rotor systems that avoids Lund's linear force coefficient (FC) method. This research is motivated by the fact that the FC method eliminates the air film and foil state variables through an assumption that is considered to be the cause of reported discrepancies between transient nonlinear dynamic analysis (TNDA) and the FC method for free low-amplitude dynamic analysis of FAB-rotor systems. The new method is based on an analysis of the Jacobian matrix of the dynamical system representation of the fully-coupled rotor/air film/foil structure domains. Through appropriate scaling of the eigenvectors and the application of a minimum journal amplitude criterion, the multitude of eigenvectors of the Jacobian can be filtered to extract the relevant modes. Where appropriate, a maximum damping criterion should also be applied. Each extracted mode is precisely verified using TNDA through mode-specific initial conditions. The methodology is successfully tested on three cases in the literature, which respectively illustrate its applicability to complex bearing types (3-pad), complex foil models, and flexible rotors. Abstract: Campbell diagrams describe the variation with speed of the modes of low amplitude (linear) free vibration of a rotordynamic system about its static equilibrium condition, and are an important tool that aids the interpretation of the nonlinear response. However, Campbell diagrams for foil-air bearing (FAB) rotor systems are currently derived using the linear force coefficients (FC) method and individual modes have not been verified by transient nonlinear dynamic analysis (TNDA) at low amplitudes. In fact, significant discrepancies between FC and TNDA predictions for the stability of the leading (i.e. least damped) mode have been reported. This paper establishes for the first time a method for deriving Campbell diagrams of a generic FAB-rotor model that avoids the FC method. It is based on an eigenvalue analysis of the Jacobian matrix of the dynamical system. Through suitable scaling and the application of appropriate criteria, the multitude of eigenvectors of the Jacobian can be filtered to extract the relevant modes. Each mode is verified using TNDA with mode-specific initial conditions derived from the Campbell diagram analysis. The methodology is successfully tested on three cases in the literature, which respectively illustrate its applicability to complex bearing types (3-pad), complex foil models, and flexible rotors.