The extraction of Campbell diagrams from the dynamical system representation of a foil-air bearing rotor model

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.

The effects of air film pressure constraints and top foil detachment on the static equilibrium, stability and modal characteristics of a foil-air bearing rotor model

“…The rotor, each air film, and each foil structure, are domains represented by time-based differential equations [2]. The dynamical system representation of the resulting coupled nonlinear model follows the generic state space format [3] 𝐬 ′ = 𝛘(𝜏, 𝐬)…”

“…where ( ) ′ denotes differentiation with respect to the time variable 𝜏, 𝐬 is the state vector and 𝛘 is a vector of nonlinear functions of 𝐬 and 𝜏 (the explicit dependency on 𝜏 arising from the external dynamic excitation 𝐟 ext e.g. rotating unbalance) [3]. The system is typically analysed for the following:…”

“…Campbell diagrams [3]. As far as nonlinear response analysis is concerned, in order to reduce the computational burden, it has been the practice for many years to adopt an approximate non-simultaneous solution approach wherein the air film, foil and rotor equations were decoupled by approximating the air film equations as algebraic, rather than differential, equations e.g.…”

“…[2,7,[10][11][12][13] and these stem from their violation of the fully coupled nature of the dynamical system (eq. ( 1)) [3].…”

Wet Modal Analyses of Various Length Coaxial Sump Pump Rotors with Acoustic‐Solid Coupling

“…is study used an acoustic-solid coupling model to separately determine the flow in the impeller water region and the dynamic characteristics of the coupled fluidsolid model. e wet modal analysis of the various length rotors in the sump pump used water as the working medium with the results closer the results of the natural frequency and critical speed of the actual operation of the sump pump rotor [23,24] is shown in Table 8. e predicted and measured first-order critical speeds are compared, as shown in Figure 10 which shows little difference between the predictions and the manufacturer data with very similar trends.…”