Numerical Method and Bifurcation Analysis of Jeffcott Rotor System Supported in Gas Journal Bearings

Abstract

From the viewpoint of nonlinear dynamics, the stability and bifurcation of the rotor dynamical system supported in gas bearings are investigated. First, the dynamical model of gas bearing-Jeffcott rotor system is given, and the finite element method is used to approach the unsteady Reynolds equation in order to obtain gas film forces. Then, the method for stability analysis of the unbalance response of the rotor system is developed in combination with the Newmark-based direct integral method and Floquet theory. Finally, a numerical example is presented, and the complex behaviors of the nonlinear dynamical system are simulated numerically, including the trajectory of the journal and phase portrait. In particular, the stabilities of the system’s equilibrium position and unbalance responses are studied via the orbit diagram, phase space, Poincaré mapping, bifurcation diagram, and power spectrum. The results show that the numerical method for solving the unsteady Reynolds equation is efficient, and there exist a rich variety of nonlinear phenomena in the system. The half-speed whirl encountered in practice is the result from Hopf bifurcation of equilibrium, and the numerical method presented is available for the stability and bifurcation analysis of the complicated gas film-rotor dynamic system.

Efficient Techniques for the Computation of the Nonlinear Dynamics of a Foil-Air Bearing Rotor System

“…[1][2][3][4]). This means that the pressure at each mesh point is a state variable and the discretised RE will be a set of temporal differential equations (TDEs i.e.…”

Dynamic behaviour of rotors supported by fluid-film bearings operated close to fluid-induced instability

“…(18) has to be transformed to the state-space in order to be integrable by standard numerical solvers. The transformation was briefly described in section 2.3.…”

“…A transient state which occurs in rotor-bearing system when a journal pass the threshold speed is studied in [13], where small amplitude perturbations are assumed in the bearing. The transient state of Jeffcott rotor supported by a gas bearing and accompanying bifurcations are studied in [18].…”

A Neural Network Identification Technique for a Foil-Air Bearing Under Variable Speed Conditions and Its Application to Unbalance Response Analysis

“…As observed in [1,2], due to the computational burden so introduced, the simultaneous solution of the state equations of the air film, foil and rotor has typically been avoided. In works such as [3][4][5][6][7][8] the air-film ODEs are uncoupled from the foil and rotor ODEs and approximated as algebraic equations; these latter equations were solved iteratively for the current pressure distribution using the rotor state variables at the previous time step [1,2].…”

“…The use of Finite Difference (FD)/Finite Element (FE)/Control Volume methods [3][4][5][6][7][8][9] to discretize the RE over the air film, creates a grid of points representing the pressure field, turning the RE into a set of first order ordinary differential equations (ODEs) with time as the independent variable (state equations). As observed in [1,2], due to the computational burden so introduced, the simultaneous solution of the state equations of the air film, foil and rotor has typically been avoided.…”