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

Abstract

The foil-air bearing (FAB) plays a key role in the development of high speed, economical and environmentally friendly oil-free turbomachinery. However, FABs are known to be capable of introducing undesirable nonlinear effects into the dynamic response of a rotor-bearing system. This necessitates a means for calculating the nonlinear response of rotor systems with FABs. Up to now, the computational burden introduced by the interaction of the dynamics of the rotor, air film and foil structure has been overcome by uncoupling these three subsystems, introducing the potential for significant error. This paper performs the time domain solution of a simple rotordynamic system without uncoupling the state variables. This solution is then used as a reference for the verification of two proposed novel methods for reducing the computational burden: (a) use of harmonic balance; (b) use of Galerkin transformation. The applicability and accuracy of these two methods is illustrated on a simple symmetric rotor-FAB system.

Nonlinear Dynamic Analysis of a Turbocharger on Foil-Air Bearings With Focus on Stability and Self-Excited Vibration

“…With regards to the first aspect of the problem, as discussed in [1,2], in the case of compressible fluid bearings the RE is a state equation since it includes time as an independent variable [3][4][5][6][7][8]. 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) [1,2].…”

“…Additionally, the air film gap at a given location is a function of the foil deformation there, apart from the journal displacement. Hence, a further state equations are introduced and the total number of state equations to be solved would be equal to where is the number of rotor modes and casing modes (if considered), and is the number of bearings [1,2]. Such a large nonlinear system would be numerically "stiff", requiring very small time-steps to maintain a given accuracy if an explicit numerical integration scheme is used [10].…”

“…In previous research [1,2], the authors developed two alternative techniques (respectively based on FD and Galerkin Reduction) to preserve the state equations of the air films and solve them simultaneously with the other state equations. In the present paper, the state equations are similarly preserved but only FD is used and symbolic computing is used as an alternative to the vectorised formulation adopted in [1,2] for the efficient computation of the Jacobian.…”

“…For example, the above-mentioned works [1][2][3][4][5][6][7][8] assumed a simple symmetric rigid rotor-bearing system. Other works considered rigid rotors with four degrees of freedom (DOFs) (corresponding to translation and rotation in each of the xy, yz planes) [11] or five DOFs (where an additional axial DOF was considered to account for an air foil thrust bearing) [12].…”

“…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) [1,2]. Additionally, the air film gap at a given location is a function of the foil deformation there, apart from the journal displacement.…”