Nowadays, air foil bearings find widespread use in very high speed, lightly loaded oil-free rotating turbomachineries such as compressors and microgas turbines because they have theoretically no speed limitations and they are environmentally benign. In the design of such bearings, it is of cardinal importance to enhance their steadystate and dynamic performance characteristics for the safety operation, especially against the external dynamic excitations. Most of elasto-aerodynamic approaches under dynamic conditions proposed in the technical literature include only the static pressure induced deformation of foils. This paper presents a theoretical investigation on the effects of both static and dynamic deformations of the foils on the dynamic performance characteristics and stability of a self-acting air foil journal bearing operating under small harmonic vibrations. For the dynamic deformations of foils to be taken into account, the perturbation method is used for determining the gas-film stiffness and damping coefficients for given values of excitation frequency, compressibility number and compliance factor of the bump foil. The rotor-dynamic coefficients serve as input data for the linear stability analysis of rotor-bearing system. The nonlinear stationary Reynolds' equation is solved by means of the Galerkin's finite element formulation, whereas the finite differences method are used to solve the first-order complex dynamic equations resulting from the perturbation of the transient compressible Reynolds' equation. As a first approximation, the corrugated subfoil is modelled as a simple elastic foundation, i.e. the stiffness of a bump is uniformly distributed throughout the bearing surface. It was found that the dynamic properties and stability of the compliant finite length journal bearing are significantly affected by the compliance of foils especially when the dynamic deformation of foils is considered in addition to the static one by applying the principle of superposition.This step is necessary to solve the first-order complex dynamic Equations (22) and (23). METHOD OF SOLUTION UNDER DYNAMIC CONDITIONSThe procedure for obtaining a numerical solution to the elasto-aerodynamic problem under dynamic conditions is outlined as follows:Set the initial value of the frequency ratio g (e.g. g = 1 corresponding to a synchronous excitation). Solve the linear partial differential Equations (22) and (23) over the finite difference grid with successive over-relaxation scheme to obtain the complex dynamic pressuresQ e andQ f (Figure 8). It should Figure 13. Example of the coded half fluid-film domain. (À) Ambient or subambient static pressure region. (+) Over-ambient static pressure region.Figure 12. 3D view of the steady-state pressure field calculated without the subambient pressures.

The Phan-Thien Tanner (PTT) model is one of the most widely used rheological models. It can properly describe all the common characteristics of viscoelastic non-Newtonian fluids. Synovial fluid of human joints, which also lubricates artificial joints, is well known to be highly viscoelastic. Thus it is reasonable to attempt to describe such joint behavior using non-Newtonian flow models. Modeling the geometry of the total hip replacement, the PTT model is applied in spherical coordinates to a thin confined fluid film. As an illustrative problem, the case of a sudden impulsive start of simple squeezing motion is solved, similar to landing on one’s feet after a vertical jump. The phenomena of shear thinning, stress relaxation, and stress overshoot are all exhibited.