The recently-developed simultaneous solution scheme for solving nonlinear rotordynamic systems running on foil-air bearings (FABs) has overcome the practice of decoupling the air film, foil and rotor equations that has been typically followed to reduce computational burden at the expense of accuracy. However, the published works using the simultaneous solution technique have been limited to a simple bump foil model in which the individual bumps were modelled as independent spring-damper (ISD) subsystems. The overall aim of this thesis is to present methods that enable more realistic FAB models to be integrated into the simultaneous solution scheme, without compromising its efficiency. Two such alternative approaches are presented: (1) the full foil structure modal model (FFSMM) of the bump foil structure; (2) non-parametric system identification of the entire FAB i.e. foil and air film. The FFSMM provides a more realistic model of the bump foil structure since it considers the interaction between the bumps and foil inertia. Although the foil damping is still assumed to be linear, the model presented is adaptable to nonlinear friction forces. The dynamics of the bump foil structure are studied by finite element methods and experimentally validated using a purpose-made corrugated foil structure. The FE result shows that the effect of bump interaction increases the effective stiffness of the FAB. Foil inertia is not important for the range of speeds considered in the thesis, but the experimentally validated fundamental foil resonance of around 2 kHz is within the operating speed range of high-speed turbomachinery. The FFSMM can take into account the curvature of the bearing sleeve, but the effect of this feature is proven to be negligible for the size of bearing used in the study. The FFSMM simulation results are correlated against ISD model results and published experimental maximum film thickness and locus of the journal response. The results of the FFSMM were then compared against experimental results under unbalance response conditions measured from a purpose-built test rig. The rotor was mathematically modelled using rigid body equations of motion, which were validated by modal analysis. The unbalance rotor response results obtained from the FFSMM and experiment both show that the sub-synchronous motion is not only mainly influenced by the increment of unbalance mass, but, to a greater extent, the increment of rotor speed. The findings show good agreement between the model and experimental results. This thesis also presents the non-parametric system identification of a FAB, which is also adaptable to the simultaneous solution scheme. This work is motivated by two advantages: (a) it removes computational limitations by replacing the whole bearing equations by a displacement/force relationship, where the air film effect is taken into account; (b) it can capture complications that cannot be easily modelled, if the identification is based on empirical data. A Recurrent Neural Network (RNN) is trained to identify the full numerical model of a FAB over a wide range of speeds. The identified model of the FAB is adapted into the frequency domain rotor-dynamic simultaneous solution technique by using harmonic balance (HB) methods, thus directly producing the steady-state orbit response. Excellent correlation is demonstrated between the identification technique and the full numerical model under two validation processes: (i) using different sets of input/output data; (ii) the application of the identified RNN-FAB model to HB analysis in lieu of the full numerical model of the FAB.