# A Numerical Analysis of the Nonlinear Dynamics of Shimmed Conical Gas Foil Bearings
###### Abstract  
Conical gas foil bearings (CGFBs) have the potential to halve the number of necessary bearings in a conventional rotor system supported by gas foil bearings. Transient simulations of gas foil journal bearings and gas foil thrust bearings have proven their necessity to accurately predict the safe operating range of such bearings. This work presents the first transient model able to simulate the three-dimensional dynamics of CGFBs. The static behaviour of a single CGFB with a uniform bearing clearance is compared to a CGFB modified by thin metal shims and the resulting advantages of shimmer CGFBs are discussed. An investigation of the linear stability behaviour shows that the axial load of the bearing determines the stability of the equilibrium position. Furthermore, three transient simulations demonstrate the capability of the presented model to describe the nonlinear dynamics of a shimmed CGFB such as the occurrence of stable limit cycles and self-excited sub- and super-synchronous vibrations with and without a rotor unbalance. Additionally, waterfall diagrams are used to investigate the frequency response for different rotational speeds. The novel findings of this work are the importance of a non-uniform bearing clearance for the functionality of a CGFB and the identification of the axial force as a critical factor in maintaining bearing stability. These findings are specific to CGFBs and have not been discussed or mentioned in previous works. c pacs: 05.40.-a, 05.40.-a, 05.40.-a, 05.40.-a, 05.4  
The aim of these modifications is to enhance the static and dynamic behaviour of GFBs. A well-known disadvantage of GFBs is the occurrence of sub-synchronous vibrations [5; 6]. These vibrations occur at the onset speed of sub-synchronous vibration (OSSV) and can lead to an instability of the rotor at the onset speed of sub-synchronous instability (OSSI) [7]. A known way to enhance the dynamic performance of a GFB is to insert thin metal shims between the bearing sleeve and the foil structure. This modification yields a non-uniform clearance around the circumference of the bearing [8; 9; 10; 11; 12]. The numerical analysis of the dynamical behaviour of GFBs can be categorized into two different approaches. Classically, a perturbation approach based on the work of [PERSON] [13] in 1968 is used to predict linearised bearing parameters. These parameters can then be used to calculate the OSSV. Discrepancies between this OSSV and the OSSV obtained by transient analysis have been reported [14; 15]. Additionally, [PERSON] and [PERSON] [16] have shown the inability of [PERSON]'s method to detect certain instabilities due to the omission of state variables associated with the foil and air film domains. In 2014, [PERSON] and [PERSON] [17] presented a model which allows the simultaneous solution of the governing equations. The analysis of the Jacobian of this fully coupled system of equations yields the exact OSSV as obtained by a transient nonlinear dynamic analysis. In contrast to the perturbation approach, this model can furthermore predict the nonlinear stability behaviour beyond the OSSV. [PERSON] et al. [18] and [PERSON] et al. [19] showed that supercritical as well as subcritical bifurcation can occur in GFBs. While subcritical bifurcations lead to stable limit cycles (sub-synchronous vibration), supercritical bifurcations lead to unstable limit cycles which can cause instability of the rotor even before the OSSV. Regardless of the approach, different models to capture the behaviour of the compliant foil structure have been presented [20]. A model referred to as the \"simple elastic foundation model\" (SEFM) has been used and adapted repeatedly [17; 21; 22; 23; 24; 25; 26; 27; 28; 29; 30; 31]. It provides a linear stiffness model and considers equivalent viscous damping by including a loss factor. One of the biggest flaws of the SEFM is the modelling of dissipation due to friction by using viscous damping. It is generally known that this representation is not adequate.