# A Numerical Analysis of the Nonlinear Dynamics of Shimmed Conical Gas Foil Bearings
###### Abstract
Hence, a lot of models have been presented that not only provide a better modelling of friction but also include more of the nonlinear properties of the foil structure. These models include the nonlinear bump stiffness, the interaction between bumps, and the modelling of the top foil as a beam, plate, or shell element, all while modelling the friction between top foil, bump foil and bearing sleeve. All these techniques have been applied to GFJBs or GFTBs. Although several patents for CGFBs can be found only a few research papers have been published [32; 33; 34; 35; 36; 37; 38]. In 2013, [PERSON] and [PERSON] [32] presented an experimental work on CGFBs. A bearing with unusually large bumps was used and the experiments proved the ability of CGFBs to support radial and axial loads simultaneously. In recent studies from 2020 and 2021, [PERSON] et al. [34; 35; 36; 37; 38] theoretically investigated the static and dynamic characteristics of CGFBs considering misalignment, and roundness and taper error. In these works, [PERSON]'s perturbation approach was used in combination with a nonlinear stiffness model of the compliant foil structure. In [34; 38], the authors presented experimental results to verify the functionality of the presented CGFBs. These studies put the main research focus onto the compliant foil structure. They presented a foil structure model considering coulomb friction [37] and proposed guidelines for the design of bump-type CGFBs [35]. Although the transient analyses of GFJBs have proven their necessity to calculate an accurate OSSV [14; 15], no transient investigation of CGFBs has yet been presented in the literature. Furthermore, transient simulations have shown that the nonlinear stability behaviour, e.g., stable, and unstable limit cycles are crucial for the accurate prediction of the safe operating range of GFBs [18; 19]. The aim of this work is to present a fully coupled three-dimensional transient nonlinear dynamic model for CGFBs, able to predict the static characteristics, as well as the linear and nonlinear stability behaviour of CGFBs. The novelty of this work is the investigation of CGFBs using a fully coupled model. Neitheran investigation of the static nor dynamic behaviour of CGFBs has been presented in previous works using this simulation approach. To date, a particular phenomenon related to CGFBs has not been discussed in the literature. The governing equations presented in Section 2 yield the problem that the air film pressure is not directly coupled with the axial displacement of the rotor, if the rotor is in the radially centrical position. In that case, the bearing would provide no static axial force and stiffness. To solve this problem, it is common for conical oil bearings to use spiral grooves on the conical journal [39]. Inspired by that approach, this work investigates the use of thin metal shims between the bearing sleeve and bump foil. As stated, shimming is a common modification to enhance dynamic performance [8; 9; 10]. Furthermore, shimming could greatly enhance the axial load carrying capacity of CGFBs and solve the aforementioned problem.