# A Numerical Analysis of the Nonlinear Dynamics of Shimmed Conical Gas Foil Bearings
## 2 Theory
### Fluid Film  
The fluid film between the journal and the compliant foil can be described by the compressible, isothermal, isoviscous Reynolds equation. The following assumptions are commonly used to describe the behaviour of GFBs and have shown good agreement with experimental results. 1. The influence of inertia forces and the force of gravity on the fluid can be neglected. 2. The fluid flow is laminar. 3. Wall adhesion is assumed, and no-slip flow is considered. 4. The pressure is constant over the thickness of the air film. 5. The air can be described as an ideal gas. 6. The temperature of the fluid film is constant. Assumptions 1-5 are standard for the simulation of fluid film bearings and a detailed analysis of the limitation and justification of these assumptions can be found in [40]. The assumption of a constant temperature is commonly used but partly questionable. The inclu  
Figure 1: Geometry of a conical gas foil bearing; Parameters marked with (*) are projected onto the x-y-plane for clarity. sion of a non-constant temperature greatly increases the complexity of the model and brings a lot of additional uncertainties [41]. For this reason and due to experimentally verified results of isothermal simulations, this assumption is also used for the presented model. With these assumptions the Reynolds equation for a conical shaped air gap can be written as shown in Equation (1), where the non-dimensional pressure \(\widetilde{p}=p/p_{0}\) and the non-dimensional film thickness \(\widetilde{h}=h/c_{0}\) are defined by the air film pressure \(p\), the air gap thickness \(h\), the ambient pressure \(p_{0}\) and the nominal bearing clearance \(c_{0}\). For a bearing with the half-cone angle \(\varphi\), a maximum radius \(r_{1}\), and a length \(L\), the coordinates \(\theta\in[0,2\pi]\) and \(\zeta=\zeta/\xi_{1}\) are defined by a coordinate \(\zeta\in[\xi_{0},\xi_{1}]\) along the air gap with \(\xi_{1}=r_{1}/\sin(\varphi)\) and \(\xi_{0}=\xi_{1}-L\). Time \(t\) is represented by dimensionless time \(\tau=(\Omega/2)t\) using the angular velocity \(\Omega=2\pi n\) and the rotational speed \(n\) of the conical journal. \[\frac{1}{\sin^{2}\left(\varphi\right)\zeta}\ \frac{\partial}{\partial\theta} \left(\widetilde{p}\widetilde{h}^{3}\frac{\partial\widetilde{p}}{\partial \theta}\right)+\frac{\partial}{\partial\zeta}\left(\zeta\widetilde{p} \widetilde{h}^{3}\frac{\partial\widetilde{p}}{\partial\zeta}\right)=\Lambda \zeta\left(\frac{\partial\widetilde{p}\widetilde{h}}{\partial\theta}+\frac{ \partial\widetilde{p}\widetilde{h}}{\partial\tau}\right) \tag{1}\]  
The bearing number \(\Lambda=\frac{6\pi n\Omega\xi_{1}^{2}}{p\sigma_{0}^{2}}\) includes the viscosity \(\mu\) of air. This equation can be derived from the generalized Reynold's equation presented by [PERSON] et al. [42]. The stationary form of this equation was also presented by [PERSON] [43] in 1993 and used by [PERSON] et al. [34; 35; 36; 37; 38]. To the authors knowledge this equation has not yet been used for transient simulations of CGFBs or integrated into a fully coupled GFB system. The introduction of the new state variable \(\psi=\widetilde{p}\widetilde{h}\) yields Equation (2), and allows the transformation into a form which enables a simultaneous solution of the three domains. \[\frac{\partial\psi}{\partial\tau}=\frac{1}{\Lambda\zeta}\left\{\frac{1}{\sin^ {2}\left(\varphi\right)\zeta}\frac{\partial}{\partial\theta}\left[\psi\left( \widetilde{h}\frac{\partial\psi}{\partial\theta}-\psi\frac{\partial\widetilde{ h}}{\partial\theta}\right)\right]+\frac{\partial}{\partial\zeta}\left[\zeta \psi\left(\widetilde{h}\frac{\partial\psi}{\partial\zeta}-\psi\frac{\partial \widetilde{h}}{\partial\zeta}\right)\right]\right\}-\frac{\partial\psi}{ \partial\theta} \tag{2}\]  
With the commonly used assumption of a constant film height in the axial direction applied Equation (2) can be simplified to Equation (3).