# A Numerical Analysis of the Nonlinear Dynamics of Shimmed Conical Gas Foil Bearings
## 2 Theory
### Numerical Solution  
To solve this coupled system of differential equations, different approaches can be used. The use of finite difference method (FDM), finite element method and Galerkin reductions have been shown to be applicable. In this work, the FDM is used to approximate the derivatives in Equation (3). Equation (3) can be converted into a form where the derivatives only occur separated from each other. \[\frac{\partial\psi}{\partial\tau}=\frac{1}{\Lambda\zeta}\left[\frac{1}{\sin^{2}{( \varphi)}\zeta}\left(\psi\tilde{h}\frac{\partial^{2}\psi}{\partial\theta^{2}}- \psi\psi\frac{\partial^{2}\tilde{h}}{\partial\theta^{2}}+h\frac{\partial\psi}{ \partial\theta}\frac{\partial\psi}{\partial\theta}-\frac{\partial\tilde{h}}{ \partial\theta}\psi\frac{\partial\psi}{\partial\theta}\right)+\zeta\psi\tilde{h }\frac{\partial^{2}\psi}{\partial\xi^{2}}+\zeta\tilde{h}\frac{\partial\psi}{ \partial\xi^{2}}+\psi\tilde{h}\frac{\partial\psi}{\partial\xi}+\psi\tilde{h} \frac{\partial\psi}{\partial\xi}\right]-\frac{\partial\psi}{\partial\theta} \tag{9}\]  
The FDM is applied to approximate the first and second-order derivatives in Equation (9) by using the central difference formulations shown in Equation (10) where \(f\) is \(\psi\) or \(\widetilde{w}\) and \(\
u\) is \(\zeta\) or \(\theta\), respectively. The error of these approximations is of order \((\Delta\
u)^{2}\). To calculate the central differences a \(N_{\mathrm{z}}\cdot N_{\theta}\) FD-grid in shape of the air gap of the CGFB is used. The axial bearing edges are excluded from the grid. \[\frac{\partial f}{\partial\
u}\bigg{|}_{f=f_{ij}}=\frac{f_{ij,j+1}-f_{ij,-1}}{ 2\Delta\
u};\left.\frac{\partial^{2}f}{\partial\
u^{2}}\right|_{f=f_{ij}}= \frac{f_{ij,+1}-2f_{ij,j}+f_{ij,-1}}{\Delta\
u^{2}} \tag{10}\]  
The boundary condition at the bearing edges is enforced by including \(\widetilde{p}=1\) in the central differences of the outer grid points as illustrated in Figure 2. In circumferential direction, a periodic boundary condition is used. The commonly used Guembel condition is applied by setting sub-ambient pressure to ambient pressure while integrating within Equation (8). The integrals are approximated by applying the trapezoidal rule on the FD-grid. Equation (5) can be rewritten as the equivalent system of first order differential equations shown in Equation (11). \[\frac{\partial}{\partial\tau}\left[ \begin{array}{c}\left(\varepsilon_{ \mathrm{x}}\right)\\ \varepsilon_{\mathrm{y}}\\ \varepsilon_{\mathrm{z}}\end{array} \right)=\left[ \begin{array}{c}\frac{ \partial}{\partial\tau}\begin{pmatrix}\varepsilon_{\mathrm{x}}\\ \varepsilon_{\mathrm{y}}\\ \varepsilon_{\mathrm{z}}\end{pmatrix}\\ \frac{\partial}{\partial\tau}\begin{pmatrix}\varepsilon_{\mathrm{x}}\\ \varepsilon_{\mathrm{y}}\\ \varepsilon_{\mathrm{z}}\end{pmatrix}\\ \frac{4}{m_{\mathrm{z}}\epsilon_{\mathrm{0}}\Omega^{2}}\begin{pmatrix}F_{ \mathrm{xF}}+m_{\mathrm{f}}\Omega^{2}u\cos(2\tau)\\ F_{\mathrm{yF}}+F_{\mathrm{G}}+m_{\mathrm{f}}\Omega^{2}u\sin(2\tau)\\ F_{\mathrm{zF}}-F_{\mathrm{ax}}\end{pmatrix}\end{array} \right] \tag{11}\]  
Substituting the FD-formulation into Equations (4) and (9) yields the vector functions \(\mathbf{g}_{\psi}\) and \(\mathbf{g}_{\widetilde{w}}\). Together with Equation (11) this results in a system of \(N_{\theta}(N_{\mathrm{z}}+1)+6\) coupled ordinary first order differential equations. The state vector \(\mathbf{s}\) and the vector function \(\mathbf{g}(\tau,\mathbf{s})\) are shown in Equation (12). The vector function \(\mathbf{g}_{\varepsilon}\) results from Equation (11) and \(\mathbf{J}\) represents the Jacobian of the dynamical system. Figure 2: FD-grid of the unwound conical air gap with a representation of the used boundary conditions. \[\mathbf{g}(\tau,\mathbf{s})=\frac{\partial\mathbf{s}}{\partial\tau};\mathbf{s}= \begin{bmatrix}\boldsymbol{\psi}_{\text{FD}}\\ \widetilde{\mathbf{w}}_{\text{FD}}\\ \boldsymbol{\varepsilon}\\ \frac{\partial\mathbf{s}}{\partial\tau}\end{bmatrix} ;\boldsymbol{\varepsilon}=