## 5 Performance Analysis of C-GFB
### Static Analysis of Bearing Performance  
An unsteady compressible Reynolds equation for an isothermal and isoviscous ideal gas with the normalized terms \(P=p/\!p_{a}\), \(H=h/\!C\), \(\
u=\omega_{e}\omega\), \(\Lambda=6\mu\omega R^{2}/\!p_{a}C^{2}\), and \(\tau_{e}=\omega_{e}t\) is expressed as  
\[\frac{\partial}{\partial X}\bigg{(}PH^{3}\frac{\partial P}{\partial X}\bigg{)} +\frac{\partial}{\partial Z}\bigg{(}PH^{3}\frac{\partial P}{\partial Z}\bigg{)} =\Lambda\frac{\partial(PH)}{\partial X}+2\Lambda\
u\frac{\partial(PH)}{ \partial\tau_{e}} \tag{4}\]  
where \(\omega\) is the rotational speed, \(\omega_{e}\) is the excitation frequency, \(P\) is the normalized gas-film pressure, and \(H\) is the normalized gas-film thickness. The film thickness considering journal eccentricities, mechanical preloads due to PZT actuators, and bump deflection is as follows:  
\[H=1+\varepsilon_{X}cos\theta+\varepsilon_{Y}sin\theta-r_{p}(\theta)/C+\delta_ {bump}/C \tag{5}\]  
where \(\varepsilon_{X}=e_{X}/C\) and \(\varepsilon_{Y}=e_{Y}/C\); \(r_{p}(\theta)\) is the approximated preload function caused by the piezoelectric preload \(\delta_{PZT}\), composed of the piezoelectric and structural deformations as given in Equation (3b), \(C\) is the nominal radial clearance, and \(\delta_{bump}\) is the local mechanical deformation of the bump foil. Approximated preload function: Among the several methods for approximating the mechanical preloads in bump-type GFBs based on the variation in film thickness ([8; 17]), the fully sinusoidal approximated model [15] is adopted for determining the approximated preload function for C-GFBs considering:  
\[r_{p}(\theta)=\delta_{PZT}/2\big{[}\cos\ 3\big{(}\theta-\theta_{p,n}\big{)}+ \sin\pi/2\big{]},\] (6a) where \[\theta_{p,n}\] is the angular position of \[n\] - th PZT actuator: \[n=\left\{ \begin{array}{ll}1:&\theta_{p,n}=60^{\circ}\ for\ 0<\theta\leq 2/3\pi\\ 2:&\theta_{p,n}=180^{\circ}\ for\ 2/3\pi<\theta\leq 4/3\pi\\ 3:&\theta_{p,n}=300^{\circ}\ for\ 4/3\pi<\theta\leq 2\pi\end{array}  \right.. \tag{6b}\]  
**Structural stiffness function:** A structural stiffness function is defined as a total assembly stiffness of the PZT actuators and foil structure along the circumferential coordinate, composed of \(k_{b}\), \(k_{p}\), and \(4k_{sp}\). Overall, the function is identical to \(k_{b}\), while a combined stiffness of the bump foil and of PZT actuators applies at the upper singer locations. It can be expressed by performing simple mathematical operations on Equation (3b) as follows:  
\[k_{\text{assy}}=\frac{k_{b}\big{(}k_{p}+4k_{sp}\big{)}}{k_{b}+(k_{p}+4k_{sp})}. \tag{7}\]  
A discontinuity exists in the structural stiffness function at each end of the upper stingers. In this study, an arc tangent function is introduced to smooth the structural stiffness function at each end of the upper stingers, as follows:  
\[k(\theta)=sgn(\theta)tan^{-1}\bigg{\{}\alpha\bigg{[}\theta-\bigg{(}\theta_{p, n}+sgn(\theta)\frac{\theta_{s}}{2}\bigg{)}\bigg{]}\bigg{\}}\frac{k_{b}-k_{ \text{assy}}}{\pi}+\frac{k_{b}+k_{\text{assy}}}{2} \tag{8}\]  
where  
\[ \begin{array}{ll}sgn(\theta)=&-1\text{ for }0<\theta\leq 1/3\pi,\ 2/3\pi<\theta\leq\pi,\ \text{and}\ 4/3\pi<\theta\leq 5/3\pi\\ &1\text{ for }1/3\pi<\theta\leq 2/3\pi,\ \pi<\theta\leq 4/3\pi,\ \text{and}\ 5/3\pi< \theta\leq 2\pi\end{array} \]  
\(\theta_{s}\) is the angular length of the mechanical preload. \(\alpha\) represents a smoothness constant, such that a large value results in an accurate albeit insufficiently smooth function, whereas a small value results in a sufficiently smooth albeit inaccurate function [22]. Figure 7 shows predicted structural stiffness function versus circumferential coordinate with respect to smoothness constant.