## 3 Structural Model of PZT Actuator and Foil Structures
### Structural Model of PZT Actuator and Foil Structures  
Figure 3 illustrates a schematic view of an equivalent spring model for the assembly of a PZT actuator and a foil structure. The PZT actuator, with the PZT stack and four holder springs, and the foil structure have stiffness coefficients per unit area of \(k_{p}\), \(4k_{sp}\), and \(k_{b}\), respectively. In the local area (\(\Delta\theta\ \times\ l_{b}\)), where \(\Delta\theta\) indicates a discretized single grid in the angular direction of the bearing and \(l_{b}\) stands for the bearing axial length, the PZT stack and holder springs are in parallel, whereas the bump foil and PZT actuator are in series. They deform when the local gas-film pressure \(p_{A}\) acts on the local top foil; the pressure is regarded as the average gas-film pressure across the bearing axial length \(l_{b}\)[15; 17], i.e.,  
\[p_{A}=1/l_{b}\int_{-\frac{1}{2}l_{b}}^{\frac{1}{2}l_{b}}(p-p_{a})dz\]  
The local top-foil displacement \(\Delta w_{t}\) is the sum of deformations for the PZT actuator and bump, and is given as:  
\[\Delta w_{t}=\delta_{PZT}+\delta_{bump}\] (3a) where \[\delta_{PZT}=\frac{k_{p}}{k_{p}+4k_{sp}}\Big{(}D_{33,1}V_{3}^{2}\ +\ D_{33,2}V_{3}\Big{)}-\frac{4k_{sp}\delta_{pre}}{k_{p}+4k_{sp}}-\frac{p_{A} }{k_{p}+4k_{sp}},\delta_{bump}=-\frac{p_{A}}{k_{b}}. \tag{3b}\]  
In particular, the PZT actuator has three components of deformations from stack piezoelectricity, stack stiffness, and holder spring stiffness. The no-load deformation of the PZT actuator is determined by the stiffness ratio of the PZT stack and holder springs under the fixed driving voltage. The holder spring for tightening the moving stinger to the stack has a slight pre-extension of \(\delta_{pre}\), which is designed to be sufficiently so as small not to constrain the piezoelectric deformation. As a result, \(\delta_{PZT}\) acts as a mechanical preload for conventional bump-type GFBs; therefore, it is called to the piezoelectric preload of C-GFBs in this paper. The piezoelectric preload is also variable depending on film pressures as well as applied voltages.  
A complex stiffness of the bump foil \(k_{b}^{\prime}=k_{b}(1+j\gamma)\) is introduced to consider the dynamic effects of the bump foil structure. Here, the loss factor \(\gamma\) arises from the frictional motion between the bump foil and the surrounding structures plus the hysteresis damping effect that occurs during the repeated deformation-restoration process of the bump foil. The loss factor must be determined from characterization tests of the bump foil [17]; it ranges from 0.05 to 0.2 for dimensionless static loads of 0.03 to 0.14 [18], and 0.1 is used in  
Figure 3: Equivalent spring model for PZT actuators and foil structures in the local area (\(\Delta\theta\ \times\ l_{b}\)) when local gas-film pressure acts on the local top foil.  
this study. On the other hand, the loss factor of the PZT stacks and holder springs are not considered because the material damping is not significant.  
Consequently, \(k_{sp}\) is a known parameter provided by the manufacturer, and \(k_{b}\) can be calculated using [PERSON]'s formula for an individual bump foil under free-free condition [16]. However, the remaining parameters of \(D_{33,1}\), \(D_{33,2}\), and \(k_{p}\) are unknown, and thus are identified via experiments. Note that the manufacturer provides a piezoelectric coefficient of \(D_{33}\), but it is not accurate for a system with significant nonlinearities.