## 4 Development of C-GFB and Parameter Identifications  
The unknown parameters of PZT actuators, \(k_{p}\), \(D_{33,1}\), and \(D_{33,2}\), are identified by measuring the deformations of PZT stacks under external load and no-load conditions with various piezoelectric voltages applied. First of all, we fabricated a test C-GFB with PZT actuators, and identified its geometric bearing clearance. Figure 4 illustrates a fabricated C-GFB with PZT actuators. The dimensions of the bearing and PZT actuators are listed in Table 1. The stiffness coefficient per unit area of the holder springs (\(k_{sp}\)) is specified as 3.95 \(\times\) 10\({}^{6}\) N/m\({}^{3}\), which is calculated by dividing the stiffness of 1.4 \(\times\) 10\({}^{6}\) N/m by the contact area between the upper stinger and the bottom foil of 3.54 \(\times\) 10\({}^{-4}\) m\({}^{2}\). All the parts of the bearing, except for the PZT stack and foil structures, were fabricated via precision machining. Meanwhile, the Mos2 lubricant coatings with thickness of 0.02 mm are sprayed on one end-surface of the top foil to reduce bearing friction and wear, particularly in the low-speed regions. The bump foil is fabricated by the press process using a flat-shaped tool (die) with a forming load of 40 MPa. This is because, when analyzing the moldability of the bump foil according to the forming loads, 40 MPa was the most suitable pressure. Note that [PERSON] et al. [19] reported that a load of at least 28 MPa is required to achieve the designed bump shapes. The bump, top, and bottom foil layers are affixed to the bearing key, which is inserted into a slot inside the bearing housing. The bearing housing was manufactured to have the inner radius of 20.945 mm for an assumed rotor radius of 20 mm, considering the designed foil structure thickness of 0.782 mm with the top, fabricated bump, and bottom foils, yielding a geometric radial bearing clearance of 0.163 mm. However, we used 0.100 mm as a nominal bearing clearance for simulations in this paper, based on experimental experiences that the large bearing clearance causes excessive rotordynamic instability for a rotor with the radius of 20 mm up to a rotating speed of 90 kpm [20]. Note that the bearing clearance is sometimes able to be adjusted by inserting thin foils over the circumference to stabilize rotordynamic responses during experiments. Figure 5 illustrates the PZT coefficient measurement of the PZT stack: (a) depicts the measurement setup with a micrometer, and (b) presents measured PZT deformations versus applied driving voltages. The measurement is performed in the no-load condition, i.e., Equation (3) yields \(\delta_{3}=D_{33}V_{3}\) or \(\delta_{3}=D_{33,1}V_{3}^{2}+D_{33,2}V_{3}\) depending on the system linearity. The micrometer measures the thickness variation of the PZT stacks with respect  
Figure 4: Photographic images of the fabricated C-GFB with PZT actuators. to the applied driving voltages. The thickness variation can be regarded as the PZT stack displacement \(\delta_{3}\). In Figure 5b, the dashed line represents the predicted PZT stack displacement with the manufacturer \(D_{33}\) of 0.467 \(\upmu\)m/V for a linear system. Overall, the measured displacements of the PZT stacks #1-#3 do not agree with the predicted curve. Furthermore, they exhibit significant nonlinear behavior. This result demonstrates why the PZT coefficient measurement is necessary, although its ideal value is provided by the manufacturer. Consequently, a second-order polynomial function is used to express a representative curve for the relationship between the PZT stacks' displacement and driving voltage. The function is \(\delta_{3}=-0.0028V_{3}^{2}+0.9122V_{3}\), with \(R^{2}=0.9926\). Thus, \(D_{33,1}=-0.0028\) and \(D_{33,2}=0.9122\). Meanwhile, the PZT stacks generate marginally different displacements.