## 2 Concept and Design of C-GFB  
Figure 1 presents a detailed illustration of the proposed C-GFB with PZT actuators: (a) overall configuration of C-GFB with PZT actuators and (b) bearing preload principle when actuators are electrified with certain driving voltages. The origin of the stationary \(X\), \(Y\), \(Z\) coordinate system is located at the center of the bearing. The circumferential coordinate \(\theta\) is defined counterclockwise from the negative \(X\)-axis. The detailed design parameters of the C-GFB are listed in Table 1.  
The bearing consists of a bearing housing, bearing key, foil structures, and three PZT actuators. The foil structures are laminated structures including a top, bump, and bottom foil. A PZT actuator generates mechanical preloads with a PZT stack generating piezoelectric displacement in its thickness direction. The PZT actuators are assembled with the bearing housing at the angular locations of 60\({}^{\circ}\), 180\({}^{\circ}\), and 300\({}^{\circ}\). The angular length of the upper stingers is 27.7\({}^{\circ}\). Figure 1: Schematic views of a controllable gas foil bearing (C-GFB) with the ability of mechanical-preload control: (**a**) overall configuration of C-GFB with piezoelectric (PZT) actuators and (**b**) bearing preload principle when actuators are electrified with certain driving voltages. The PZT actuator is composed of the moving parts (upper and lower stingers) and the stationary parts (PZT case and PZT cap). The upper stinger is in contact with the foil structures and pushes the foils toward a bearing center when the PZT actuators are electrified with certain driving voltages, thereby providing a mechanical preload. The lower stinger is connected to the PZT case which is fixed to the bearing housing by four holder springs that provide the PZT stack with tension. The holder springs provide tension to allow the upper stinger to return to its original position when the stroke of the PZT stack decreases. This facilitates the moving parts to move accurately in line with the PZT displacement. This mechanism of mechanical preload generated by the PZT actuator is analogous to the principle of shimmed gas foil bearings in which the mechanical preload is produced by three metal shims. The magnitude of mechanical preload is defined as the distance between the lobe arc center and bearing center in lobed GFBs [9], and as the inserted shim thickness in shimmed GFBs [8], as given in Figure 1b. Meanwhile, the mechanical preload magnitude is quantified by the displacement of PZT actuator toward the bearing center at each location. Figure 2 illustrates a schematic view of a PZT stack and a PZT element in axial configuration: (a) a PZT stack of ceramic PZT materials separated by metallic electrodes and (b) a unit PZT element composed of a material and two electrodes. A number of PZT elements constitute a PZT stack in the direction 3. \(N_{p}\) layers of PZT materials and \(N_{p}+1\) electrodes are stacked along axis 3. Local axes 1, 2, and 3 are placed along the length (\(l_{p}\)), width (\(w_{p}\)), and thickness (\(t_{p}\)), respectively, of the PZT material. Note that the \(XYZ\) coordinate system is a global coordinate system in bearing units, and the 123 coordinate system is a local coordinate system in PZT element units. The electrical input lines of all the PZT elements are connected in parallel to an external power source to apply an identical driving voltage (\(V_{3}\)) to each element, generating a sizable piezoelectric deformation of the stack. In addition, the stack also has its own mechanical compliance, yielding a mechanical deformation under external loads.