## 2 Materials and Methods  
### Theoretical Modeling of Integral Squeeze Film Bearing Damper  
The structure of the integral squeeze film bearing damper is mainly divided into four parts: an outer flange, an elastic body, a squeeze film area, and an inner flange. The outer flange is fitted and fixed to the bearing housing. The inner flange fits snugly against the outer ring of the rolling bearing. The elastic body can change the stiffness of the system, and its structure is similar to the elastic support and has the function of vibration isolation. The squeeze film area can produce a squeezing film effect to generate damping and dissipate vibration energy. These special structures achieve the purpose of vibration control, as shown in Figure 1. The two main structures are theoretically modeled below. First, the stiffness damping model of the integral squeeze film bearing damper is shown in Figure 2. When it was subjected to an external force of \(\mathrm{F_{0}}\mathrm{sin}(\omega t)\), the vibration isolation function of the elastic body was analyzed [15; 16; 17; 18]. We made the following assumptions: (1) the elastic body is unitized; (2) the structural damping of the elastic body is neglected; (3) the bearing housing is considered as a rigid body. For the vibration isolation structure dynamics, there is the following equation:  
Figure 1: Integral squeeze film bearing damper (ISFBD) structure. Figure 2: Vibration isolation model. \[m\ddot{x}+kx=F_{0}\sin(\omega t). \tag{1}\]  
Then, the displacement of the bearing housing is:  
\[x_{0}=\left|\frac{F_{0}}{k}\frac{1}{(1-s^{2})}\right| \tag{2}\]  
where s represents the frequency ratio, that is, \(\omega/\omega_{0}\). The force that can be transmitted to the bearing housing is:  
\[F_{f}=|kx_{0}|. \tag{3}\]  
The force transfer coefficient is then obtained as follows:  
\[\beta=\left|\frac{F_{f}}{F_{0}}\right|=\left|\frac{1}{1-s^{2}}\right|. \tag{4}\]  
It can be seen that when s \(\geq\sqrt{2}\), the elastic body begins to exert the vibration isolation effect, so the exciting force received at the journal is isolated to prevent transmission to the bearing housing and protect the stability of the rotor system. A damping dissipation theoretical model diagram of the squeeze film area is shown in Figure 3. The squeeze film area of the integral squeeze film bearing damper can dampen vibration control and dissipate the energy of vibration [19; 20]. The following assumptions were made for the damping fluid: (a) incompressible Newton fluid is used; (b) laminar flow thin film is used; (c) there is no slip at surfaces; and (d) there is no inertia effect or other external body forces. The oil film of the novel integral squeeze film bearing damper was analyzed, and the Reynolds equation was used for model analysis [21; 22]:  
\[\frac{1}{R^{2}}\frac{\partial}{\partial\theta}(h^{3}\frac{\partial p}{ \partial\theta})+\frac{\partial}{\partial z}(h^{3}\frac{\partial p}{\partial z })=-12\mu\omega\frac{\partial h}{\partial\theta} \tag{5}\]  
where \(R\) is the radius of the squeeze film zone, \(p\) is the oil film pressure, \(\mu\) is the damping fluid viscosity coefficient, \(z\) is the axial coordinate, and \(\theta\) is the circumferential coordinate. Since the novel i damper has a seal at both ends for oil seal, it is considered that there is no pressure flow in the axial direction, that is, \(\partial p/\partial z\approx 0\). Then Equation (5) can be simplified to:  
Figure 2.0  
Figure 2.1 Damping vibration absorption model. Performing one and two integrals on \(\theta\), respectively, we can obtain:  
Figure 2.2  
Figure 2.3  
where \(C\) is the gap, and \(\varepsilon\) is the eccentricity. The Sommerfeld is applied, that is:  
Figure 2.4  
Using Equation (9), we can derive:  
\[\left\{ \begin{array}{l}1+\varepsilon\cos\theta=\frac{1-\varepsilon^{2}}{1- \varepsilon\cos\alpha}\\ d\theta=\frac{\left(1-\varepsilon^{2}\right)^{1/2}}{1-\varepsilon\cos\alpha}d \alpha\end{array} \right. \tag{10}\]  
Substituting the Sommerfeld equation for Equation (8) gives:  
\[