# Study on the Effect Mechanism of MRD to Rotor System
## 2 System Model and Basic Theory  
### Theory and Model of MRD  
Figure 1 shows a schematic diagram of MRD. \(R_{o}\) and \(R_{i}\) are the outer and inner diameters of MRD, respectively. \(C=R_{o}-R_{i}\) is the average clearance of MRD. \(e_{B}\) is the eccentricity. \(I\) is the external current applied on the clearance of MRD. \(xoy\) is the fixed coordinate system and the origin \(o\) of it is located at the intersection point of the axis of the outer ring and the symmetric section of MRD. \(o_{1}\) is the center of the inner ring. \(XOY\) is fixed to the outer ring. Oil film thickness \(h\) can be approximately expressed as  
\[h=c-e_{B}\cos\theta \tag{1}\]Assuming that the fluid is incompressible and satisfies the short bearing theory, the balance equation and continuous equation of MRF can be expressed as:  
\[\frac{\partial p}{\partial Z}=\frac{\partial\tau}{\partial Y} \tag{2}\]  
\[\frac{\partial v}{\partial Y}+\frac{\partial w}{\partial Z}=0 \tag{3}\]  
where \(p\) is the film pressure. \(v\) and \(w\) are the velocity of the fluid along the \(Y\) and \(Z\) directions, respectively. The bilinear constitutive law of MRF can be expressed as:  
\[\left\{ \begin{array}{ll}\tau=\tau_{y}+\eta\frac{\partial w}{\partial Y}& \text{for }\tau>\tau_{c}\\ \tau=\alpha_{\eta}\eta\frac{\partial w}{\partial Y}&\text{for }-\tau_{c}\leq \tau\leq\tau_{c}\\ \tau=-\tau_{y}+\eta\frac{\partial w}{\partial Y}&\text{for }\tau<-\tau_{c} \end{array} \right. \tag{4}\]  
where \(\tau\), \(\tau_{y}\) and \(\tau_{c}=\frac{\alpha_{\eta}}{\alpha_{\eta}-1}\tau_{y}\) are shear stress, yield stress and stress at the core boundary, respectively. \(\eta\) is the viscosity of MRF without external magnetic field. \(\alpha_{\eta}=\eta_{c}/\eta\) is the viscosity ratio in which \(\eta_{c}\) is the viscosity of core. Under the action of a magnetic field, MRF behaves as a non-newtonian fluid which contains yield zone and unyielding zone as shown in Figure 2. In the unyielding zone, MRF behaves as a solid. \(L\) is the axial length of MRD. \(Zc\) is the point in the \(Z\)-direction at which the core begins to fill the entire clearance of MRD. Figure 1: Schematic diagram of magnetorheological damper (MRD). Figure 2: State characteristic of magnetorheological fluid (MRF) in the clearance of MRD. Based on the bilinear constitutive law, a general mathematical model of MRD was proposed by [PERSON] et al. [22; 23; 24]. Using their method, the implicit Reynolds equation of MRD can be given as:  
\[\left\{ \begin{array}{rl}\frac{1}{\eta}(h^{3}p^{\prime}+3h^{2}x_{y}+8\frac{x_{c} ^{3}}{p^{\prime 2}}-12\frac{xy_{c}^{2}}{p^{\prime 2}})-\frac{8x_{c}^{3}}{p^{ \prime 2}}=12\dot{h}Z+C_{1}\quad\text{for}\quad\dot{h}<0\\ \frac{1}{\eta}(h^{3}p^{\prime}-3h^{2}\tau_{y}-8\frac{x_{c}^{3}}{p^{\prime 2}}+12 \frac{xy_{c}^{2}}{p^{\prime 2}})+\frac{8x_{c}^{3}}{\eta_{c}p^{\prime 2}}=12 \dot{h}Z+C_{2}\quad\text{for}\quad\dot{h}>0\end{array} \right.\quad Z_{c}<Z\leq L/2 \tag{5}\]  
where \(Z_{c}=-\frac{x_{c}\dot{h}^{2}}{6\eta_{c}\dot{h}}\) and \(p^{\prime}{}_{c}=-\frac{2x_{c}}{\dot{h}}\). \(\dot{h}\) is the first derivative of film thickness \(h\) with respect to time \(t\). \(p^{\prime}\) is the first derivative of film pressure \(p\) respect to \(Z\). By solving Equation (5), \(p^{\prime}\) can be obtained. Then integrating \(p^{\prime}\), film pressure \(p\) can be given as:  
\[p=\int\limits_{L/2}^{Z}p^{\prime}dZ+p_{A}\quad Z_{c}<Z\leq L/2 \tag{6}\]  
where \(p_{A}\) is the atmospheric pressure. When \(0\leq Z\leq Z_{c}\), the core fills the entire clearance of MRD and MRF behaves as a Newtonian fluid with viscosity \(\eta_{c}\).