## 2 Theoretical Analysis and Design of the MRD-Based Roller Cage Shoe
### Mechanical Model of MRD-Based Roller Cage Shoes  
#### 2.2.1 Rheological Mechanical Model of MRDs  
Figure 4a illustrates a simplified model of the magnetic circuit within the MRD. The magnetorheological fluid damping buffer employs a shear valve mode, with the magnetic field's direction perpendicular to the MRF movement. This configuration creates a ring-shaped damping channel between the sides of the damper's piston rod and the inner wall of the cylinder. When the piston rod experiences force from the roller, it causes a change in the volume of the upper and lower fluid chambers due to axial movement. This prompts MRF to flow from the lower chamber into the upper chamber through the damping channel. Upon energizing the coil, electromagnetic induction principles come into play. This results in the generation of magnetic flux density across the resistive gap between the inner damping cylinder and the outer piston head. The applied magnetic field induces the magnetorheological effect, creating resistance to motion. This, in turn, hinders the movement of the piston, leading to effective vibration damping. Figure 4: Rheological mechanical model of the MRD. (**a**) Simplified model; (**b**) shear valve mode. Figure 3: Structure of an MRD-based buffer: 1. disc spring baffle; 2. sleeve; 3. damping baffle; 4. bottom cover; 5. cylinder barrel; 6. piston channel; 7. topside cover; 8. piston rod; 9. guide bar; 10. disc spring; 11. seal; 12. coil; 13. damping gap; 14. sealing assembly. The shear valve mode is a hybrid mode that combines shear and valve characteristics. In this mode, there is parallel relative motion between the upper and lower pole plates of the MRD. Simultaneously, the MRF in the gap between the pole plates flows in a direction parallel to the plates. At this juncture, the applied magnetic field's direction is perpendicular to both the motion of the pole plates and the flow of the MRF [17], as depicted in Figure 4b. The damping force of the MRD in this mode is the combined result of both the shear and valve components. Therefore, the damping force of the MRD can be mathematically represented as  
\[F_{1}=\frac{12\eta Lv}{\pi Dh^{3}}A_{p}^{2}+\frac{3l}{h}\tau_{y}A_{p} \tag{1}\]  
\[F_{2}=\frac{\eta bL}{h}v-2 bL\tau_{y} \tag{2}\]  
where \(F_{1}\) is the damping force provided in valve mode and \(F_{2}\) is the damping force provided in shear mode; \(\eta\) is the zero-field viscosity of the MRF; \(L\) is the effective length of the piston; \(A_{p}\) is the area of the piston; \(h\) is the damping channel gap between the piston and the cylinder; \(D\) is the outer diameter of the piston; \(\tau_{y}\) is the shear yield strength of the MRF; and \(v\) is the speed of movement of the piston relative to the cylinder. Superimposing the two equations gives the damping force in the shear valve mode as  
\[F=F_{\eta}+F_{\tau}=(\frac{12\eta LA_{p}^{2}}{\pi Dh^{3}}+\frac{\eta bL}{h})v+ (\frac{3 LA_{p}}{h}-2 bL)\tau_{y} \tag{3}\]  
where \(F_{\eta}\) is the velocity-dependent viscous damping force; \(F_{\tau}\) is the Coulomb damping force that varies with the magnetic field. Because of the small width of the damping channel, the damping force generated by the MRF during flow is much smaller than that due to shear, so in the preliminary design, the damping force in the mixed mode can be approximated using the following equation:  
\[F=F_{\eta}+F_{\tau}=\frac{12\eta LA_{p}^{2}}{\pi Dh^{3}}v+\frac{3L\tau_{y}A_{ p}}{h}\text{sgn}v \tag{4}\]  
The dynamic tunability factor can be approximated based on the ratio of the Coulomb force to the viscous force:  
\[\lambda=\frac{F_{\tau}}{F_{\eta}}=\frac{\tau_{y}Dh^{2}}{4\eta A_{p}v} \tag{5}\]  
The MRF used in the MRD designed in this paper is MRF-J25T, which has a large shear yield strength and good settling stability. The basic parameters are shown in Table 1.  
The relationship between the shear yield stress (\(\tau\)) and the magnetic field strength, \(B\), is shown in Figure 5, which shows nonlinearity.