## 2 Theoretical Analysis and Design of the MRD-Based Roller Cage Shoe
### Mechanical Model of MRD-Based Roller Cage Shoes
#### 2.2.2 Calculation of Disc Spring Stiffness  
Disc springs, serving as prevalent non-linear vibration-damping elements, possess the capability to bear a substantial load with minor deformation. Disc springs in use often need more than one disc spring in a certain combination, mainly in the forms of pairs, stacks, and their composite form. For the design of the disc spring structure in the magnetorheological damping buffer proposed in this paper, considering the working environment of the disc spring, the structure of the space, and size requirements, a series of disc springs with the material of 60 Si2 MnA was selected. The selected disc spring parameters are shown in Table 2.  
Currently, the load capacity, \(F\), and stiffness, \(F^{\prime}\), of disc springs are usually calculated using the approximation provided by the Almen-Laszlo formula in the national standard. This method was developed by [PERSON] and [PERSON] [18] in 1936, based on the assumption of a curved beam, as a formula for calculating the characteristics of a butterfly spring with a uniform cross-section. After many years of practical testing, the method has proved to be relatively stable and reliable in general engineering applications and is therefore widely used in several international and industry standards. The A-L formula ignores the effects of frictional radial stress, friction, contact nonlinearity, etc. The relationship between load and deformation can be calculated according to the geometrical parameters and material properties of the disc-shaped single spring. The advantage of this formula is that it is \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Symbol** & **Dimensions** \\ \hline External diameter & \(D\) & 56 mm \\ Inside diameter & \(d\) & 28.5 mm \\ Thicknesses & \(t\) & 3 mm \\ Free height & \(H_{0}\) & 4.3 mm \\ Extreme travel & \(h_{0}\) & 1.3 mm \\ Density & \(\rho\) & 7850 kg/m\({}^{3}\) \\ Modulus of elasticity & \(E\) & 206 GPa \\ Poisson’s ratio & \(\mu\) & 0.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters of disc springs. Figure 5: The characteristic curve of MRF-J25T: (**a**) \(\tau\)-B curve; (**b**) H-B curve. simple and easy to use, and it is suitable for general engineering applications with small deformations, simple structures, and low accuracy requirements. Therefore, this formula is chosen for calculation based on the operation of the MRD. \[F^{\prime}=\frac{4E}{1-\mu^{2}}\cdot\frac{t^{3}}{K_{1}D^{2}}\cdot K_{4}^{2}\left\{ K_{4}^{2}[{(\frac{h_{0}}{t})}^{2}-3\cdot\frac{h_{0}}{t}\cdot\frac{f}{t}+\frac{3}{2 }{(\frac{f}{t})}^{2}]+1\right\} \tag{7}\]  
\[F=\frac{4E}{1-\mu^{2}}\cdot\frac{t^{4}}{K_{1}D^{2}}\cdot K_{4}^{2}\cdot\frac{f }{t}[K_{4}^{2}\cdot(\frac{h_{0}}{t}-\frac{f}{t})(\frac{h_{0}}{t}-\frac{f}{2t}) +1] \tag{8}\]  
where  
\[K_{1}=\frac{1}{\pi}\cdot\frac{[(C-1)/C]^{2}}{(C+1)/(C-1)-2/\ln 2} \tag{9}\]  
\[K_{4}=\sqrt{-\frac{C_{1}}{2}+\sqrt{(\frac{C_{1}}{2})}^{2}+C_{2}} \tag{10}\]  
\[C=\frac{D}{d} \tag{11}\]  
\[C_{1}=\frac{(t^{\prime}/t)^{2}}{[(1/4)\cdot(H_{0}/t)-t^{\prime}/t+3/4][(5/8) \cdot(H_{0}/t)-t^{\prime}/t+3/8]} \tag{12}\]  
\[C_{2}=\frac{C_{1}}{(t^{\prime}/t)^{3}}[\frac{5}{32}(\frac{H_{0}}{t}-1)^{2}+1] \tag{13}\]  
where \(t^{\prime}\) is the thinning thickness of the disc spring with supporting surfaces. The springs involved in this paper have supporting surfaces, so there are \(t=t^{\prime}\). The combination of disc springs selected in this paper is in the form of pairs, as shown in Figure 3, which consists of five pairs of disc springs of the same specification. In the absence of friction, there are  
\[F_{z}=F \tag{14}\]  
Figure 5:0  
Figure 5:1  
From the above formula, it can be seen that the disc spring pairs are similar to the series form of the spring. The load capacity, \(F_{z}\), of the combined disc spring is equal to the load capacity, \(F\), that each disc spring can provide, and the deflection, \(f_{z}\), is the sum of the deflections, \(f\), of all the constituent disc springs.