## 3 Design Process of Single Suspension Unit  
### Parameters of the Bogie Arm  
As mentioned earlier, the bogie arm features the length of \(BC\) and \(CD\) links, the angle \(\gamma\), and the two-dimensional position of the connected point \(C(x_{C},z_{C})\), which strongly depends on the working limitations of the contact point. It is essential to first determine the range of the contact point corresponding to the lowest and highest altitudes of the freewheel. It is easy to see that the determination of parameters for the middle suspension unit is the simplest. In fact, it can be seen from Figure 8 that the lowest altitude of \(G_{0}\) is the minimum value of the curve \(\Delta\), defined as \(z_{G_{0}}=f(x_{G_{0}})\) with \(f(x)\triangleq a_{n}x^{n}+\ldots+a_{1}x+a_{0}\). As a result, the value of \(x_{G_{0}}\) must satisfy the following condition:  
\[f^{\prime}(x_{G_{0}})\equiv\sum_{i=0}^{n-1}a_{n-i}(n-i)x_{G_{0}}^{n-i-1}=0 \tag{2}\]  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Description** \\ \hline \(L\) & Displacement of the compressed spring \\ \(k\) & Spring stiffness \\ \(L_{0}\) & Basis length of compressed spring \\ \(N_{a}\) & Number of turns of compressed spring \\ \(D_{c}\) & Mean diameter coil of compressed spring \\ \(d\) & Wire diameter of compressed spring \\ \(l_{B}C\) & Length of the link \(BC\) \\ \(l_{C}D\) & Length of the link \(CD\) \\ \(\gamma\) & Angle formed by the links \(BC\) and \(CD\) \\ \(l_{d}\) & Distance between two adjacent suspensions \\ \hline \hline \end{tabular}
\end{table}
Table 2: Parameters for the design process of single suspension unit.  
Figure 7: Comparison of contact points: (**a**) Practice; (**b**) equivalent model.  
where \(f^{\prime}(\square)\) denotes the first-time derivative of the function \(f(\square)\) along with its variable.  
The lowest altitude (\(z_{D_{0}}^{l}\)) and highest altitude (\(z_{D_{0}}^{h}\)) of the center axis of the freewheel \(D_{0}\) are determined as follows:  
\[\begin{cases}z_{D_{0}}^{l}=-R-h_{1}\\ z_{D_{0}}^{h}=h_{2}-R\end{cases} \tag{3}\]  
Let \(\gamma_{1}\) and \(\gamma_{2}\) be the angles formed by the BC and CD links of the suspension unit with the horizontal axis, introducing the constraint \(\gamma=\gamma_{1}+\gamma_{2}\), which depends on the position of the freewheel. Referring to [14] about the golden ratio suggested for the design of the tank suspension, denoted as \(\varepsilon=2.75\), in this case, the golden ratio defined as the division of the vertical travel of the bogie arm by its length projected to the \(z\)-axis. Consequently, the constraint of \(l_{BC}\), \(l_{CD}\), \(\gamma_{1}\), and \(\gamma_{2}\) is expressed as follows:  
\[\varepsilon=\frac{l_{BC}\sin  
the theory of plane geometry, the lower limitation of the length \(l_{d}\) must satisfy the condition of Equation (8) to ensure that the collision of two adjacent suspension units does not occur.  
\[l_{d}>l_{CD}\left[\cos\left(\sin^{-1}\left(\frac{z_{C_{1}}-z_{D_{1}}^{l}}{l_{CD} }\right)\right)-\cos\left(\sin^{-1}\left(\frac{z_{C_{0}}-z_{D_{0}}^{h}}{l_{CD}} \right)\right)\right]-2r \tag{8}\]  
where \(z_{D_{1}}^{l}\) is the largest vertical coordinate of the point \(D_{1}\). In addition, the length of mounting space of the suspension units (\(l_{l}\)), as shown in Figure 9b, must be less than the free space between the two pulleys (\(l_{l}<l_{w}-2R\)). Consider the case of the freewheels in the highest position, i.e., the extended stroke of the bogie arms exceeds farthest. The upper limitation of the length \(l_{d}\) must then satisfy the following constraint:  
\[l_{d}<\frac{l_{w}-2R-r-\cos\left(\sin^{-1}\left(\frac{z_{C_{0}}-z_{D_{0}}^{h}} {l_{CD}}\right)\right)l_{CD}}{n-1} \tag{9}\]  
where \(n\) is the total number of suspension units.