## 3 Design Process of Single Suspension Unit
### Coordinate of Connected Point \(A\) and Length \(l_{AB}\)  
Referring to assumption (2), determining the coordinates of connection point \(A\) at any suspension unit is sufficient to achieve the coordinates for all other connected points \(A\). Because the parameters of the middle suspension unit are clearly defined in the preceding sections, it is treated as a representative position to design connected point \(A\) in this section. The location of connection point \(A\) is determined to satisfy the following two conditions:  
* The operation of the bogie arm produces the greatest reaction at the freewheel;
* The coordinates of point \(A\) are in the mounting area of the track-wheeled. In the first condition, the highest reaction force at the freewheel occurs when the moment of the center \(C\) is at its maximum. To achieve this, the direction of the spring compression force vector must be perpendicular to the \(BC\) link. Therefore, it is necessary to form the \(AB\) link to be perpendicular to the \(BC\) link within the working area of the bogie arm. Additionally, this ensures that the angle \(\widehat{ABC}\) remains at approximately 90 degrees even when the bogie arm is in other positions, thereby maximizing the moment magnitude. In the second condition, the location of connection point \(A\) must be guaranteed to be below the upper branch of the timing belt. Figure 10 illustrates the case where the position of \(A\) has satisfied two conditions, i.e., the moment at which the \(AB\) link is perpendicular to the \(BC\) link (1 st condition) and \(A\) is in the mounting area (2 nd condition). Let the length \(l_{A_{0}B_{0}}\) be the distance from the connected point \(A_{0}\) on the whole frame to the connected point \(B_{0}\) on the bogie arm. It is equivalent to the sum of the compression spring length, the length of the slide tube, and the piston rod. Applying the theory of plane geometry to Figure 10, the following inequality must occur:  
\[z_{B_{0}}+l_{A_{0}B_{0}}\sin(90-\gamma_{1})<R \tag{10}\]  
Figure 9: Design parameters of the bogie arm: (**a**) the most probable collision scenario for two adjacent suspension units when the freewheels are at their lowest and highest positions; (**b**) the length of mounting space of the suspension units. The angle \(\gamma_{1}\) specifies the position of the bogie arm such that the angle \(\widehat{ABC}\) is equal to 90 degrees. The value of \(z_{B_{0}}\) is the vertical coordinate of \(B_{0}\) determined through the constraint with the connected point \(C_{0}\) using the following equation with the index (\(j=0\)):  
\[ \begin{cases}x_{B_{j}}=x_{C_{j}}+l_{CB}\cos(\gamma_{1})\\ z_{B_{j}}=z_{C_{j}}+l_{CB}\sin(\gamma_{1})\end{cases}  \tag{11}\]  
Substitute Equation (11) into Equation (10) to specify the preliminary of \(l_{A_{0}B_{0}}\) and \(\gamma_{1}\). The coordinates of the point \(A_{0}\) are determined as follows:  
\[ \begin{cases}x_{A_{0}}=x_{B_{0}}-l_{A_{0}B_{0}}\cos(90-\gamma_{1})\\ z_{A_{0}}=z_{B_{0}}+l_{A_{0}B_{0}}\sin(90-\gamma_{1})\end{cases}  \tag{12}\]  
The coordinates of connected points \(A\) belong to the other suspension units are determined as follows:  
\[ \begin{cases}x_{A_{j}}=x_{A_{0}}+jl_{d}\\ z_{A_{j}}=z_{A_{0}}\end{cases}  \tag{13}\]  
Determination of the coordinates of the other points \(A\) using Equation (13) ensures that the second condition remains satisfied in the remaining suspension units. Furthermore, the first condition is guaranteed due to the bolt-nut adjustment. The motion of the freewheel represents the working area of the single suspension unit. Therefore, the length \(l_{AB}\) was calculated based on the position of the freewheel as follows:  
\[l_{AB_{j}}=\sqrt{(x_{A_{j}}-x_{B_{j}})^{2}+(z_{A_{j}}-z_{B_{j}})^{2}} \tag{14}\]  
where \(l_{AB_{j}}\) is length \(l_{AB}\) of the \(j^{th}\) suspension unit.