# Development of a Novel Variable-Diameter Wheel
## 5 Modified PRBM and Validation
### Validation  
To validate the effectiveness of the modified PRBM, an application example is presented, in which only the contraction process was analyzed. This example clarifies the contraction process using seven loading cases. Figure 6 shows the initial position (i.e., the first case) of a variable-diameter wheel. The right wheel leg is presumed to touch the terrain and the angle between its \(Z\) and \(x\)-axis near the terrain is \(60^{\circ}\). Based on Section 3, the initial angle between the two cranks in the example is \(\alpha_{2}\) (\(67^{\circ}\)). Therefore, when the mechanism exits the initial position, \(\theta_{10}=93.5^{\circ}\) and \(\theta_{20}=26.5^{\circ}\). According to Equation (45), \(\theta_{30}\) can be obtained as  
\[r_{2}\times\sin\left(\frac{\theta_{10}-\theta_{20}}{2}\right)=r_{3}\times\sin \left(\theta_{30}-\frac{\theta_{10}+\theta_{20}}{2}\right)+\frac{r_{4}}{2}. \tag{58}\]  
The springs are made of steel (60 Si2 MnA). [PERSON]'s ratio is 0.3, and [PERSON]'s modulus is 206 GPa. Table 2 presents the geometric parameters of helical torsion springs [9]. Based on Equation (57) and the parameters in Table 2, the \(i\)th torsion spring constant \(k_{i}\) can be obtained. The concentrated forcesare hypothesized to be \(F_{N}=57.17\) N and \(F_{q}=40.02\) N [9; 33]. On the basis of these parameters, the deflections in the wheel caused by \(M_{1}\) and \(M_{2}\) can be calculated using Equations (13)-(49). To validate the modified PRBM, the finite-element software ABAQUS was used. The angles (\(\theta_{1}\) and \(\theta_{2}\)) were given, and the other five DOFs at the wheel center were fixed. Furthermore, \(F_{N}\) and \(F_{q}\) acted on the hinge points of the two wheel feet in the wheel leg in contact with the ground. Linear Timoshenko beam elements were applied to the springs. The three hubs and the 12 wheel feet in the wheel were treated as the rigid bodies. The contraction process of the wheel with six loading cases is described in Figure 6. For the first case, deflections existed only in the wheel leg in contact with the terrain. The wheel did not generate any mechanism movement. However, each wheel leg almost had the same deflections in the other loading cases, and the wheel was gradually contracting with the hub rotations. After departing from the initial position, \(M_{1}\) increased the rotation (\(\theta_{1}\)) of the left crank in a clockwise direction, whereas \(M_{2}\) decreased the rotation (\(\theta_{2}\)) of the right crank in the opposite direction. As a result, the difference between \(\theta_{1}\) and \(\theta_{2}\) gradually increased in the latter five cases (Figure 6), while the angle between the two spoke links gradually increased. Consequently, the wheel diameter decreased. Finally, the wheel was folded into a circular wheel. For the six load conditions (Table 3), the moments acting on the left and right cranks during the mechanism motion were calculated using Equations (13)-(49) and compared with the corresponding FEA results in Table 3. However, the modified PRBM was not suitable for the first case, which is a structural analysis and is not included in Table 3. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Values** \\ \hline \(\theta_{10}\) & 93.5\({}^{\circ}\) \\ \(D_{\text{eq}}\) & 182 mm \\ \(\theta_{20}\) & 26.5\({}^{\circ}\) \\ \(r_{1}\) & 97 mm \\ \(r_{2}\) & 97 mm \\ \(r_{3}\) & 140 mm \\ \(r_{4}\) & 140 mm \\ \(d\) & 4 mm \\ n\({}_{1}\) & 10 \\ n\({}_{2}\) & 6 \\ \(D\) & 14 mm \\ \(l\) & 140 mm \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters used in modified PRBM. Figure 6: Contraction process of wheel. The results in Table 3 indicate that the absolute values of the moments increased when the wheel folded. The elastic potential energy of these springs always increases due to wheel contraction, which can be attributed to the work of \(M_{1}\) and \(M_{2}\). Hence, the input moments during the mechanism movement should increase.