## 4 Dynamic Modeling and Test Verification  
### Dynamic Modeling and Verification of Leaf Springs  
Adams is recognized as a professional software for the vehicle dynamics modeling. The dynamic models based on the software can truly reflect the dynamic performance of vehicles [38; 39]. Adams/Leaf tool uses discrete beam elements to simulate leaf springs. In other words, each piece of leaf springs is divided into several sections, which are connected by flexible beams without mass. The flexible beam is a kind of flexible connection. The force and moment between two marked points of two elements are calculated by the theory of Timoshenko beam [40]. In this study, according to the structural parameters values of the multi-leaf spring and the taper-leaf spring, the simulation models were established based on Adams/Leaftool software, as shown in Figure 5. Because the friction force between the adjacent pieces is very small [37], it is ignored in the simulation models. Moreover, the previous tests in [37]  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Parameter** & **Value** & **Unit** & **Parameter** & **Value** & **Unit** \\ \hline \(n\) & 3 & pieces & \(h_{11}\) & 5.6 & mm \\ \(\sigma_{\text{max}}\) & 460.2 & MPa & \(h_{21}\) & 5.6 & mm \\ \(b\) & 63.0 & mm & \(h_{31}\) & 5.6 & mm \\ \(l_{1}\) & 123.5 & mm & \(h_{12}\) & 11.0 & mm \\ \(l_{2}\) & 475.0 & mm & \(h_{22}\) & 11.0 & mm \\ \(show that the load-stiffness characteristics for multi-leaf springs and taper-leaf springs are almost linear. Thus, their vertical stiffness and the torsional stiffness are often used to verify the model accuracy [9].  
To verify the accuracy of the models, the vertical and torsional stiffness tests were conducted. The test equipment is the TYE-W400I hydraulic testing machine produced by Jinan Shidai Shijin Group, as shown in Figure 6. It uses the programmable logic controller. It can measure the load and the displacement through sensors. The load was measured by a NS-WL1 tension-compression sensor. The displacement was measured by a SDVG20 displacement sensor. The control system automatically collected and processed the test force and the deformation of leaf springs, and directly output the stiffness value through the microcomputer display screen. For the models in Figure 5, the vertical forces were applied to simulate the deformation, respectively. The simulated stiffness values from the simulation results were calculated. A comparison of the stiffness values is shown in Table 2.  
From Table 2, it can be seen that the vertical stiffness values of the multi-leaf spring and the taper-leaf spring are almost the same. The result shows that the taper-leaf spring can equivalently replace the original multi-leaf spring. From Table 2, it also can be seen that the torsional stiffness of the multi-leaf spring is obviously larger than that of taper-leaf spring. Moreover, the test values are close to the simulation values. The results show that the simulation models can effectively reflect the stiffness characteristics of the multi-leaf spring and the taper-leaf spring, respectively. In addition, the experimental results indicate that the weight of the taper-leaf spring is less 13.4% than that of the multi-leaf spring.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{3}{c}{**Vertical**} & \multicolumn{3}{c}{**Torsional**} \\ \cline{2-7}
**Type** & **Test (kN/m)** & **Simulation (N/m)** & **Error (\%)** & **Test (kN/m/rad)** & **Simulation (kN/rad)** & **Error (\%)** \\ \hline The multi-leaf spring & 112.3 & 109.0 & 2.9 & 10.6 & 10.1 & 4.7 \\ The taper-leaf spring & 113.1 & 109.2 & 3.5 & 8.4 & 8.2 & 2.4 \\ \hline \hline \end{tabular}
\end{table}
Table 2: A comparison of the stiffness values.  
Figure 5: The simulation models: (**a**) The multi-leaf spring; (**b**) the taper-leaf spring.  
Figure 6: The test equipment.