# Evaluation of Mechanical Performance of a New Disc Spring-Cable Counter Pressure Shock Absorber
## 3 Numerical Simulation and Modal Validation
### Model Validation  
Figure 7 and Table 3 show the results of the DSCCP shock absorber model under the two static loadings. It can be seen that the relationship is basically proportional between loading and deformation under linear static loading. When the maximum deformation of 44.8 mm is reached, the flattening load is 643,225 N, which has an error of only 0.59% with the theoretical calculation. When sin-function static loading is adopted, there is a phenomenon of non-coincidence of loading and unloading. The reason is that there is friction behavior between the disc spring plates, as well as between the disc spring plates and the upper and lower floating pressure plates. This frictional behavior is the main contributions for the energy consumption of disc springs. Under this static loading, the flattening force at maximum deformation is 635,456 N, which has an error of 1.82% with the theoretical calculation. Therefore, the effectiveness and rationality of the numerical model of the DSCCP shock absorber can be guaranteed. \begin{table}
\begin{tabular}{c c c c c c} \hline
**Loading** & \multirow{2}{*}{\(\mu_{1}\)} & \multirow{2}{*}{\(\mu_{2}\)} & **Maximum Deformation** & **Numerical Simulation I (kN)** & **Theoretical Calculation II (kN)** & **Error Calculation II (I \(-\) II)/I \(\times\) 100\%** \\ \hline \multirow{6}{*}{Linear loading} & \multirow{6}{*}{0.005} & 0.2 & 44.8 mm & 727 & 647 & 11.01\% \\ \cline{3-6}  & & 0.3 & 44.8 mm & 678 & 647 & 4.58\% \\ \cline{3-6}  & & 0.4 & 44.8 mm & 685 & 647 & 5.48\% \\ \cline{3-6}  & & 0.5 & 44.8 mm & 732 & 647 & 11.60\% \\ \cline{3-6}  & & 0.6 & 44.8 mm & 736 & 647 & 12.05\% \\ \cline{3-6}  & & 0.2 & 44.8 mm & 670 & 647 & 3.33\% \\ \cline{3-6}  & & 0.3 & 44.8 mm & 675 & 647 & 4.14\% \\ \cline{3-6}  & & 0.4 & 44.8 mm & 681 & 647 & 5.04\% \\ \cline{3-6}  & & 0.5 & 44.8 mm & 688 & 647 & 5.97\% \\ \cline{3-6}  & & 0.6 & 44.8 mm & 696 & 647 & 7.51\% \\ \hline \end{tabular}
\end{table} 
Table 2: Flattening force comparisons for DSCCP shock absorber with different friction coefficient \(\mu_{2}\). Figure 6: Relationship between force and deformation under sin-function loading. Stress contours of the profile and top surface of the DSCCP shock absorber model are shown in Figure 8 under the deformations of \(0.75h_{0}=33.6\) mm and \(h_{0}=44.8\) mm, respectively. It can be seen that the stress contours of the disc springs are basically the same between the two static ladings. The maximum stress occurs at the upper inner edge of disc spring, where the stress gradually decreases from inside to outside. The stress contours of the top plate are relatively consistent and uniform. Under the vertical deformation of 33.6 mm, the maximum and minimum stress values are 96.75 MPa and 0.044 MPa for the linear loading, 93.22 MPa and 0.042 MPa for the sin-function loading. Under the vertical deformation of 44.8 mm, the maximum and minimum stress values are 114.2 MPa and 0.32 MPa for the linear loading, 114.1 MPa and 0.42 MPa for the sin-function loading. It can be concluded that differences of the stresses between the two loadings are very small, supporting the rationality of the numerical model. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Loading** & \multirow{2}{*}{\(\mathbf{\mu_{1}}\)} & \multirow{2}{*}{\(\mathbf{\mu_{2}}\)} & **Maximum** & **Numerical** & **Theoretical** & **Error** \\
**Type** & & & **Deformation** & **Simulation/I** & **Calculation/II** & **\(\mathbf{|(I-II)/I|\times 100\%}\)** \\ \hline Linear & \multirow{2}{*}{0.005} & \multirow{2}{*}{0.3} & \multirow{2}{*}{44.8 mm} & \multirow{2}{*}{643,225 N} & \multirow{2}{*}{647,028 N} & \multirow{2}{*}{0.59\%} \\ loading & & & & & & \\ \hline Sin-function & \multirow{2}{*}{0.005} & \multirow{2}{*}{0.3} & \multirow{2}{*}{44.8 mm} & \multirow{2}{*}{635,456 N} & \multirow{2}{*}{647,028 N} & \multirow{2}{*}{1.82\%} \\ loading & & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Flattening force comparison between numerical simulation and theoretical calculation.