# A Novel Hierarchical Multi-Stable Cylindrical Structure with Superior Energy Trapping
## 5 Results and Discussion  
### Energy Trapping and Load-Bearing Behavior  
The trapping energy mechanism of multi-stable cylindrical structures is illustrated in Figure 8a. Due to the bi-stable nature of multi-stable cylindrical structures, a portion of the energy is stored within the elements during the transition between stable states. Therefore, the concept of energy trapping (ET) is employed to evaluate the structure's ability to retain the absorbed energy. Energy absorption [52] refers to the total mechanical work exerted on the structure during the loading process and is typically associated with mono-stable structures. In contrast, energy trapping specifically quantifies the portion of energy that remains stored in the system, which is particularly relevant to bi-stable and multi-stable systems. Figure 8: _Cont._  
Figure 7: Comparison of simulation and experimental results between HMCS and TMCS. In order to assess the structure's ability to trap energy, the force-displacement curve is integrated along the displacement axis, from which two energy regions, \(E_{in}\) and \(E_{out}\), are obtained. These two regions represent the energy absorbed and released by the structure during the transition between stable states. In the following analysis, the trapped energy refers to the difference between these two quantities, specifically \(E_{in}\) minus \(E_{out}\). The area under the curve can be calculated using the following Equation:  
\[ET=\int_{0}^{d}f(x)dx \tag{5}\]  
\[SET=\frac{\int_{0}^{d}f(x)dx}{M} \tag{6}\]  
In the Equation, \(ET\) denotes the total energy trapping, \(SET\) represents the specific energy trapping, which is the energy trapping per unit mass, \(x\) is the displacement variable, \(d\) is the maximum displacement, \(f(x)\) is the external force corresponding to the displacement \(x\), and \(M\) is the mass of the multi-stable structure. The energy trapping performance of the HMCSs was compared with that of the TMCSs with only an outer ring, as reported in reference [59], as shown in Figure 4. We conducted a comparative study on four different cases with varying structural parameters of the HMCSs and TMCSs. The detailed configurations of the four cases are listed in Table 1.  
The specific capacity (\(SC\)), defined as \(SC=P/M\) (where \(P\) is the peak load and \(M\) is the structural mass) [52], is adopted in this study as a standard metric for evaluating the load-carrying capability of energy-absorbing structures. As shown in Figure 8b, at \(Q=3.5\), the \(SC\) value of the TMCS is 1.84, whereas that of the HMCS reaches 4.06. At \(Q=5\), the SC increases to 2.96 for the TMCS and 4.51 for the HMCS. These results indicate that across different height-to-thickness ratios, the HMCS consistently exhibits significantly \begin{table}
\begin{tabular}{c c c c c} \hline  & **Case** & \(\mathbf{h}\) & \(\mathbf{t}\) & \(\mathbf{Q}\) \\ \hline Hierarchical multi-stable & 1 & 7 mm & 2 mm & 3.5 \\ cylindrical structure & 2 & 10 mm & 2 mm & 5 \\ Traditional multi-stable & 3 & 7 mm & 2 mm & 3.5 \\ cylindrical structure & 4 & 10 mm & 2 mm & 5 \\ \hline \end{tabular}
\end{table} 
Table 1: Structural parameters of the four cases. Figure 8: (**a**) Typical force–displacement relationship of mono-stable and bi-stable structures; (**b**) force–displacement curves of the four cases; (**c**) trapped energy; (**d**) specific trapped energy. higher SC values than the TMCS, clearly demonstrating its superior load-carrying capacity. Furthermore, from a mechanical perspective, the slope of the force-displacement curve at any given point corresponds to the stiffness of the structure at that deformation state. As illustrated in Figure 8b, the HMCS consistently exhibits steeper slopes compared to the TMCS across the entire loading process, indicating significantly higher structural stiffness. This further confirms the superior load-bearing characteristics of the HMCS design. The total trapped energy and specific trapped energy results are presented in Figure 8c,d, respectively.