# A Novel Hierarchical Multi-Stable Cylindrical Structure with Superior Energy Trapping
## 5 Results and Discussion
### Tuning Snap-Through Sequence  
As illustrated in Figure 11, the HMCS contains four layers, from top to bottom, named as Layer I, Layer II, Layer III, and Layer IV. The corresponding thicknesses of each layer are denoted as \(t_{1}\), \(t_{2}\), \(t_{3}\), and \(t_{4}\). To investigate the effect of different thickness configurations on the snap-through sequence, two configurations were designed and tested, with varying values of thickness \(t\). The detailed structural configurations are listed in Table 2.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Case** & \(\mathbf{t_{1}}\) & \(\mathbf{t_{2}}\) & \(\mathbf{t_{3}}\) & \(\mathbf{t_{4}}\) \\ \hline
1 & 1.3 mm & 1.2 mm & 1.1 mm & 1.0 mm \\
2 & 1.2 mm & 1.3 mm & 1.1 mm & 1.0 mm \\ \hline \hline \end{tabular}
\end{table}
Table 2: Structural parameters of the two cases.  
Figure 10: (a) Relationship between the peak force and the geometric parameters t and h; (b) relationship between the valley force and the geometric parameters t and h; (c) relationship between the energy trapping performance and the geometric parameters t and h.  
When the HMCS is loaded, the snap-through response sequence occurs as shown in Figure 11, where (a) corresponds to Case 1 and (b) corresponds to Case 2. The results show that when the HMCS begins to compress, the layer with the lowest thickness begins to deform first. Once this layer is fully deformed, the remaining layers are captured sequentially based on the increasing thickness of their respective beams. As shown in Figure 12, the four local maxima and minima in the force-displacement curve correspond to the four curved beams with different thicknesses within the multi-layer HMCS. The deformation process of each layer can be divided into three stages, consistent with the structural response described in Section 2. Initial deformation occurs during the first stage, where the structure primarily experiences combined compression and bending. The required force rapidly increases with increasing displacement. Hence, this stage is referred to as the positive stiffness stage. In the second stage, the structure mainly undergoes compression deformation, and the force-displacement curve exhibits elastic instability, revealing a negative stiffness effect. Thus, this stage is defined as the negative stiffness stage. The third stage marks the densification of the structure, during which the curved beams gradually become compacted. As displacement further increases, the required force rises again, and the structure once more exhibits positive stiffness behavior.  
Figure 11: Snap-through sequences of two HMCSs.  
The deformation behavior of each layer in the structure corresponds to the elastic instability phenomenon occurring in the curved beams, causing each layer to exhibit a negative stiffness effect. During compression, deformation initially occurs in the thinnest layer (1.0 mm), exhibiting a peak force of 59.3 N, and subsequently propagates to the thickest layer (1.3 mm), reaching a peak force of 91.1 N. As shown in Figure 12, the experimental results agree well with the finite element simulation results, demonstrating similar trends and characteristic peak forces. The overall consistency validates the accuracy of the finite element analysis. From the two structures tested above, it can be learned that the thickness of the curved beams influences the sequence of the snap-through between each layer, so the thickness of the curved beams can be adjusted to tune the snap-through behaviors of the overall structure to obtain metamaterials with customizable properties.