# A Novel Hierarchical Multi-Stable Cylindrical Structure with Superior Energy Trapping
## 5 Results and Discussion
### The Effect of Structural Parameters  
Previous studies [59] indicated that the mechanical performance of multi-stable structures is closely related to their geometric parameters. Therefore, the effects of height \(h\) and thickness \(t\) were investigated separately.  
As illustrated in Figure 9a,b, the peak force exhibits an increasing trend with the curved beam height \(h\), indicating enhanced structural stiffness. Notably, when \(h=6\), the structure demonstrates no energy trapping capability, suggesting the absence of bi-stability. In contrast, for \(h>7\), the structure not only exhibits bi-stable behavior but also shows a marked increase in trapped energy with further increases in \(h\). As shown in Figure 9c,d, both the peak force and energy trapping capacity increase significantly with the thickness \(t\). Additionally, thicker curved beams tend to exhibit mono-stable behavior, thereby lacking effective energy trapping capability, as exemplified by the HMCS with \(t=2.5\). Conversely, thinner curved beams, such as those with \(t=1\), 1.5, and 2, display multi-stability. Therefore, optimal multi-stability in HMCSs requires a critical thickness parameter \(t\).  
Figure 9: _Cont._  
In summary, height \(h\) and thickness \(t\) are key parameters influencing the mechanical performance of HMCSs. It is crucial to consider their synergistic effects in the design and application stages to identify boundaries between different response types.  
To investigate the combined effects of height \(h\) and thickness \(t\) and determine the boundaries of different response types, simulations of the HMCSs were performed across a range of geometric parameters: thickness \(t\) varied from 1 to 2.2 with an increment of 0.1, and height \(h\) varied from 6 to 10 with an increment of 0.1. The parameter \(l\) was fixed at 20 mm.  
The peak load, representing the maximum load before structural snap-through, typically characterizes the structure's ability to resist external loading, i.e., initial stiffness. As seen in Figure 9(a), the peak load of the HMCSs increases with increases in parameters \(h\) and \(t\). Within the selected parameter range, the maximum peak force is approximately 20.7 times greater than the minimum.  
Valley load responses are presented in Figure 9(b). The valley load corresponds to the minimum load after structural snap-through and is commonly used to identify the bi-stable behavior of negative-stiffness structures. Structures exhibiting negative valley loads are typically considered bi-stable. In Figure 9(b), positive and negative valley loads are separated by a red curve, with the negative region shown in yellow-white to the right of this boundary. It can be seen that the valley load exhibits a relatively simple relationship with the height \(h\): the load decreases as \(h\) increases. In contrast, the relationship between the valley load and thickness \(t\) is more complex. Within the bi-stable region, the valley load first decreases and then increases with increasing \(t\). Conversely, in the mono-stable region, the valley load continuously decreases as \(t\) increases.  
Figure 9(c) shows the relationship between energy trapping ability and parameters \(t\) and \(h\) for the HMCSs. It can be seen that the structure's energy trapping ability increases with higher \(t\) and \(h\) values, and parameter \(t\) exerts a more pronounced influence on energy trapping compared to parameter \(h\). Maximum energy trapping occurs near the boundary between response types. Although the boundary between mono-stable and bi-stable response types is not constant, the critical parameter ratio remains relatively consistent around \(h/t\approx 3.69\), serving as a reasonable criterion for response type selection.  
Figure 9: Effect of height and thickness on load–displacement curves, peaks, and energy absorption.