# A Novel Hierarchical Multi-Stable Cylindrical Structure with Superior Energy Trapping
## 2 Structural Design  
### Design of the Bi-Stable Unit Cell  
When a straight beam is subjected to axial compression, the structure undergoes a sudden transition from one stable state to another upon reaching the critical load. Therefore, establishing the buckling deformation mechanism of the curved beam model is of great importance. As illustrated in Figure 1, three typical buckling modes of a clamped beam can be observed.  
When both ends of the beam segment are constrained, the vertical displacement at the center of the curved beam is governed by the transverse force applied at its midpoint. According to the literature [32], the solutions corresponding to the three buckling modes are given as follows:  
\[F_{1}=\frac{3\pi^{4}Q^{2}}{2}\Delta\left(\Delta-\frac{3}{2}+\sqrt{\frac{1}{4}- \frac{4}{3Q^{2}}}\right)\left(\Delta-\frac{3}{2}-\sqrt{\frac{1}{4}-\frac{4}{3Q^ {2}}}\right) \tag{1}\]  
\[F_{2}=\frac{N^{2}\left(N_{2}^{2}-N_{1}^{2}\right)}{8}\left(\frac{N_{2}^{2}}{N_ {2}^{2}-N_{1}^{2}}-\Delta\right)=4.18\pi^{4}-2.18\pi^{4}\Delta \tag{2}\]  
\[F_{3}=\frac{N_{1}^{2}\left(N_{3}^{2}-N_{1}^{2}\right)}{8}\left(\frac{N_{3}^{2} }{N_{3}^{2}-N_{1}^{2}}-\Delta\right)=8\pi^{4}-6\pi^{4}\Delta \tag{3}\]  
In the expressions, \(F_{1}\) and \(\Delta\) denote the force and displacement variables, respectively. The geometric parameter \(Q=h/t\), where \(h\) and \(t\) are the height and thickness of the first buckling mode.  
As shown in Figure 2, the mechanical response curve \(F_{1}\) exhibits a cosinusoidal pattern. Its peak value varies with the geometric parameter \(Q\), indicating that the structure possesses nonlinear and variable stiffness characteristics. The slope of the curve undergoes a \"positive-negative-positive\" transition, which further confirms the presence of a stiffness variation pattern following the same trend. For \(Q>2.31\) (e.g., \(Q=3\)), the force curve shows negative values, indicating that the structure exhibits bi-stable behavior. In this case, the structure cannot spontaneously return to its original state after unloading and must be restored by applying a reverse external force. When \(Q=2.31\), the structure reaches a critically bi-stable condition, where the slope balances between positive and negative. In contrast, for \(Q<2.31\) (e.g., \(Q=1.67\)), the structure exhibits mono-stable behavior and is capable of automatically returning to its original configuration upon the removal of external force.  
\(F_{3}\) is tangent to the critical bi-stable curve \(F_{1}\) (\(Q=2.31\)) at \(\Delta=1\), intersects the bi-stable curve \(F_{1}\) (\(Q=3\)) at three distinct points, and intersects the mono-stable curve \(F_{1}\) (\(Q=1.67\)) only at \(\Delta=1\). Under this condition, the second-order asymmetric mode is effectively suppressed, resulting in a pronounced bi-stable effect. Inspired by previous studies [54; 55; 56; 57], an additional beam is incorporated at the midpoint of the cosinusoidal curved beam, as illustrated in Figure 3a. By applying a vertical load at the center of the beam, the structure transitions between two stable configurations. This geometric constraint transforms the second-order bending mode of the curved beam into axial deformation. The design not only prevents the occurrence of asymmetric torsional modes but also maintains geometric symmetry during loading, thereby improving the mechanical stability of the structure.  
Figure 1: Three buckling modes of a clamped beam.  
The predefined shape function of the cosinusoidal curved beam can be expressed as  
\[\omega(x)=\frac{h}{2}\Big{[}1-\cos 2\pi\frac{x}{l}\Big{]} \tag{4}\]  
The geometric parameters are defined as follows: \(l\) is the span of the curved beam, and \(h\) is the height of the cosinusoidal curved beam as shown in Figure 3b. \(t\) is the thickness of the beam. The width of the curved beam is denoted by \(b\). The proposed cell consists of a cosinusoidal curved beam and several straight supporting beams, as shown in Figure 3c, and is used to design the multi-stable structure.