# A Novel Hierarchical Multi-Stable Cylindrical Structure with Superior Energy Trapping
## 1 Introduction  
Mechanical metamaterials [1, 2] are composed of periodically arranged building blocks and carefully designed unit cells, whose properties primarily derive from structural geometry rather than the inherent composition of the base materials [3, 4]. This characteristic enables them to achieve functionalities beyond the reach of traditional materials, including negative stiffness [5, 6], a negative Poisson's ratio [7, 8], negative thermal expansion [9], and programmability [10, 11]. As an important branch of mechanical metamaterials, multi-stable metamaterials have gained widespread attention for their superior performance in energy trapping [12, 13, 14], vibration control [15], wave manipulation [16], and smart deformable structure [17, 18]. Multi-stable metamaterials typically originate from the arrangement and configuration of specific units, such as bi-stable beams [19, 20], origami structures [21, 22, 23, 24], composite laminates [25, 26], or flexible units [27, 28]. Among these, bi-stable curved beams exhibitingelastic instability are the most typical negative stiffness unit, rapidly switching between two stable states under external loading, thereby enabling energy absorption and release [29]. Based on load-displacement curves, curved beams can be classified into two types: multi-stable and mono-stable [30; 31]. Bi-stable curved beams maintain their deformed shapes at a local minimum of elastic energy, requiring no external energy input, whereas mono-stable curved beams spontaneously revert to their original configuration once external loading is removed. By adjusting geometric parameters (e.g., thickness, span, and amplitude) [32; 33; 34], researchers can effectively regulate the buckling behavior and mechanical response of these beams, providing extensive possibilities for constructing multi-stable structures. In recent years, numerous studies have conducted in-depth analyses and experiments on the mechanical properties of one-dimensional, two-dimensional, and three-dimensional bi-stable units from various perspectives, including mechanical response [35], thermal behavior [36], dynamic behavior [37], magnetic multi-stable metamaterials [38], and deformation applications [39]. These investigations reveal that geometric modifications [40], Young's modulus variations [41], temperature dependencies [42], and fiber reinforcement [43] can significantly influence the mechanical properties of planar bi-stable structures. Owing to their unique nonlinear characteristics and high-efficiency energy trapping capabilities, multi-stable structures find broad applications in aerospace [44] and automotive safety and protective systems [45; 46]. For instance, they can serve as deployable systems and high-energy-trapping components in aerospace [47] and as efficient cushioning and collision protection devices in automotive applications [48]. Despite recent advances in multi-stable design, traditional multi-stable cylindrical structures (TMCSs) based on curved beam configurations still exhibit critical limitations that hinder their broader application in engineering contexts. Firstly, many TMCS designs are planar or unidirectional in deformation [49; 50], restricting their suitability for complex, spatially reconfigurable systems. Secondly, the energy trapping capacity is often constrained by single-layer bi-stable configurations, leading to insufficient energy dissipation under high impact conditions [51]. Additionally, their snap-through responses are typically fixed and untunable, and the lack of structural hierarchy reduces adaptability and system-level integration. Moreover, purely elastic deformation mechanisms in traditional TMCSs often result in limited energy absorption per unit mass, especially at elevated strain rates [52; 53]. The present study proposes a hierarchical multi-stable cylindrical structure (HMCS). This novel design aims to simultaneously improve load-bearing capacity, energy trapping performance, and configurational tunability.