Theory and Mechanisms of Mechanical Equilibrium Stability and Vibration Control in a Novel Tristable Meta-structure

• Proposes a novel axial tristable metastructure based on slender rod buckling. • Establishes a general-purpose analytical model with high accuracy for reliable design. • Constructs a universal phase diagram for precise parameter optimization and efficient engineering. • Derives a natural frequency equation for predictable and tunable vibration control. Multistable mechanical meta-structures offer promising potential for energy dissipation and vibration control. However, existing analytical models still face some essential challenges and difficulties, including (i) providing explicit local-design guidance, (ii) characterizing tristability, (iii) constructing general phase diagrams, and (iv) quantifying vibration-control performance. To address these challenges, this paper proposes a novel axial tristable metamaterial structure by exploring the controlled buckling of a slender rod. The analytical model is established through equilibrium-stability analysis and validated using an energy-based formulation, enabling direct interpretation of parameter responses for engineering design. By introducing a dimensionless representation, a general phase diagram is constructed to illustrate the tristable response and rapidly optimize the design under prescribed variables. In addition, a frequency expression is derived through dynamic analysis to realize vibration tuning. Finite element simulations show close agreement with theoretical predictions, which validates the accuracy and applicability of the proposed framework. The results provide a general theoretical basis and practical design guidance for developing high-performance self-centering dampers with programmable tristable behavior.