Improvement of Bandgap Properties in Finite Metamaterial Beam Structures by Local Damping Measures

Metamaterial structures are characterized by a periodic arrangement of the so-called unit cells. These can be of a wide variety of types, e.g., (alternating) geometry or material variations. A characteristic feature of these structures is that a bandgap can occur in a relatively wide frequency range. In this chapter, we investigate a metamaterial structure which consists of unit cells which are themselves resonators. When considering infinite unit cell repetitions, complex boundary conditions (Bloch–Floquet theorem) may be applied to a single unit cell to reduce the computational cost. By modeling an infinite structure, boundary reflections are not taken into account. However, these reflections play an important role regarding a real structure with a finite number of repetitions of the unit cell. In the present work we analyze the vibration behavior of beam-like 2D periodic metamaterial. The focus is on wave reflections and corresponding edge effects, which can be particularly pronounced in metamaterial structures. In the bandgap, there are vibration modes in which almost apparently just the edge regions of the entire structure oscillate, while the central part hardly shows any deflection. The local response at the edges occurs due to superposition of waves with different traveling directions. These vibration phenomena prevent direct transfer of metamaterial design based on infinite wave propagation analysis to finite structures. For metamaterial design, we propose a fast dynamic analysis of finite structures using state-of-the-art model order reduction techniques. This way the local vibration effects are included in the steady-state response and allow for vibration analysis of the real structure. Furthermore, we tackle the edge reflections in the real structure by local dissipation measures.