Flexural Frequency Bandgaps in a Foldable Metamaterial Structure

Abstract

This manuscript investigates the flexural wave propagation behavior of a foldable metamaterial structure. Origami-inspired foldable structures are making inroads into many engineering applications — deployable solar cell arrays, foldable telescope lenses, foldable automotive airbags, to name a few; driven primarily by some of the remarkable mechanical properties (high stiffness, negative Poisson’s ratio, bistability etc.) of these structures. The chief motivation of this research is a comprehensive analysis of flexural wave propagation in such foldable structures. The repeating unit cell of the structure consists of an Euler-Bernoulli beam and a torsion spring. Transfer Matrix (TM) method is used to analyze the vibration attenuation properties of the structure and it is shown that the structure exhibits bandgap behavior. The obtained bandgaps are validated using Finite Element Analysis (FEA). Using the characteristic equation of the transfer matrix, we derive a transcendental equation for the bandgap edge frequencies. We show that for the nth band gap, the second band edge frequency is always equal to the natural frequency of the nth modeshape of the constituent beam under the simply supported condition. This frequency, therefore, is independent of the torsion spring constant. In addition, a detailed parametric study of the variation in band edge frequencies when the geometric and material parameters of the structure (Young’s modulus of beam, torsional spring constant, width and thickness of beam etc.) are varied is conducted. It is concluded that the ratio of flexural rigidity of the beam to the torsion spring constant (EI/kt) is an important parameter affecting the width of the bandgap. For low values of the ratio, i.e., low beam flexural rigidity and high torsional stiffness, the first band edge frequency is almost equal to the second band edge and, effectively, no bandgap exists. As the stiffness ratio increases, i.e. high flexural rigidity (EI of the beam) and low torsional stiffness kt, the first band edge frequency assumes progressively lower values relative to the second band edge and we obtain a relatively large bandgap over which no flexural waves propagate. This has important ramifications for the design of foldable metamaterial structures.