Frequency filter for elastic bending waves: Poincaré map method and experiment

Abstract

In this work, the Poincaré map numerical method was successfully developed to solve the fourth-order differential equation that describes the flexural vibrations of a beam, within the Timoshenko beam theory. The Euler-Bernoulli continuity conditions were considered, which are valid for frequencies smaller than the critical frequency. As an example, this method was used to design a complex elastic structure, characterized by a flexural frequency spectrum with a broad band gap. Such structure consists of two coupled phononic crystals, which were designed with filling factor values in such a way that in their bending frequency spectra, an allowed band of the first part, overlaps with a band gap of the second one and vice versa. The resulting composed system has a much wider effective gap than its original components, between 4 and 10.5 kHz. This system works as an elastic bending wave filter. Finally, these three structured elastic systems were constructed, and characterized by the acoustic resonance spectroscopy technique. The natural flexural frequencies as well as the corresponding wave amplitudes of each structured beam were measured. The experimental measurements show excellent agreement with the numerical simulation.