IMPEDANCE COMPUTATION USING THE SPECTRAL ELEMENT METHOD AND THE RICCATI EQUATION
Assis, G.F.C.A.1; Dos Santos, J. M. C.1; Camino, J. F.1; Arruda, J. R. F.1
1 Universidade Estadual de Campinas
doi:10.20906/CPS/CILAMCE2017-0836
Resumo
The reduction of structural vibrations has always been a difficult task. One approach to achieve this goal using a wave approach consists in reducing wave reflections at the boundaries. Without reflections, standing waves cannot build up and, therefore, vibration is attenuated. Reflection can be avoided by using anechoic devices at the structure extremities. The Acoustic Black Hole (ABH) theory was established based on the property that longitudinal and flexural waves do not reflect from boundaries with thickness decaying with a power law. The dynamics of tapered structures can be formulated using a state space approach. The elastodynamic equations in the frequency domain (spectral formulation) are written as a first order state space differential equation system. For a tapered structure, the space state formulation leads to a Riccati equation that can be numerically solved. Thus, any geometrical law for the structure thickness can be treated. The intent of this paper is to investigate the numerical solution of the Riccati equation. Furthermore, the ABH effect using graded material properties instead of tapered geometries is proposed. Elementary rod model are treated. Results using the proposed formulation are validated by comparison with semi analytical results obtained using the Spectral Element Method (SEM) for constant and linearly varying properties. SEM solutions for rods with linearly varying coefficients formulated using Bessel functions are recalled. Numerical issues when using the Ricatti equation are addressed.
Palavras-chave: Impedance; Rod; Spectral Element Method; Riccati Equation