A symplectic analytical wave propagation model for forced vibration of thin plate with acoustic black hole indentation

Abstract

In recent years, acoustic black holes (ABHs) have been revealed as an effective bending wave control technology and widely used to regulate the vibration and noise reduction performance of plate structure. However, the isolation effect of most traditional ABH plate can only be aroused at frequencies above the cut-off frequency. In order to widen the frequency range of the ABH effect, this paper embedded single and multiple ABH indentations at the end of plate. For the forced vibration response of the ABH indentation plate, an analytical wave propagation model was proposed based on the symplectic method. The governing equations of bending vibration of thin plate was first introduced into the symplectic dual system, then the symplectic analytical solution of wave propagation parameters and waveforms were obtained. A Gaussian-discretization scheme was used to discretize the ABH indentation, whereas the proposed method was used to evaluate the wave amplitude. The vibration response was finally obtained using the superposition principle and wave propagation, reflection, and transmission relationships. The effectiveness of the proposed method was verified using the finite element results. The results showed that the ABH indentation plate, especially the plate with multiple ABH indentations, had significant vibration reduction performance. Considering that embedding ABH indentation can weaken the structural rigidity of the plate, the ABH indentation plate with reinforcing stiffeners was then proposed to alleviate this limitation, and the influence of the width and quantity of reinforcing stiffeners on the vibration response was studied. The results indicated that the increase of width will to some extent weaken the ABH effect, and the quantity of stiffeners had a significant impact on the ABH effect.