Gaussian expansion for the vibration analysis of plates with multiple acoustic black holes indentations

Figure. Energy focusing at the center of ABH indentations in a plate for two different frequencies.

The acoustic black hole (ABH) effect can be achieved embedding cuneate indentations with power-law profile in plates. That results in remarkable properties like the reduction of plate vibrations, the potential for energy harvesting thanks to the focalization of energy, or the design of new metamaterials to manipulate acoustics waves. The analysis of such phenomena demands simulating the modal shapes and response to external excitations of ABH plates. This is usually done by means of numerical approaches, like the finite element method (FEM). However, if one is interested in performing long parametric analyses and in capturing the ABH plate behavior at high frequencies, the computational cost associated to numerical methods may become too demanding. In this work, a semi-analytical approach is suggested to circumvent the situation. The Rayleigh-Ritz method is applied using two-dimensional Gaussian functions to expand the flexural motion of a plate with non-uniform thickness. Then, a matrix-replacing strategy is proposed to embed the multiple ABHs in the plate. That results in low dimensional matrix systems, which yet provide very accurate solutions.

After presenting all theoretical developments, the semi-analytical method is first applied to analyze the performance of a single ABH when varying several of its parameters. Then, various configurations involving multiple ABHs are considered. Those range from long strips exhibiting frequency attenuation bands, to plates containing ABH indentations in many shapes and sizes.

This research has emerged as a collaboration between the Automobile College from the Chongqing University, China, and the Engineering Department of the La Salle, Universitat Ramon Llull.

Link:  https://www.sciencedirect.com/science/article/pii/S0888327019303346

Jie Deng, Ling Zheng, Oriol Guasch, Hang Wu, Pengyun Zeng, Yifang Zuo (2019), "Gaussian expansion for the vibration analysis of plates with multiple acoustic black holes indentations". Mechanical Systems and Signal Processing, 131, pp. 317-334.

This work has been funded by the National Natural Science Foundation of China under Grant (51875061) and the China Scholarship Council (CSC No.201806050075). The authors gratefully acknowledge this support as well as the in-kind assistance from La Salle, Universitat Ramon Llull, and the Chongquing University to make that collaboration possible. The third author would also like to thank l'Obra Social de la Caixa and the Universitat Ramon Llull for their support under grant 2018-URL-IR2nQ-031.