Wang G., Liu Y., Shao L., Yu D., Wen J.

The locally resonant phononic crystals can obtain the low-frequency gaps with a structure of small dimensions, which can lead to promising applications such as low-frequency vibration/noise insulations. Based on its resonant characteristic, the vibration modes corresponding to its band gaps are studied in several references. Moreover, Hirsekorn proposed some simple models that can be used to simulate the lower edge of the locally resonant band gap in two-dimensional phononic crystals, and the parameters of the model are calculated based on reasonable simplifications. [M. Hirsekorn, Appl. Phys. Lett.,84,3364]. However, he has not found the corresponding model that describing the upper edge of the locally resonant band gap, which is equally important for the simple design of locally resonant phononic crystals. Furthermore, the three-dimensional case has not been studied. In this paper, two simple spring-mass models are constructed successfully to estimate the frequencies on BOTH the lower and upper edges of the lowest locally resonant band gap in BOTH two- and three-dimensional ternary phononic crystals respectively, based on the lucubrating on their corresponding lattice vibration modes. The analytical estimations with the models and the theoretical predictions with the lumped-mass method are in very good agreements. The proposal of these models are heuristic and helpful for better understanding of the locally resonant band gap mechanism as well as better estimation of the corresponding band edge frequencies, which will be useful for the application of the locally resonant phononic crystals.