**5.2 The effect of the stubs on the broad complete BGs**

In order to investigate the effect of single "hard" stubs on the complete phononic bandgaps of the novel metal-matrix PCs, we studied the effect of the steel-stub height on the first complete bandgap. **Figure 15** displays the evolution of the first complete bandgap as a function of the steel-stub height *h*.

We find that, with the increase of the steel-stub height, the location of the first complete bandgap shifts to a lower frequency range before it shifts to a higher frequency range. Furthermore, the bandwidth of the bandgaps becomes broad. For example, when the steel-stub height is below or equal to 3.5 mm, after increasing the steel-stub height, the location of the bandgaps shifts to lower frequencies. However, when the steel-stub height exceeds 3.5 mm, the location of the bandgaps shifts to higher frequencies as the steel-stub height increases.

#### **Figure 14.**

*The evolution of the first complete BG in the lower frequency complete bandgap metal-matrix embedded phononic crystals as a function of the steel-stub height with D = 8 mm, dup = 9 mm, dup = 5 mm, h = 5 mm, e = 1 mm, and a = 10 mm, respectively.*

**Figure 15.**

*The evolution of the first bandgap in the proposed phononic-crystal plate with "hard" stubs as a function of steel-stub height with D = 8 mm, d = 7.5 mm, e = 1 mm, and a = 10 mm, respectively.*

## **6. Conclusions**

In this chapter, metal-matrix embedded phononic crystals consisting of double-sided stubs (single stubs and composite stubs), which are deposited on a two-dimensional locally resonant phononic crystal plate that consists of an array of rubber fillers embedded in a steel plate, are introduced. The following summaries were drawn [14, 18]:

The spring-mass system of the resonator can be decoupled by introducing the rubber filler, and then the out-of-plane bandgap and the in-plane bandgap can be adjusted into the same lowest frequency range. The out-of-plane bandgap and the in-plane bandgap can be overlapped with each other. As a result, a lower frequency complete bandgap which ranges from 59 to 93 Hz is obtained in the metal-matrix embedded phononic crystals.

Both the out-of-plane and the in-plane phononic bandgap increase after introducing single "hard" cylinder stubs. When introducing the rubber filler and the single "hard" stub simultaneously, two new kinds of resonance modes are produced: an in-plane analogous-rigid modes, where the whole stub vibrates in-plane with the plate, and the other new resonance mode is the out-of-plane analogous-rigid mode, where the whole stub vibrates out-of-plane with respect to the plate. The out-of-plane bandgap increases for the out-of-plane analogous-rigid mode, and the in-plane bandgap increases for the in-plane analogous-rigid mode. Both the out-ofplane and the in-plane analogous-rigid modes are mainly formed due to introducing the single "hard" cylinder stub. The in-plane and out-of-plane bandgaps overlap and produce a broad complete bandgap in the metal-matrix embedded phononic crystals. Within this broad complete bandgap, both the in-plane and out-of-plane Lamb waves are prohibited. The absolute bandwidth of the bandgap for the proposed structure is five times higher than for the classic double-sided stubbed metal-matrix phononic-crystal plate.

The effect of the stub on the bandgaps is investigated. Results show that the location of the bandgaps can be modulated in a significant lower frequency range, and the bandwidth can be expanded in a considerable large frequency range by introducing different composite taper stubs, and the bandwidth can be expanded to a larger frequency range by introducing different "hard" stubs.

The proposed structure provides an effective way for phononic crystals to obtain wide complete bandgaps and lower frequency complete bandgaps (below 100 Hz), which have a potential application in the low-frequency vibration reduction in a practical case.

**47**

**Author details**

\*, Yihua Dou1

and Linkai Niu2

\*Address all correspondence to: ziyedeyan@stu.xjtu.edu.cn

1 School of Mechanical Engineering, Xi'an Shiyou University, Xi'an, China

Suobin Li1

China

provided the original work is properly cited.

*Metal-Matrix Embedded Phononic Crystals DOI: http://dx.doi.org/10.5772/intechopen.80790*

This chapter was supported by the Project of the National Science Foundation of

**Acknowledgements**

China (No. 51674199).

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

2 College of Mechanical Engineering, Taiyuan University of Technology, Taiyuan,
