# Analysis of Longitudinal Waves in Rod-Type Piezoelectric Phononic Crystals

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## Abstract

**:**

## 1. Introduction

## 2. Analysis of Longitudinal Waves in Rod-Type Piezoelectric Phononic Crystals with the MTMM

#### 2.1. Basic Model

#### 2.2. Coordinate Systems and Physical Variables

#### 2.3. Governing Equations and Wave Solutions of a Constituent Rod

#### 2.4. Transfer Matrix of a Member

#### 2.5. Transfer Matrix of a Joint

#### 2.6. Global Transfer Matrix of the Unit Cell

#### 2.7. Dispersion Relation of Infinite Periodic Structures

#### 2.8. Global Transfer Relation of Finite Periodic Structures

## 3. Numerical Examples

#### 3.1. Validation of the Proposed MTMM

#### 3.2. Passive Control of Longitudinal Waves in Rod-Type Piezoelectric Phononic Crystals

#### 3.2.1. Influence of the Electrode’s Thickness

#### 3.2.2. Influence of the Rod’s Cross-Sectional Dimension

#### 3.2.3. Influence of the Elastic Rod Insert

#### 3.3. Active Control of Longitudinal Waves in Rod-Type Piezoelectric Phononic Crystals

#### 3.4. Dispersion Properties of Longitudinal Waves in Rod-Type Piezoelectric Phononic Crystals

- (1)
- The eigenvalue’s amplitude spectra, which cannot be obtained by MRRM [5], demonstrate clearly the width, the central frequencies and the bounding frequencies of the pass-bands ($\left|\mu \right|=1$) and the stop-bands ($\left|\mu \right|\ne 1$). They also reflect the attenuation amplitudes ${\mathrm{e}}^{{q}_{I}L}$ of waves in the stop-bands, which are verified by the attenuation constant (${q}_{I}L$) spectra. The eigenvalue’s amplitude spectra cannot indicate the properties of waves in the pass-bands, but the phase constant spectra do.
- (2)
- In these frequency-related dispersion curves, the bounding frequencies of the odd and even order stop-bands correspond to $qL={q}_{R}L=(2n+1)\pi $ and $qL={q}_{R}L=2n\pi $ ($n$ is a natural number), respectively. Within the stop-bands, the real part (${q}_{R}L$) of the complex wavenumber $qL$, which cannot be computed by the MRRM [5] but obtained here by the MTMM, have the same phases as their boundaries. In the wavelength spectra, the representations corresponding to these two kinds of phases are horizontal lines $\lambda /L=2/(2n+1)$ and $\lambda /L=1/n$. In the phase velocity spectra, they correspond to inclined lines that pass through the origin and have slopes $c/\omega =L/[(2n+1)\pi ]$ $c/\omega =L/[2n\pi ]$, respectively. The lines determined by the above formulas and the lines of bounding frequencies form the grids to draw the spectra in the corresponding frequency-related dispersion curves.

## 4. Conclusions

- (1)
- The proposed analytical MTMM provides an alternative analysis method for the complex band structures and transmission spectra till to $0.9{\omega}_{0}$ (${\omega}_{0}$ is the minimum critical frequency) within which the Love rod theory is valid. Its effectiveness is validated by some numerical examples.
- (2)
- In passive mode, the electrode’s thickness and the rod’s cross-sectional dimension can be used to slightly adjust the band structures of the rod-type piezoelectric phononic crystals, while the elastic rod insert is able to enormously alter the band structures.
- (3)
- In active mode, the switchable electrical boundaries among electric-short, applied electric capacity, electric-open and applied feedback control conditions is effective for modulating some of the band structures that are related to the electromechanical coupling of the rod-type piezoelectric phononic crystals. The tunable capacity and control gain in the applied electric capacity and applied feedback control cases, respectively, can also be used for tuning the propagation of longitudinal waves. The band structures of the electric-short condition play a referential role for designing the active control scheme.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

MTMM | Modified Transfer Matrix Method |

MRRM | Method of Reverberation-Ray Matrix |

TMM | Transfer Matrix Method |

FEM | Finite Element Method |

AM | Analytical Method |

## Appendix A

## Appendix B

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**Figure 2.**The description of the unit cell: (

**a**) the global and local coordinates; (

**b**) the convention of generalized displacements and forces; and (

**c**) the convention of wave amplitudes.

**Figure 3.**The phase constant spectra of the homogeneous PZT-5H piezoelectric rod with periodic electrical boundaries: (

**a**) electric-open condition; (

**b**) applied electric capacity condition with $C=\alpha A/l=1.02\times {10}^{-12}\mathrm{F}$; (

**c**) electric-short condition; (

**d**) applied feedback control condition with ${K}_{g}=5\times {10}^{8}\mathrm{V}/\mathrm{m}$; and (

**e**) all of the four abovementioned conditions in high frequency range.

**Figure 4.**The phase constant spectra of the periodic PZT-5H piezoelectric rod covered by brass electrodes with periodic electrical boundaries: (

**a**) electric-open condition; (

**b**) applied electric capacity condition with $C=\alpha A/l=1.02\times {10}^{-10}\mathrm{F}$; (

**c**) electric-short condition; and (

**d**) applied feedback control condition with ${K}_{g}=5\times {10}^{8}\mathrm{V}/\mathrm{m}$.

**Figure 5.**The transmission (in dB units) in a finite periodic PZT-5H piezoelectric rod covered by brass electrodes consisting of 14 unit cells with electric-open and electric-short boundaries from our MTMM and their comparison with the corresponding ones by Degraeve et al. [20]: (

**a**) comparison with the numerical transmission; and (

**b**) comparison with the experimental transmission.

**Figure 6.**Influence of the electrode’s thickness on the propagation constants spectra of the periodic PZT-5H piezoelectric rod covered by brass electrodes with periodic electrical boundaries: (

**a**) electric-open condition; (

**b**) applied electric capacity condition with $C=\alpha A/l=1.02\times {10}^{-12}\mathrm{F}$; (

**c**) electric-short condition; and (

**d**) applied feedback control condition with ${K}_{g}=5\times {10}^{8}\mathrm{V}/\mathrm{m}$.

**Figure 7.**Influence of the rod’s cross-sectional dimension on the propagation constants spectra of the periodic PZT-5H piezoelectric rod covered by brass electrodes with periodic electrical boundaries: (

**a**) electric-open condition; (

**b**) applied electric capacity condition with $C=\alpha A/l$; (

**c**) electric-short condition; and (

**d**) applied feedback control condition with ${K}_{g}=5\times {10}^{8}\mathrm{V}/\mathrm{m}$.

**Figure 8.**Influence of the elastic rod insert on the propagation constants spectra of the periodic PZT-5H piezoelectric rod covered by brass electrodes with periodic electrical boundaries: (

**a**) electric-open condition; (

**b**) applied electric capacity condition with $C=\alpha A/l=1.02\times {10}^{-12}\mathrm{F}$; (

**c**) electric-short condition; and (

**d**) applied feedback control condition with ${K}_{g}=5\times {10}^{8}\mathrm{V}/\mathrm{m}$.

**Figure 9.**Influence of the electrical boundaries on the propagation constants spectra of the periodic Epoxy-Brass-PZT-5H-Brass composite rod: (

**a**) applied electric capacity condition; and (

**b**) applied feedback control condition.

**Figure 10.**Various frequency-related dispersion curves of the periodic Epoxy-Brass-PZT-5H-Brass composite rod in low frequency range: (

**a**) eigenvalue’s amplitude spectra; (

**b**) wavenumber spectra; (

**c**) wavelength spectra in logarithmic coordinate; (

**d**) phase velocity spectra in logarithmic coordinate; and (

**e**) phase velocity spectra in a small scale.

**Figure 11.**Various frequency-related dispersion curves of the periodic Epoxy-Brass-PZT-5H-Brass composite rod in high frequency range: (

**a**) eigenvalue’s amplitude spectra; (

**b**) wavenumber spectra; (

**c**) wavelength spectra in logarithmic coordinate; (

**d**) phase velocity spectra in logarithmic coordinate; and (

**e**) phase velocity spectra in a small scale.

Electrical Boundary Conditions | Associated Mathematical Formulas | Expressions of $\widehat{V}$ | Expressions of $\widehat{Q}$ |
---|---|---|---|

Electric-open | $\widehat{Q}=0$ | $\frac{e}{\alpha}\left[\widehat{u}(l)-\widehat{u}(0)\right]$ | $0$ |

Applied electric capacity | $\widehat{Q}=-C\widehat{V}$ | $\frac{eA}{\alpha A+Cl}\left[\widehat{u}(l)-\widehat{u}(0)\right]$ | $-\frac{eCA}{\alpha A+Cl}\left[\widehat{u}(l)-\widehat{u}(0)\right]$ |

Electric-short | $\widehat{V}=0$ | $0$ | $-\frac{eA}{l}\left[\widehat{u}(l)-\widehat{u}(0)\right]$ |

Applied feedback control | $\widehat{V}=-{K}_{g}\left[\widehat{u}(l)-\widehat{u}(0)\right]$ | $-{K}_{g}\left[\widehat{u}(l)-\widehat{u}(0)\right]$ | $-\frac{\alpha A}{l}\left(\frac{e}{\alpha}+{K}_{g}\right)\left[\widehat{u}(l)-\widehat{u}(0)\right]$ |

Electrical Boundary Conditions | Expressions of $\mathrm{d}\widehat{\phi}/\mathrm{d}x$ | Expressions of $B$ | |

Electric-open | $\frac{e}{\alpha}\frac{\mathrm{d}\widehat{u}}{\mathrm{d}x}$ | $0$ | |

Applied electric capacity | $\frac{e}{\alpha}\frac{\mathrm{d}\widehat{u}}{\mathrm{d}x}-\frac{e}{\alpha}\frac{C}{\alpha A+Cl}\left[\widehat{u}(l)-\widehat{u}(0)\right]$ | $-\frac{{e}^{2}CA}{{\alpha}^{2}A+\alpha Cl}$ | |

Electric-short | $\frac{e}{\alpha}\frac{\mathrm{d}\widehat{u}}{\mathrm{d}x}-\frac{e}{\alpha l}\left[\widehat{u}(l)-\widehat{u}(0)\right]$ | $-\frac{{e}^{2}A}{\alpha l}$ | |

Applied feedback control | $\frac{e}{\alpha}\frac{\mathrm{d}\widehat{u}}{\mathrm{d}x}-\frac{1}{l}\left(\frac{e}{\alpha}+{K}_{g}\right)\left[\widehat{u}(l)-\widehat{u}(0)\right]$ | $-\frac{eA}{l}\left(\frac{e}{\alpha}+{K}_{g}\right)$ |

Materials | Stiffness Constants ($\text{GPa}$) | |||||||||

${c}_{11}$ | ${c}_{22}$ | ${c}_{33}$ | ${c}_{12}$ | ${c}_{13}$ | ${c}_{23}$ | |||||

PZT-5H | 117.0 | 126.0 | 126.0 | 84.1 | 84.1 | 79.5 | ||||

Brass | 162.46 | 162.46 | 162.46 | 82.58 | 82.58 | 82.58 | ||||

Epoxy | 6.98 | 6.98 | 6.98 | 3.76 | 3.76 | 3.76 | ||||

Materials | Poisson’s Ratios | Piezoelectric Constants ($\mathrm{C}/{\mathrm{m}}^{2}$) | Dielectric Constants ($\times {10}^{-9}\mathrm{F}/\mathrm{m}$) | |||||||

${\nu}_{12}$ | ${\nu}_{13}$ | ${e}_{11}$ | ${e}_{12}$ | ${e}_{13}$ | ${\alpha}_{11}$ | |||||

PZT-5H | 0.41 | 0.41 | 23.3 | −6.5 | −6.5 | 13.02 | ||||

Brass | 0.337 | 0.337 | — | — | — | — | ||||

Epoxy | 0.35 | 0.35 | — | — | — | — | ||||

Materials | Mass Density ($\text{kg}/{\mathrm{m}}^{3}$) | Length ($\text{mm}$) | Cross-Sectional Area ($\times \pi {\text{mm}}^{2}$) | Cross-Sectional Moments of Inertia ($\times \pi {\text{mm}}^{4}$) | ||||||

$\rho $ | $l$ | $A$ | ${I}_{y}$ | ${I}_{z}$ | ||||||

PZT-5H | 7500 | 10 | 1/4 | 1/64 | 1/64 | |||||

Brass | 8320 | 0.025 | 1/4 | 1/64 | 1/64 | |||||

Epoxy | 1180 | 10 | 1/4 | 1/64 | 1/64 |

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**MDPI and ACS Style**

Li, L.; Guo, Y.
Analysis of Longitudinal Waves in Rod-Type Piezoelectric Phononic Crystals. *Crystals* **2016**, *6*, 45.
https://doi.org/10.3390/cryst6040045

**AMA Style**

Li L, Guo Y.
Analysis of Longitudinal Waves in Rod-Type Piezoelectric Phononic Crystals. *Crystals*. 2016; 6(4):45.
https://doi.org/10.3390/cryst6040045

**Chicago/Turabian Style**

Li, Longfei, and Yongqiang Guo.
2016. "Analysis of Longitudinal Waves in Rod-Type Piezoelectric Phononic Crystals" *Crystals* 6, no. 4: 45.
https://doi.org/10.3390/cryst6040045