# Vibration and Bandgap Behavior of Sandwich Pyramid Lattice Core Plate with Resonant Rings

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Model and Theory

_{0}of the plate, an input acceleration excitation (denoted as a

_{in}) is introduced, and the resulting acceleration response (denoted as a

_{out}) is measured at the response acquisition point P

_{1}, as depicted in the inset on the right of Figure 2. Ultimately, the vibration attenuation T can be obtained by altering the frequency of the input acceleration excitation:

## 3. Results and Discussion

#### 3.1. Simulation Parameters

_{f}‘, and the core height is denoted by h

_{c}. The length, radius and inclination angle of the truss core and the resonant ring are represented by l, h

_{m}, r

_{c}, r

_{m}and θ, respectively. The geometrical parameters and material properties of the model are listed in Table 1 and Table 2.

_{1}and E

_{2}represent the eigenmodes of the fourth to eighth dispersion curve. The face-sheets of the SPLCRR primarily exhibit lateral motion, with negligible movement in the vertical direction. The motion characteristics of the complete bandgap and the flexural wave bandgap will significantly affect vibration attenuation, as we will show later.

#### 3.2. Experimental Verification

#### 3.3. Effect of Geometric Parameters on Bandgap

_{c}= 1 mm–3 mm) and angle values (θ = 20–32°), and face-sheet thickness values (h

_{f}= 2 mm–3 mm) are chosen. The bandgap behaviors of SPLCRR are significantly influenced by changes in the lattice core and face-sheet dimensions, as evident from Figure 7, wherein the bandgaps move to higher frequencies with the increase in the rod radius and face-sheets thickness, but the increase in the lattice core rod angle causes the bandgaps to move to low frequencies. This is expected, because the change in lattice core parameters and the face-sheets parameter results in a modification stiffness of lattice sandwich layer (eq. X). It is worth noting that the width of the second bandgap of our interest reaches its maximum when r

_{c}= 2.5 mm, θ = 22° and h

_{f}= 2.5 mm, respectively. In addition, the change in rod radius is more sensitive to width of the second bandgap as compared to other two parameters.

_{m}and h

_{m}(defined in Section 3.1) of resonant ring on bandgaps are discussed in detail here, where nine different geometric parameters (r

_{m}= 4–6 mm) and (h

_{m}= 5–15 mm) chosen in this example. From Figure 8, it can be observed that the bandgap behaviors are strongly influenced by resonant ring radius and height, and increasing the radius and height leads to the bandgaps moving to lower frequency. This is expected, because the increase in resonant ring radius and height results in a modification of equivalent mass of lattice sandwich layer. In addition, the change in the resonant ring radius and height has no significant effect on the width of the second bandgap. In other words, the bandgap frequency range can be adjusted by changing geometric parameters of the resonant ring without affecting the bandgaps width.

#### 3.4. Effect of Material Parameters on Bandgap

_{c}= 2.5–250 GPa) and face-sheets (E

_{f}= 0.25–25 GPa) are chosen in this example; the other properties are the same as those in Section 3.1. From Figure 9, it can be observed that the change in the elastic modulus of lattice rod and face-sheets exerts a significant impact on the bandgap behaviors of SPLCRR, wherein the bandgaps move to higher frequencies with the increase in the elastic modulus of lattice rod and face-sheets. This is reasonable that the increase in the elastic modulus of lattice rod and face-sheets results in the increase in equivalent stiffness of lattice sandwich layer. It is worth noting that the width of the second bandgap of our interest reaches its maximum when E

_{c}= 7.5 GPa and E

_{f}= 2.5 GPa, respectively. In addition, the width of the second bandgap first increases and then gradually disappears with the increase in the elastic modulus E

_{c}, while the increase in the elastic modulus E

_{f}shows the opposite trend.

_{m}and η

_{m}(defined in Section 3.1) of resonant ring on bandgaps are discussed in detail here, where nine different material parameters (E

_{m}= 2.5–250 GPa) and (η

_{m}= 0.01–0.3) are chosen in this example. From Figure 10, it can be obtained that with the increase of Young's modulus and damping ratio of the resonant ring, there is no significant change in the upper and lower edge frequencies of the band gap and no significant change in the bandgap width. Therefore, the Young's modulus and damping ratio of the resonant ring has little effect on the band gap of the metamaterial plate.

#### 3.5. Effect of Period Parameters on Bandgap

## 4. Conclusions

- (a)
- Remarkable vibration suppression and bandgaps were verified by comparisons between numerical and experimental results.
- (b)
- Different geometric parameters were discussed. The thickness of face-sheets and the rod radius of the core had significant effects on the frequency range and width of the bandgap, which moved to higher frequency with the increase in the two values. The rod radius was more sensitive to width of the second bandgap compared to other parameters.
- (c)
- Different material parameters were discussed. The elastic modulus of lattice core and face-sheets had significant effects on the frequency range and width of the bandgap, wherein the bandgaps moved to higher frequencies with the increase in the elastic modulus of lattice rod and face-sheets.
- (d)
- Different period parameters were discussed. The frequencies of the bandgap gradually decrease as the period length increases. The frequency of the upper and lower edges of the bandgap under the square distribution was higher than that of the hexagonal distribution.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**(

**a**) Typical lattice sandwich plate, (

**b**) typical lattice sandwich unit cell, (

**c**) SPLCRR unit-cell.

**Figure 2.**Dispersion curves of the sandwich pyramid lattice core plates without and with resonant rings on the left and right panels, respectively. Inset: The left panel is the first irreducible Brillouin zone, and the right panel is the input/output setting of numerical simulation.

**Figure 3.**Eigenmodes of the SPLCRR unit cell at frequency (

**a**) f

_{A}= 502 Hz (

**b**) f

_{B}= 596 Hz (

**c**) f

_{C}= 625 Hz (

**d**) f

_{D}= 874 Hz (

**e**) f

_{F}= 964 Hz and (

**f**) f

_{E}= 1003 Hz.

**Figure 6.**Comparison between the typical sandwich plate and SPLCRR plate for vibration transmission spectra.

**Figure 7.**Effect of geometric parameters of lattice core and face-sheets on bandgaps of the SPLCRR. (

**a**) lattice core rod radius values r

_{c}(

**b**) angle values θ (

**c**) face-sheet thickness values h

_{f}.

**Figure 8.**Effect of geometric parameters of resonant ring on bandgap of the SPLCRR. (

**a**) resonant ring radius values r

_{m}(

**b**) resonant ring height values h

_{m}.

**Figure 9.**Effect of material parameters of lattice core and face-sheets on bandgaps of the SPLCRR. (

**a**) elastic modulus of lattice rod E

_{c}(

**b**) elastic modulus of face-sheets E

_{f}.

**Figure 10.**Effect of material parameters of resonant ring on bandgaps of the SPLCRR. (

**a**) elastic modulus of resonant ring E

_{m}(

**b**) damping ratio of the resonant ring η

_{m}.

**Figure 11.**Effect of period parameters on bandgap of the SPLCRR. (

**a**) period length (

**b**) different distribution forms.

**Figure 12.**Distribution forms of SPLCRR plate for (

**a**) square and (

**b**) triangular lattices. First irreducible Brillouin zone in shaded region for (

**a**) square and (

**b**) triangular lattices.

a | h_{c} | h_{f} | h_{m} | r_{c} | r_{m} | θ |
---|---|---|---|---|---|---|

40 mm | 38 mm | 2 mm | 10 mm | 2 mm | 5 mm | 30° |

**Table 2.**Material parameters of the SPLCRR [27].

Density ρ (kg/m^{3}) | Modulus E (GPa) | Poisson’s Ratio μ | |
---|---|---|---|

Nylon | 1200 | 3.5 | 0.37 |

Steel | 7800 | 210 | 0.3 |

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**MDPI and ACS Style**

Li, C.; Chen, Z.; Jiao, Y. Vibration and Bandgap Behavior of Sandwich Pyramid Lattice Core Plate with Resonant Rings. *Materials* **2023**, *16*, 2730.
https://doi.org/10.3390/ma16072730

**AMA Style**

Li C, Chen Z, Jiao Y. Vibration and Bandgap Behavior of Sandwich Pyramid Lattice Core Plate with Resonant Rings. *Materials*. 2023; 16(7):2730.
https://doi.org/10.3390/ma16072730

**Chicago/Turabian Style**

Li, Chengfei, Zhaobo Chen, and Yinghou Jiao. 2023. "Vibration and Bandgap Behavior of Sandwich Pyramid Lattice Core Plate with Resonant Rings" *Materials* 16, no. 7: 2730.
https://doi.org/10.3390/ma16072730