# Free Vibrations of a New Three-Phase Composite Cylindrical Shell

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## Abstract

**:**

## 1. Introduction

## 2. Mechanical Properties of the Three-Phase Composite Material

#### 2.1. Halpin-Tsai Model for the Hybrid Matrix

#### 2.2. Mori-Tanaka Method for Three-Phase Composites

## 3. Governing Equations of Motion

#### 3.1. Governing Equations for Cylindrical Shells

#### 3.2. Solution Procedure for Free Vibrations

## 4. Validation of the Calculation Results

#### 4.1. Convergence Analysis

#### 4.2. Mode Comparison

## 5. Free Vibrations of a Three-Phase Composite Cylindrical Shell

#### 5.1. Effects of Boundary Spring Stiffness on the Natural Frequencies

#### 5.2. Effects of Reinforcements on the Natural Frequencies

## 6. Conclusions

- (1)
- Boundary spring stiffness has a significant impact on the natural frequencies of the three-phase composite cylindrical shell. The natural frequencies of the three-phase composite cylindrical shell reach a maximum under the sufficiently large spring stiffness corresponding to the boundary condition when clamped at both ends. When the boundary spring stiffness is 0, that is, the corresponding free boundary condition is at both ends, then the natural frequencies of the FG three-phase composite cylindrical shell is minimum, which is far lower than the natural frequency of the three-phase composite cylindrical shell under the other two boundary conditions of clamped and simple-support.
- (2)
- The synergistic enhancement of GPLs and carbon fibers can greatly increase the natural frequencies of the composite cylindrical shell. Among these, GPLs can play a more obvious strengthening role than carbon fibers. We know that too many GPLs may increase the possibility of aggregation, but only a small amount of GPLs can significantly enhance the composite cylindrical shell.
- (3)
- The four FG GPL distributions have different effects on the natural frequencies of the FG three-phase composite cylindrical shell. When GPLs are distributed in the X-type FG form, each order of the natural frequency of the three-phase composite cylindrical shell is higher than those of the other three FG distributions, whereas each order of the natural frequency of the three-phase composite cylindrical shell is lower when GPLs are distributed in the O-type FG form.
- (4)
- Carbon fiber is another important reinforcement material in the three-phase composite cylindrical shell, and its laying angle can also significantly affect the natural frequencies of the three-phase composite cylindrical shell. If carbon fibers are distributed along the axis of this cylindrical shell, each order of the natural frequency of the three-phase composite cylindrical shell is higher, whereas each order of the natural frequency of the three-phase composite cylindrical shell is lower when the carbon fibers are distributed along the circumferential direction.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**The FG three-phase composite cylindrical shell model reinforced synergistically with GPLs and carbon fibers.

**Figure 2.**The model of the three-phase composite cylindrical shell under arbitrary boundary conditions in a column co-ordinate system.

**Figure 4.**The relationships between the number of layers and the rate of change for the natural frequencies of the FGM cylindrical shell.

**Figure 5.**The natural frequencies of the three-phase composite cylindrical shells corresponding to different boundary spring stiffness.

**Figure 6.**The natural frequencies of the three-phase composite cylindrical shells under three types of boundary conditions.

**Figure 7.**The changes for natural frequencies of the three-phase composite cylindrical shells with the increase in GPL mass fraction.

**Figure 8.**The natural frequencies of the three-phase composite cylindrical shells corresponding to four forms of GPL FG distributions.

**Figure 9.**The changes of the natural frequencies of the three-phase composite cylindrical shells with the increase in carbon fiber content. (

**a**) FG-X three-phase composite cylindrical shell. (

**b**) FG-V three-phase composite cylindrical shell. (

**c**) FG-A three-phase composite cylindrical shell. (

**d**) FG-O three-phase composite cylindrical shell.

**Figure 10.**The natural frequencies of the three-phase composite cylindrical shells corresponding to six types of carbon fiber layup angles.

Material | ${\mathit{E}}_{\mathit{e}}\left(\mathit{G}\mathit{P}\mathit{L}\right)$ | ${\mathbf{\nu}}_{\mathit{e}}$ | ${\mathbf{\rho}}_{\mathit{e}}\left(\mathbf{k}\mathbf{g}/{\mathbf{m}}^{3}\right)$ |
---|---|---|---|

8551-7 epoxy polymer (matrix) | 4.08 | 0.38 | 1272 |

Graphene platelets | 1010 | 0.186 | 1062.5 |

Material | ${\mathit{L}}_{\mathit{G}\mathit{P}\mathit{L}}\left(\mathbf{\mu}\mathbf{m}\right)$ | ${\mathit{W}}_{\mathit{G}\mathit{P}\mathit{L}}\left(\mathbf{\mu}\mathbf{m}\right)$ | ${\mathit{h}}_{\mathit{G}\mathit{P}\mathit{L}}\left(\mathbf{\mu}\mathbf{m}\right)$ |
---|---|---|---|

Graphene platelets (GPLs) | 2.5 | 1.5 | 1.5 |

Material | ${\mathit{E}}_{\mathit{f}}^{1}\left(\mathbf{G}\mathbf{p}\mathbf{a}\right)$ | ${\mathit{E}}_{\mathit{f}}^{2}\left(\mathbf{G}\mathbf{p}\mathbf{a}\right)$ | ${\mathbf{\nu}}_{\mathit{f}}^{12}$ | ${\mathbf{\nu}}_{\mathit{f}}^{13}$ | ${\mathit{G}}_{\mathit{f}}^{12}\left(\mathbf{G}\mathbf{p}\mathbf{a}\right)$ | ${\mathit{G}}_{\mathit{f}}^{23}\left(\mathbf{G}\mathbf{p}\mathbf{a}\right)$ | ${\mathbf{\rho}}_{\mathit{f}}\left(\mathbf{k}\mathbf{g}/{\mathbf{m}}^{3}\right)$ |
---|---|---|---|---|---|---|---|

Carbon fiber | 276.0 | 19.0 | 0.2 | 0.2 | 27.0 | 7.0 | 1780 |

BC | Spring Stiffness | ||||
---|---|---|---|---|---|

${\mathit{k}}_{\mathit{u}}$ | ${\mathit{k}}_{\mathit{v}}$ | ${\mathit{k}}_{\mathit{w}}$ | ${\mathit{k}}_{\mathit{\varphi}\mathit{x}}$ | ${\mathit{k}}_{\mathit{\varphi}\mathbf{\theta}}$ | |

F | 0 | 0 | 0 | 0 | 0 |

S | ${10}^{15}$ | ${10}^{15}$ | ${10}^{15}$ | 0 | ${10}^{15}$ |

C | ${10}^{15}$ | ${10}^{15}$ | ${10}^{15}$ | ${10}^{15}$ | ${10}^{15}$ |

**Table 5.**Comparisons of the theoretical results of the natural frequency of cylindrical shells under C-C boundary conditions with finite elements and the existing literature results. (L = 0.502 m; R = 0.0635 m; h = 0.00163 m; ρ = 7800 kg/m

^{3}; ν = 0.28; E = 2.1E + 11N/m

^{2}).

n | Present | Dai et al. [51] | Abaqus | ||||
---|---|---|---|---|---|---|---|

M = 6 N = 6 | M = 8 N = 8 | M = 10 N = 10 | M = 12 N = 12 | M = 14 N = 14 | |||

1 | 905.1 | 901.9 | 900.6 | 899.8 | 899.2 | 896.6 | 900.5 |

2 | 922.91 | 915.8 | 911.9 | 909.4 | 907.5 | 898.2 | 900.9 |

3 | 1425.6 | 1414.0 | 1408.4 | 1404.9 | 1402.5 | 1388.9 | 1395.8 |

4 | 1502.3 | 1501.1 | 1500.6 | 1500.3 | 1501.6 | 1501.6 | 1516.3 |

5 | 1692.1 | 1685.1 | 1682.6 | 1681.1 | 1680.3 | 1676.0 | 1690.7 |

6 | 1911.3 | 1904.5 | 1899.8 | 1896.2 | 1893.4 | 1880.9 | 1883.2 |

7 | 2083.5 | 2060.7 | 2048.7 | 2041.4 | 2036.6 | 2014.1 | 2020.8 |

8 | 2144.4 | 2097.7 | 2080.4 | 2071.6 | 2064.9 | 2389.0 | 2064.7 |

9 | 2343.8 | 2269.0 | 2239.4 | 2223.3 | 2213.1 | 2472.6 | 2189.4 |

**Table 6.**Modal comparisons of the X-type FG cylindrical shell under C-C boundary conditions are given.

C-C | Mode No. | |||||
---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | |

Present | ||||||

Abaqus | ||||||

Present | 187.46 | 204.85 | 229.20 | 268.78 | 338.52 | 343.53 |

Abaqus | 188.27 | 206.40 | 229.94 | 273.04 | 339.18 | 346.36 |

Error | 0.43% | 0.75% | 0.32% | 1.56% | 0.19% | 0.81% |

**Table 7.**Modal comparisons of the X-type FG cylindrical shell under S-S boundary conditions are given.

S-S | Mode No. | |||||
---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | |

Present | ||||||

Abaqus | ||||||

Present | 175.92 | 194.93 | 219.61 | 261.31 | 317.10 | 327.30 |

Abaqus | 176.93 | 196.88 | 220.36 | 266.12 | 319.96 | 329.99 |

Error | 0.57% | 0.99% | 0.34% | 1.81% | 0.89% | 0.81% |

**Table 8.**Modal comparisons of the X-type FG cylindrical shell under F-F boundary conditions are given.

F-F | Mode No. | |||||
---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | |

Present | ||||||

Abaqus | ||||||

Present | 25.80 | 36.83 | 72.72 | 87.11 | 138.76 | 154.43 |

Abaqus | 25.91 | 33.90 | 73.40 | 86.03 | 141.09 | 155.56 |

Error | 0.42% | 7.95% | 0.92% | 1.25% | 1.65% | 0.72% |

**Table 9.**Modal comparisons of the composite laminated cylindrical shell under C-C boundary conditions are given.

C-C | Mode No. | |||||
---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | |

Present | ||||||

Abaqus | ||||||

Present | 121.39 | 137.84 | 147.07 | 202.43 | 203.95 | 226.69 |

Abaqus | 121.34 | 137.82 | 147.14 | 202.54 | 204.37 | 226.61 |

Error | 0.04% | 0.01% | 0.05% | 0.05% | 0.21% | 0.04% |

**Table 10.**Modal comparisons of the composite laminated cylindrical shell under S-S boundary conditions are given.

S-S | Mode No. | |||||
---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | |

Present | ||||||

Abaqus | ||||||

Present | 113.54 | 130.46 | 141.21 | 198.41 | 199.96 | 203.82 |

Abaqus | 113.56 | 130.47 | 141.37 | 198.50 | 200.50 | 203.81 |

Error | 0.02% | 0.01% | 0.11% | 0.04% | 0.27% | 0.01% |

**Table 11.**Modal comparisons of the composite laminated cylindrical shell under F-F boundary condition are given.

F-F | Mode No. | |||||
---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | |

Present | ||||||

Abaqus | ||||||

Present | 21.70 | 24.12 | 60.94 | 64.29 | 115.77 | 119.55 |

Abaqus | 21.72 | 23.11 | 61.02 | 63.85 | 116.05 | 119.51 |

Error | 0.09% | 4.37% | 0.13% | 0.69% | 0.24% | 0.03% |

**Table 12.**Modal comparisons of the FG three-phase composite cylindrical shell under C-C boundary condition are given.

C-C | Mode No. | |||||
---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | |

Present | ||||||

Abaqus | ||||||

Present | 212.30 | 239.63 | 253.98 | 320.21 | 377.26 | 390.48 |

Abaqus | 211.71 | 238.76 | 253.71 | 319.13 | 377.33 | 388.34 |

Error | 0.28% | 0.36% | 0.11% | 0.34% | 0.02% | 0.55% |

**Table 13.**Modal comparisons of the FG three-phase composite cylindrical shell under S-S boundary condition are given.

S-S | Mode No. | |||||
---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | |

Present | ||||||

Abaqus | ||||||

Present | 200.48 | 229.86 | 243.67 | 313.04 | 359.15 | 367.91 |

Abaqus | 200.22 | 229.37 | 243.59 | 312.35 | 357.71 | 367.27 |

Error | 0.13% | 0.21% | 0.03% | 0.22% | 0.40% | 0.17% |

**Table 14.**Modal comparisons of the FG three-phase composite cylindrical shell under F-F boundary condition are given.

F-F | Mode No. | |||||
---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | |

Present | ||||||

Abaqus | ||||||

Present | 32.00 | 41.84 | 90.11 | 102.85 | 171.76 | 185.56 |

Abaqus | 32.01 | 38.88 | 90.17 | 101.3 | 172.07 | 184.95 |

Error | 0.03% | 7.07% | 0.07% | 1.51% | 0.18% | 0.33% |

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## Share and Cite

**MDPI and ACS Style**

Liu, T.; Duan, J.; Zheng, Y.; Qian, Y.
Free Vibrations of a New Three-Phase Composite Cylindrical Shell. *Aerospace* **2023**, *10*, 1007.
https://doi.org/10.3390/aerospace10121007

**AMA Style**

Liu T, Duan J, Zheng Y, Qian Y.
Free Vibrations of a New Three-Phase Composite Cylindrical Shell. *Aerospace*. 2023; 10(12):1007.
https://doi.org/10.3390/aerospace10121007

**Chicago/Turabian Style**

Liu, Tao, Jinqiu Duan, Yan Zheng, and Yingjing Qian.
2023. "Free Vibrations of a New Three-Phase Composite Cylindrical Shell" *Aerospace* 10, no. 12: 1007.
https://doi.org/10.3390/aerospace10121007