# Natural Frequencies Optimization of Thin-Walled Circular Cylindrical Shells Using Axially Functionally Graded Materials

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Functionally Graded Material

_{1}(x) ≤ 1. Note that ${V}_{2}\left(x\right)=1-{V}_{1}\left(x\right)$. In addition to these laws, this work suggests using control points method for optimization problems. In this method, the volume fraction profile is controlled by the volume fractions at some control points that are distributed evenly through the length of the cylinder. The location of ith control point, along the cylinder length, is ${x}_{i}=\left(i-1\right)L/\left(N-1\right)$ in which $N$ is the number of control points. The volume fraction distribution in the cylinder, ${V}_{1}(x)$, can then be estimated by the piecewise cubic hermit interpolating polynomial (PCHIP) [41,42]. The design variables in this case are ${V}_{1,1}$, ${V}_{1,2}$, …, ${V}_{1,N}$.

_{i}is the Young’s modulus, ${\rho}_{i}$ is the density, ${\nu}_{i}$ is the Poisson’s ratio, and the subscript $i$ denotes material 1 or 2.

#### 2.2. Kinematic Relations

## 3. Results and Discussion

#### 3.1. Validation and Convergence

^{3}. The numerical results, which are summarized in Table 1, are close to those available in the literature.

_{2}) with CF boundary conditions. The material volume fractions are graded over the cylinder length according to a power Law (Equation (1)). The materials properties are listed in Table 2. In this study, the dimensions of the cylinders are R = 1 m, $h/R=0.05$, and $L/R=10$. Table 3 shows the convergence study of the fundamental frequencies with different power-law exponents ($\gamma $) and number of elements. The fundamental frequency can be considered as convergent at 36 × 40 elements (i.e., 36 elements in the circumferential direction and 40 elements in the longitudinal direction). In the following investigations, the 36 × 40 element mesh is used.

#### 3.2. Parametric Study

_{2}). The properties of aluminum and zirconia are listed in Table 2. The material constituents vary in axial direction as per of the power law (Equation (1)), where ${V}_{1}(x)$ represents the volume fraction of aluminum. The FGM cylinders have zirconia at $x=0$ and aluminum at $x=L$, as shown in Figure 2.

#### 3.3. Optimization Examples

#### 3.3.1. Maximizing the Fundamental Frequency of FGM Cylindrical Shell

^{−2}. The optimal volume fractions at the control points are summarized in Table 6, and the corresponding aluminum volume fraction distribution is shown in Figure 4. The maximum fundamental frequency is resulted by using 11 control points. The maximum fundamental frequency of the axially FGM cylinder is 29.39 Hz, which is more than the fundamental frequencies of aluminum and zirconia cylinders by 56.0% and 36.4%, respectively. To reduce the computational efforts that resulted from the high number of design variables and the interpolation method, a trigonometric law (Equation (2) is proposed to describe the volume fraction distribution in the optimization process. In this law, there are four design variables only (i.e., $\alpha ,\eta ,\varphi ,$ and $\gamma $). The design variables must be carefully selected to ensure that $0\le {V}_{1}(x)\le 1$. The constraints are assumed as $0\le \alpha \le 0.5$, $-\pi \le \eta ,\varphi \le \pi $, and $0\le \gamma \le 10$. In this case, the number of the population in each generation is assumed to be 40. The optimal design parameters in Equation (2) are listed in Table 6, and the corresponding aluminum volume fraction distributions are shown in Figure 5. The maximum fundamental frequency is 28.60 Hz, which is greater than the fundamental frequency of aluminum and zirconia cylinders by 51.8% and 32.8%, respectively. Figure 4 and Figure 5 show that assigning the stiffer zirconia near the support (i.e., high bending moment region) and the lighter aluminum near the area of high modal displacement increases the fundamental frequency.

#### 3.3.2. Maximizing the Gap between Two Adjoining Natural Frequencies in FGM Cylindrical Shell

## 4. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 4.**The optimal aluminum volume fraction profile which maximizes the fundamental frequency (control points approach).

**Figure 5.**The optimal aluminum volume fraction profile which maximizes the fundamental frequency (trigonometric law, Equation (2)).

**Figure 6.**The optimal aluminum volume fraction profile which maximizes the gap between ${f}_{1}$ and ${f}_{2}$ (control points approach).

**Figure 7.**The optimal aluminum volume fraction profile which maximizes the gap between ${f}_{1}$ and ${f}_{2}$ (trigonometric law, Equation (2)).

Frequency (Hz) | ||||||
---|---|---|---|---|---|---|

Elements | 1 | 2 | 3 | 4 | 5 | |

Present | 20 × 10 | 94.08 | 108.09 | 108.6 | 138.6 | 167.3 |

24 × 10 | 93.26 | 106.25 | 108.49 | 135.29 | 167.78 | |

36 × 20 | 92.58 | 105.28 | 108.15 | 133.95 | 168.28 | |

Ref. [49] | 92.55 | 105.05 | 108.47 | 133.60 | 169.12 | |

Ref. [31] | 95.38 | 106.33 | 114.39 | 134.23 | 171.84 |

Property | Al | ZrO_{2} |
---|---|---|

E (GPa) | 20 | 205 |

ρ (kg/m^{3}) | 2700 | 6050 |

v | 0.3 | 0.31 |

**Table 3.**Convergence of the fundamental frequency (Hz) of clamped-free FGM cylinders with number of finite elements.

$\mathit{\gamma}$ | |||
---|---|---|---|

Elements | 0.5 | 1 | 10 |

12 × 5 | 23.67 | 24.88 | 21.84 |

24 × 5 | 24.26 | 25.47 | 22.38 |

36 × 5 | 24.57 | 25.80 | 22.67 |

36 × 10 | 25.17 | 26.40 | 23.19 |

36 × 20 | 25.32 | 26.53 | 23.35 |

36 × 40 | 25.34 | 26.54 | 23.37 |

**Table 4.**Variations of the fundamental frequencies (Hz) with the power-law exponent and length to radius ratio ($L/R$ ).

$\mathit{\gamma}$ | ||||||||
---|---|---|---|---|---|---|---|---|

$\mathit{L}/\mathit{R}$ | Al | 0.5 | 1 | 2 | 5 | 10 | 100 | ZrO_{2} |

0.2 | 1347.0 | 1689.1 | 1758.1 | 1754.9 | 1678.2 | 1622.3 | 1549.0 | 1539.9 |

0.5 | 469.0 | 559.8 | 581.0 | 586.6 | 572.9 | 559.1 | 538.77 | 536.21 |

1 | 231.0 | 282.6 | 293.2 | 296.1 | 286.1 | 277.4 | 265.54 | 264.09 |

2 | 119.3 | 136.1 | 140.6 | 142.3 | 140.7 | 139.1 | 136.7 | 136.4 |

5 | 41.45 | 47.71 | 49.31 | 49.76 | 49.05 | 48.40 | 47.50 | 47.39 |

10 | 18.84 | 25.35 | 26.54 | 26.33 | 24.63 | 23.38 | 21.75 | 21.54 |

15 | 8.68 | 11.75 | 12.31 | 12.20 | 11.38 | 10.79 | 10.02 | 9.92 |

20 | 4.94 | 6.71 | 7.03 | 6.97 | 6.50 | 6.16 | 5.71 | 5.65 |

**Table 5.**Variations of the fundamental frequencies (Hz) with the power-law exponent and thickness to radius ratio ($h/R$ ).

$\mathit{\gamma}$ | ||||||||
---|---|---|---|---|---|---|---|---|

$\mathit{h}/\mathit{R}$ | Al | 0.5 | 1 | 2 | 5 | 10 | 100 | ZrO_{2} |

0.001 | 3.52 | 4.38 | 4.50 | 4.57 | 4.43 | 4.27 | 4.05 | 4.02 |

0.005 | 7.09 | 9.17 | 9.57 | 9.52 | 9.01 | 8.64 | 8.16 | 8.10 |

0.01 | 9.13 | 10.92 | 11.33 | 11.37 | 11.04 | 10.80 | 10.48 | 10.44 |

0.02 | 14.68 | 16.14 | 16.59 | 16.83 | 16.87 | 16.85 | 16.80 | 16.79 |

0.03 | 18.82 | 22.26 | 22.79 | 23.23 | 23.57 | 23.36 | 21.73 | 21.52 |

0.04 | 18.83 | 25.34 | 26.53 | 26.32 | 24.62 | 23.37 | 21.74 | 21.53 |

0.05 | 18.84 | 25.34 | 26.54 | 26.33 | 24.63 | 23.38 | 21.75 | 21.54 |

Material Type | ${\mathit{f}}_{1}\left(\mathbf{Hz}\right)$ | Optimal Design Variables |
---|---|---|

Aluminum | 18.84 | - |

Zirconia | 21.54 | - |

FGM (5 Control Points) | 28.93 | $\left\{0,0,0.39,1,1\right\}$ |

FGM (7 Control Points) | 29.25 | $\left\{0,0,0.01,0.42,0.99,1,1\right\}$ |

FGM (11 Control Points) | 29.39 | $\left\{0,0,0,0.019,0.090,0.450,0.931,0.982,1,1,1\right\}$ |

FGM (Equation (2)) | 28.60 | $\alpha =0.5$$,\eta =-0.397$$,\varphi =-2.813$$,\gamma =1.351$ |

Material Type | ${\mathit{f}}_{1}\left(\mathbf{Hz}\right)$ | ${\mathit{f}}_{2}\left(\mathbf{Hz}\right)$ | ${\mathit{f}}_{2}-{\mathit{f}}_{1}\left(\mathbf{Hz}\right)$ | Optimal Design Variables |
---|---|---|---|---|

Aluminum | 18.84 | 33.75 | 14.91 | - |

Zirconia | 21.54 | 38.59 | 17.05 | - |

FGM (5 Control Points) | 13.87 | 37.94 | 24.07 | $\left\{1,1,0.39,0,0\right\}$ |

FGM (7 Control Points) | 13.57 | 37.95 | 24.38 | $\left\{1,1,1,0.43,0.01,0,0\right\}$ |

FGM (11 Control Points) | 13.32 | 37.96 | 24.64 | $\left\{1,1,1,0.995,0.994,0.469,0.007,0.004,0,0,0\right\}$ |

FGM (Equation (2)) | 14.13 | 37.97 | 23.84 | $\alpha =0.5$$,\eta =-0.406$$,\varphi =0.473$$,\gamma =1.381$ |

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**MDPI and ACS Style**

Alshabatat, N.T.
Natural Frequencies Optimization of Thin-Walled Circular Cylindrical Shells Using Axially Functionally Graded Materials. *Materials* **2022**, *15*, 698.
https://doi.org/10.3390/ma15030698

**AMA Style**

Alshabatat NT.
Natural Frequencies Optimization of Thin-Walled Circular Cylindrical Shells Using Axially Functionally Graded Materials. *Materials*. 2022; 15(3):698.
https://doi.org/10.3390/ma15030698

**Chicago/Turabian Style**

Alshabatat, Nabeel Taiseer.
2022. "Natural Frequencies Optimization of Thin-Walled Circular Cylindrical Shells Using Axially Functionally Graded Materials" *Materials* 15, no. 3: 698.
https://doi.org/10.3390/ma15030698