# Influence of the Flexible Tower on Aeroelastic Loads of the Wind Turbine

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## Abstract

**:**

## 1. Introduction

## 2. Analytical Study

#### 2.1. Equations of Motion for Vibrating Systems

#### 2.2. Tower Finite Element Modeling

#### 2.3. Eigenvalue Analysis

#### 2.4. Solving the Nacelle Attitude Angle Using Euler Angles

## 3. Calculation Method

#### 3.1. Calculation of Aerodynamic Loads under Consideration of NAF

- The inertial coordinate system S coincides with the tower bottom coordinate system ${S}_{BOT}$, and the nacelle coordinate system ${S}_{N}$ coincides with the tower top coordinate system ${S}_{top}$.
- When the azimuth angle θ = 0, the wind turbine rotation plane coordinate system ${S}_{R}$ coincides with the hub coordinate system ${L}_{H}$, and ${S}_{R}$ rotates with the azimuth angle θ.
- The tower height is L, and the horizontal distance from the top of the tower to the hub is ${L}_{H}$.
- The spindle inclination angle and blade taper angle are ${\beta}_{T}$ and ${\beta}_{C}$.

#### 3.2. Tower Load Calculation

#### 3.3. Solving Structural Equations of Motion

- Initial Calculation
- Firstly, the overall characteristic matrices K, M and C of the vibrating system are calculated.
- From the initial conditions ${x}_{0}$ and ${\dot{x}}_{0}$, ${\ddot{x}}_{0}$ is calculated.$$\ddot{{x}_{0}}={M}^{-1}\left({F}_{0}-C\dot{{x}_{0}}-K{x}_{0}\right)$$
- Step $\mathsf{\Delta}t$ is selected with parameters $\alpha $, $\beta $ and calculate the integration constants.$${A}_{1}=\frac{1}{\alpha \u2206{t}^{2}},{A}_{2}=\frac{\beta}{\alpha \u2206t},{A}_{3}=\frac{1}{\alpha \u2206t},{A}_{4}=\left(\frac{1}{2\alpha}-1\right),{A}_{5}=\frac{\u2206t}{2}\left(\frac{\beta}{\alpha}-2\right),{A}_{6}=\left(\frac{\beta}{\alpha}-1\right)$$
- The equivalent stiffness matrix $\overline{K}$ is calculated.$$\overline{K}=K+{A}_{1}M+{A}_{2}C$$

- For each time step
- The equivalent load vector $\overline{F}$ at moment $t+\mathsf{\Delta}t$ is determined.$$F={F}_{t+\u2206t}+\left[{A}_{1}{x}_{t}+{A}_{3}{\dot{x}}_{t}+{A}_{4}{\ddot{x}}_{t}\left]M+\right[{A}_{2}{x}_{t}+{A}_{6}{\dot{x}}_{t}+{A}_{3}{\ddot{x}}_{t}\right]C$$
- The displacement at moment $t+\mathsf{\Delta}t$ is calculated.$${x}_{t+\u2206t}={\overline{K}}^{-1}{\overline{F}}_{t+\u2206t}$$
- The velocity at time $t+\mathsf{\Delta}t$ and the acceleration are solved.$${\ddot{x}}_{t+\u2206t}=\frac{1}{\alpha \u2206{t}^{2}}\left({x}_{t+\u2206t}-{x}_{t}\right)-\frac{1}{\alpha \u2206t}{\dot{x}}_{t}-\left(\frac{1}{2\alpha}-1\right){\ddot{x}}_{t}$$$${\dot{x}}_{t+\u2206t}={\dot{x}}_{t}+\left(1-\beta \right)\u2206t{\ddot{x}}_{t}+\beta \u2206t{\ddot{x}}_{t+\u2206t}$$

#### 3.4. Dynamic Response Analysis Process

## 4. Flexible Tower Vibration Feedback Analysis

#### 4.1. Tower Load Error Analysis

- The nacelle and the tower have only yaw motion.
- The center of mass of the nacelle coincides with the center of the tower top.

#### 4.2. Dynamic Response Analysis of Tower System

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 5.**Wind turbine load coordinate system [16]. (

**a**) Blade coordinate system. (

**b**) Tower coordinate system.

**Figure 7.**Relationship between tower turning angle and cabin attitude. (

**a**) Forward and backward. (

**b**) Left and right.

**Figure 10.**Relative error of tower top displacement. (

**a**) Longitudinal direction. (

**b**) Crosswind direction.

${\mathit{K}}_{\mathit{R}}\left(\mathbf{G}\mathbf{N}\mathbf{m}/\mathbf{r}\mathbf{a}\mathbf{d}\right)$ | ${\mathit{K}}_{\mathit{L}}\left(\mathbf{G}\mathbf{N}/\mathbf{m}\right)$ | 1st | 2nd | 3rd |
---|---|---|---|---|

10 | 0.5 | 0.20862 | 2.0824 | 6.8703 |

10 | 1 | 0.20874 | 2.0991 | 7.1053 |

10 | 5 | 0.20882 | 2.1124 | 7.2858 |

20 | 0.5 | 0.25229 | 2.2495 | 6.9903 |

20 | 1 | 0.25245 | 2.2725 | 7.2701 |

20 | 5 | 0.25249 | 2.2909 | 7.4874 |

50 | 0.5 | 0.29507 | 2.5064 | 7.2177 |

50 | 1 | 0.29533 | 2.5409 | 7.5925 |

50 | 5 | 0.29554 | 2.5683 | 7.8877 |

100 | 0.5 | 0.31463 | 2.6753 | 7.4038 |

100 | 1 | 0.31494 | 2.7180 | 7.8656 |

100 | 5 | 0.31519 | 2.7519 | 8.2307 |

Modal | 1st | 2nd | 3rd |
---|---|---|---|

Natural frequency | 0.3391 | 3.0634 | 9.0983 |

Name | Value | Name | Value |
---|---|---|---|

Rated power | 5000 KW | Cone angle | 2.5° |

Rated wind speed | 11.4 m/s | Rated rotor speed | 12.1 rpm |

Cut-in wind speed | 3 m/s | Rotor diameter | 126 m |

Cut-out wind speed | 25 m/s | Tower mass density | 8500 kg/m³ |

Blade length | 61.5 m | Tower’s modulus of elasticity | 2.1 × 10^{11} N/m^{2} |

Blade mass | 17,740 kg | Structural-damping ratio | 1% |

Blade number | 3 | Tower height | 86.7 m |

Hub height | 90 m | Tower-base diameter | 6 m |

Hub mass | 56,780 kg | Tower-base thickness | 0.0351 m |

Nacelle mass | 240,000 kg | Tower-top diameter | 3.87 m |

Shaft tilt | 5° | Tower-top thickness | 0.0247 |

Height (m) | $\mathbf{Modulus}\mathbf{of}\mathbf{Elasticity}(\mathbf{N}/{\mathbf{m}}^{2})$ | Diameter and Wall Thickness of Tower Top (m) | Diameter and Wall Thickness of Tower Bottom (m) | Structural Damping Ratio |
---|---|---|---|---|

86.7 | 2.1 × 10^{11} | 3.87, 0.247 | 6, 0.351 | 1% |

200 | 2.1 × 10^{11} | 3.87, 0.247 | 6, 0.351 | 1% |

300 | 2.1 × 10^{11} | 3.87, 0.247 | 6, 0.351 | 1% |

${\mathit{K}}_{\mathit{L}}(\mathbf{GN}/\mathbf{m})$ | ${\mathit{K}}_{\mathit{R}}(\mathbf{GNm}/\mathbf{rad})$ | |
---|---|---|

Case 1 | 1 | 10 |

Case 2 | 1 | 20 |

Case 3 | 1 | 50 |

Case 4 | Rigid foundation |

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**MDPI and ACS Style**

Hao, J.; Wang, Z.; Yi, W.; Chen, Y.; Chen, J. Influence of the Flexible Tower on Aeroelastic Loads of the Wind Turbine. *Appl. Sci.* **2021**, *11*, 8876.
https://doi.org/10.3390/app11198876

**AMA Style**

Hao J, Wang Z, Yi W, Chen Y, Chen J. Influence of the Flexible Tower on Aeroelastic Loads of the Wind Turbine. *Applied Sciences*. 2021; 11(19):8876.
https://doi.org/10.3390/app11198876

**Chicago/Turabian Style**

Hao, Junbo, Zedong Wang, Wenwu Yi, Yan Chen, and Jiyao Chen. 2021. "Influence of the Flexible Tower on Aeroelastic Loads of the Wind Turbine" *Applied Sciences* 11, no. 19: 8876.
https://doi.org/10.3390/app11198876