# Dynamic Study of a Rooftop Vertical Axis Wind Turbine Tower Based on an Automated Vibration Data Processing Algorithm

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## Abstract

**:**

## 1. Introduction

## 2. The VAWT Wind Turbine and Vibration Measurements

## 3. Data Analysis

#### 3.1. Frequency Domain Analyses

#### 3.2. SSI Method

#### 3.3. Automatic Pole-Picking Procedure

- (1)
- Calculate frequency distances ${d}_{ij}$ and $MA{C}_{ij}$ between each pole set, $i$ and $j$:$${d}_{ij}=\left|\frac{{f}_{i}-{f}_{j}}{{f}_{j}}\right|;$$$$MA{C}_{ij}=\frac{{\left({\mathsf{\phi}}_{i}^{T}{\mathsf{\phi}}_{j}\right)}^{2}}{\left({\mathsf{\phi}}_{i}^{T}{\mathsf{\phi}}_{i}\right)\left({\mathsf{\phi}}_{j}^{T}{\mathsf{\phi}}_{j}\right)}$$
- (2)
- If ${d}_{ij}$ and $MA{C}_{ij}$ satisfy the following criteria, then add one to the local density value ${\rho}_{i}$ of pole $i$:
- ${d}_{ij}\times 100\%<1\%$;
- $\left(1-MA{C}_{ij}\right)\times 100\%<1\%$.

- (3)
- Pick the poles with relatively high local densities.

## 4. Results and Discussion

#### 4.1. Vibration Composition Analysis

#### 4.2. Mode Identification

## 5. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## Abbreviations

HAWT | horizontal axis wind turbine; |

VAWT | vertical axis wind turbine; |

SHM | structural health monitoring; |

SSI | stochastic subspace identification; |

PSD | power spectrum density; |

MAC | modal assurance criterion. |

## Symbols

${k}_{t}$ | tensile stiffness of a single hoop; |

${k}_{s}$ | shear stiffness of a single hoop; |

${k}_{M10}$, ${k}_{M12}$ | tensile stiffness of M10 bolt and M12 bolt, respectively; |

${k}_{R}$ | compression stiffness of the rubber gasket; |

${k}_{M12,\mathrm{s}}$ | shear stiffness of the M12 bolt; |

${P}_{XX}\left(f\right)$ | PSD of the response of a structure; |

$T\left(f\right)$ | transfer function; |

$I$ | unit matrix; |

${x}_{k}$ | state vector; |

${y}_{k}$ | measurement vector; |

${w}_{k}$ | excitation vector; |

${v}_{k}$ | measurement noise vector; |

$A$ | state transition matrix; |

$C$ | output location matrix; |

${k}_{t}$ | number of measurement; |

$H$ | Hankel matrix; |

$O$ | projection matrix; |

$\mathsf{\Gamma}$ | observability matrix; |

$\widehat{X}$ | Kalman filter state sequence; |

$\lambda $ | eigenvalue of $A$; |

$\mathsf{\Psi}$ | eigenvector of $A$; |

${f}_{s}$ | sampling frequency; |

$2n$ | maximum calculation order; |

$f$ | natural frequency; |

$\xi $ | damping ratio; |

$\mathsf{\phi}$ | mode shape; |

$\rho $ | local density of a result point; |

$d$ | frequency distance between two result points; |

$MAC$ | mode shape similarity (modal assurance criterion) between two result points. |

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**Figure 2.**Ambient responses of the #6 building’s roof: (

**a**) time histories along the X-direction; (

**b**) power spectrum density (PSD) along the X-direction; (

**c**) time histories along the Y-direction; (

**d**) PSD along the Y-direction.

**Figure 4.**The H-type vertical axis wind turbine (VAWT): (

**a**) modeling of the vibration system; (

**b**) the VAWT and the details of its supports (including schematics; unit: mm).

**Figure 6.**Deployment of sensors: (

**a**) schematics of the VAWT wind turbine; (

**b**) the sensor attachments.

**Figure 8.**Typical monitoring data, recorded on on 28 August 2016: (

**a**) wind turbine rotational speed; (

**b**) temperature; (

**c**) power generation distribution.

**Figure 11.**Stabilization diagram using the stochastic subspace identification (SSI) method and identified vibration parameters: (

**a**) calculation order as a function of frequency; (

**b**) damping ratio as a function of frequency (raw results—prior to pole-picking algorithm; clean results—after application of pole-picking algorithm).

**Figure 14.**Comparison of acceleration PSDs measured from the roof of the #6 building and the VAWT: (

**a**) X-direction; (

**b**) Y-direction.

**Figure 16.**Identified mode shapes and the associated schematics of the four dominant modes along (

**a**) the X-direction, and (

**b**) the Y-direction (26 August to 4 September).

Item | Piezoelectric Sensors (Tower) | Electromagnetic Sensors (Building) |
---|---|---|

Sensitivity | 9.83 mV/(ms^{2}) | 329 mV/(ms^{2}) |

Resolution | 0.001 ms^{2} | 3 × 10^{−6} ms^{2} |

Frequency range | 1.0 to 5000 Hz (±5%) | 0.25 to 100 Hz (+1 dB to −3 dB) |

0.5 to 7000 Hz (±10%) | ||

Acceleration range | 500 ms^{2} | 200 ms^{2} |

Weight | 4.4 g | 800 g |

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

1st | 2nd | 3rd | 4th | |

X | 0.94 | 3.60 | 6.90 | 10.60 |

Y | 0.71 | 1.30 | 4.10 | 9.60 |

Parameter | Value |
---|---|

Rated power | 2 kW |

Rated spin speed | 250 rpm |

Rated wind speed | 25 m/s |

Maximum design wind speed | 32 m/s |

Blade height | 2 m |

Blade span | 2.2 m |

Weight (generator + blades) | 240 kg |

Hoop | |

Thickness | 6 mm |

Material | Q235 steel |

Bolts | M10 |

Expansion bolts | M12 |

Anchorage depth | ≥75 mm |

Rubber gasket thickness | 5 mm |

Tower | |

Thickness | 4 mm |

Material | Q235 steel |

External diameter | 250 mm |

Base | |

Bolts | M18 |

Correlation Coefficients | Rotation Speed^{1/2} | Wind Speed (Raw Data) | Temperature |

Rotation speed | 1 | 0.581 | 0.757 |

Wind speed | 0.581 | 1 | 0.516 |

Temperature | 0.757 | 0.516 | 1 |

Correlation Coefficients | Rotation Speed^{1/2} | Wind Speed (Moving Average) | Temperature |

Rotation speed | 1 | 0.888 | 0.757 |

Wind speed | 0.888 | 1 | 0.677 |

Temperature | 0.757 | 0.677 | 1 |

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

1st | 2nd | 3rd | 4th | |

x | 3.6 | 5.8 | 10.7 | |

y | 3.8 | 6.0 | 9.7 | 11.7 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Wang, Y.; Lu, W.; Dai, K.; Yuan, M.; Chen, S.-E.
Dynamic Study of a Rooftop Vertical Axis Wind Turbine Tower Based on an Automated Vibration Data Processing Algorithm. *Energies* **2018**, *11*, 3135.
https://doi.org/10.3390/en11113135

**AMA Style**

Wang Y, Lu W, Dai K, Yuan M, Chen S-E.
Dynamic Study of a Rooftop Vertical Axis Wind Turbine Tower Based on an Automated Vibration Data Processing Algorithm. *Energies*. 2018; 11(11):3135.
https://doi.org/10.3390/en11113135

**Chicago/Turabian Style**

Wang, Ying, Wensheng Lu, Kaoshan Dai, Miaomiao Yuan, and Shen-En Chen.
2018. "Dynamic Study of a Rooftop Vertical Axis Wind Turbine Tower Based on an Automated Vibration Data Processing Algorithm" *Energies* 11, no. 11: 3135.
https://doi.org/10.3390/en11113135