# Effects of the Configuration of Trailing Edge on the Flutter of an Elongated Bluff Body

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## Abstract

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_{s}, gradually increases, meaning the flutter stability gradually increases. The results reveal that LCF may still occur to the bridge section with a streamlined front edge, and, in some cases, it also may have a range of wind speeds in which LCF occurs.

## 1. Introduction

## 2. Experiment Setup

#### 2.1. Model Details

_{v}and f

_{t}, representing the vertical and torsional damping ratio, respectively, are both less than 0.2%, and the details of the model’s dynamic characteristics are shown in Table 1. The blockage ratio is 4.25%.

#### 2.2. Measurement Details

_{in}, was gradually increased to the model’s critical wind speed of flutter, and the laser-displacement system was used to collect the displacement response of the model. The laser-displacement system includes two IL-100 sensors produced by Keyence, USB-6288 produced by National Instruments, and a personal computer. The sampling frequency for all tests is 1000 Hz. The torsional and vertical responses are calculated as follows:

## 3. Results and Discussions

#### 3.1. Dynamic Response of Flutter with Different Trailing Configurations

#### 3.1.1. 30c-30 Case

_{r}is higher, it develops into bending-torsional coupling vibration.

_{r}is increased to 124.25, the LCF develop into hard flutter. Figure 5 shows this process. The amplitudes of the vertical and torsional degrees of freedom gradually increase to divergence.

#### 3.1.2. 30c-45 Case

_{r}is increased to 87.50, the bridge begins to undergo torsional vibration, while the vibration is low-amplitude in the vertical degree of freedom and the effect of static wind is relatively large. When U

_{r}is in the range of 87.50~96.25, there is stable single-frequency vibration on the torsional freedom of the bridge deck, while there are some low frequency components in the vertical degrees of freedom. In terms of vertical degrees of freedom, the effect of static wind takes a larger proportion. The coupling degree of vertical and torsional vibration is low.

_{r}is increased from 96.25 to 97.13. There is no stable vibration in the vertical degree of freedom, which means the bridge deck’s vibration is pure torsional vibration during the process of LCF to hard flutter.

#### 3.1.3. 30c-60 Case

_{r}is increased from 127.75 to 128.63, the vibration is developed from stable LCF to hard flutter. In this process, both vertical degree of freedom and torsional degree of freedom have large amplitudes, and both have trend of divergence.

#### 3.1.4. 30c-90 Case

_{r}is increased to 140.00, large-amplitude and bending-torsion coupling LCF occurs to the bridge deck. The amplitude is greatly increased compared to the case of ${U}_{r}<131.25$. The vibration frequency of the vertical degrees of freedom and the torsional degrees of freedom are equal, indicating that the vibrations of the two degrees of freedom are completely coupled.

#### 3.1.5. 30c-180 Case

#### 3.2. Frequency Domain Analysis

#### 3.3. Wavelet and Hilbert Analysis

#### 3.4. Phase Lag between Torsional and Vertcial Vibration

#### 3.5. Discussion of Effect of TE

## 4. Conclusions

- (1)
- Limit cycle flutter may still occur in the bridge section with a streamlined front edge and, in some cases, it also has a wider limit cycle flutter wind speed range.
- (2)
- When the bending-torsional coupling flutter is fully developed, the torsional degree of freedom slightly lags behind the vertical degree of freedom.
- (3)
- As the trailing edge becomes more and more blunt, the critical wind speed U
_{s}gradually increases, and the flutter stability gradually increases. The wake vortex motion has a certain correlation with the flutter stability of the bridge deck.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Details of the test model. (

**a**) Overview test model (case 30c-30). (

**b**) Cross section. (

**c**) Details about FE. (

**d**) Details about the masses added inside TE.

**Figure 3.**The setup of the vibration system. The steel strand is tightened by bolts. The model of bridge deck is placed in the center of the test section.

**Figure 4.**Time history of displacement of the bridge deck in case 30c-30. (

**a**) Torsional degree of freedom; and (

**b**) vertical degree of freedom.

**Figure 5.**Time history of vibration divergence in 30c-30 case. (

**a**) Torsional degree of freedom; and (

**b**) vertical degree of freedom.

**Figure 6.**Time history of displacement of the bridge deck in case 30c-45. (

**a**) Torsional degree of freedom; and (

**b**) vertical degree of freedom.

**Figure 7.**Time history of vibration divergence in 30c-45 case. (

**a**) Torsional degree of freedom; and (

**b**) vertical degree of freedom.

**Figure 8.**Time history of displacement of the bridge deck in case 30c-60. (

**a**) Torsional degree of freedom; and (

**b**) vertical degree of freedom.

**Figure 9.**Time history of vibration divergence in 30c-60 case. (

**a**) Torsional degree of freedom; and (

**b**) vertical degree of freedom.

**Figure 10.**Time history of displacement of the bridge deck in case 30c-90. (

**a**) Torsional degree of freedom; and (

**b**) vertical degree of freedom.

**Figure 11.**Time history of vibration divergence in 30c-90 case. (

**a**) Torsional degree of freedom; and (

**b**) vertical degree of freedom.

**Figure 12.**Time history of displacement of the bridge deck in case 30c-180. (

**a**) Torsional degree of freedom; and (

**b**) vertical degree of freedom.

**Figure 13.**Time history of vibration divergence in 30c-180 case. (

**a**) Torsional degree of freedom; and (

**b**) vertical degree of freedom.

**Figure 14.**Vibration frequency of each case. (

**a**) 30c-30 case; (

**b**) 30c-45 case; (

**c**) 30c-60 case; (

**d**) 30c-90 case; and (

**e**) 30c-180 case.

**Figure 15.**Distribution of vibration in case 30c-30 when ${U}_{r}$ is 122.5. (

**a**) Torsional degree of freedom; and (

**b**) vertical degree of freedom.

**Figure 16.**Distribution of vibration in case 30c-30 when ${U}_{r}$ is increased from 148.75 to 153.13. (

**a**) Torsional degree of freedom; and (

**b**) vertical degree of freedom.

**Figure 17.**Phase trajectory of case 30c-30 under different wind speeds. (

**a**) Torsional degree of freedom; and (

**b**) vertical degree of freedom.

**Figure 18.**Phase trajectory of case 30c-180 in different time periods when ${U}_{r}$ is increased from 148.75 to 153.13. (

**a**) Torsional degree of freedom; and (

**b**) vertical degree of freedom.

**Figure 19.**The amplitude ratio of torsional vibration to vertical vibration at various apex angles of the trailing edge.

**Figure 20.**The phase difference between torsional vibration and vertical vibration at various apex angles of the trailing edge.

Serial Number | Fundamental Frequency (Hz) | Damping Ratio (%) | ||
---|---|---|---|---|

Vertical | Torsional | Vertical | Torsional | |

30c-30 | 2.640 | 5.066 | 0.193 | 0.141 |

30c-45 | 2.63 | 5.028 | 0.125 | 0.092 |

30c-60 | 2.625 | 5.020 | 0.104 | 0.145 |

30c-90 | 2.640 | 5.035 | 0.157 | 0.170 |

30c-180 | 2.623 | 5.005 | 0.176 | 0.126 |

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**MDPI and ACS Style**

Feng, J.; Wu, B.; Laima, S.
Effects of the Configuration of Trailing Edge on the Flutter of an Elongated Bluff Body. *Appl. Sci.* **2021**, *11*, 10818.
https://doi.org/10.3390/app112210818

**AMA Style**

Feng J, Wu B, Laima S.
Effects of the Configuration of Trailing Edge on the Flutter of an Elongated Bluff Body. *Applied Sciences*. 2021; 11(22):10818.
https://doi.org/10.3390/app112210818

**Chicago/Turabian Style**

Feng, Jie, Buchen Wu, and Shujin Laima.
2021. "Effects of the Configuration of Trailing Edge on the Flutter of an Elongated Bluff Body" *Applied Sciences* 11, no. 22: 10818.
https://doi.org/10.3390/app112210818