# Flow-Induced Vibration Hybrid Modeling Method and Dynamic Characteristics of U-Section Rubber Outer Windshield System of High-Speed Trains

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## Abstract

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## 1. Introduction

## 2. Extraction of Aerodynamic Load on U-Section Rubber Outer Windshield

#### 2.1. Geometric Model and Computational Setup

^{−5}s, and each step is iterated 20 times, for a total of 10,000 time steps.

#### 2.2. Data Analysis

#### 2.3. Aerodynamic Load Integration

## 3. Modal Experiment and Analysis of U-Section Rubber Outer Windshield Structure

#### 3.1. Test Object, Measurement Point and Excitation Point Arrangement

#### 3.2. Modal Analysis

## 4. Establishment of U-Section Rubber Outer Windshield Flow-Induced Vibration Response Analysis Method

#### 4.1. Flow-Induced Vibration Model Based on Mode Superposition Method of U-Section Rubber Outer Windshield Structure

**u**} is the generalized eigenvector of the mass matrix [

**M**] and the stiffness matrix [

**K**], and the vector consisting of $\left[{\lambda}_{1},{\lambda}_{2},\dots ,{\lambda}_{n}\right]$ is its generalized eigenvalue. Equation (13) is the general eigenvalue problem if the mass matrix is the unit matrix, and therefore the modal vectors can be normalized. The generalized mass matrix corresponding to the ith order modal vector is

#### 4.2. Dynamic Response of U-Section Rubber Outer Windshield Structure

## 5. Dynamic Response Analysis of U-Section Rubber Outer Windshield

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Schematic diagram of U-section rubber outer windshield structure flow-induced vibration analysis.

**Figure 6.**Schematic diagram of outer windshield measurement points. (

**a**) Measurement points arranged circumferentially along the windshield section. (

**b**) Lateral arrangement of measuring points on the outer surface of windshield.

**Figure 7.**Pressure diagram of outer windshield section at 400 km/h. (

**a**) Pressure diagram of head train windshield section. (

**b**) Pressure diagram of tail train windshield section.

**Figure 8.**Pressure diagram of U-section rubber outer windshield surface at 400 km/h. (

**a**) Side pressure cloud. (

**b**) Side pressure cloud. (

**c**) Top pressure cloud.

**Figure 9.**Aerodynamic pressure difference between inside and outside surface at different measuring points of head train windshield. (

**a**) The layout diagram of horizontal and longitudinal measurement points. (

**b**) Comparison of aerodynamic pressure difference at horizontal measurement points. (

**c**) Comparison of aerodynamic pressure difference at longitudinal measurement points.

**Figure 11.**U-section rubber outer windshield structure and lateral part of the measurement points (red number) and excitation points (yellow five-pointed star) schematic. (

**a**) The layout diagram of the modal experiment of the outer windshield. (

**b**) The bottom section of the outer windshield. (

**c**) The bottom of outer windshield. (

**d**) The measurement points (red number) and excitation points (yellow five-pointed star).

**Figure 12.**The first four mode shapes of the lateral part of the outer windshield structure. (

**a**) ${\omega}_{1}$ = 12.02 Hz. (

**b**) ${\omega}_{2}$ = 14.25 Hz. (

**c**) ${\omega}_{3}$ = 18.31 Hz. (

**d**) ${\omega}_{4}$ = 23.66 Hz. (

**e**) ${\omega}_{5}$ = 26.72 Hz. (

**f**) ${\omega}_{6}$ = 28.41 Hz. Nodes 1–20 correspond to the modal test points in Figure 11d and different colored lines represent the outer outline of the windshield structure.

**Figure 13.**Schematic diagram of displacement response measurement point selection. The red circles are the displacement response measurement point 1–12.

**Figure 14.**The displacement response of measurement points 1–6 (

**a**) and 7–12 (

**b**) on both sides of the U-section rubber outer windshield.

**Figure 15.**The velocity response of measurement points 1–6 (

**a**) and 7–12 (

**b**) on both sides of the U-section rubber outer windshield.

**Figure 16.**The displacement modal contribution of U-section rubber outer windshield measuring point 6 (

**a**) and measuring point 10 (

**b**).

Mode | Natural Frequency (Hz) | Damping Ratio (%) |
---|---|---|

1 | 12.02 | 6.25 |

2 | 14.25 | 6.92 |

3 | 18.31 | 6.37 |

4 | 23.66 | 8.63 |

5 | 26.72 | 6.21 |

6 | 28.41 | 5.36 |

7 | 31.78 | 5.72 |

8 | 33.46 | 5.70 |

9 | 39.57 | 4.10 |

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**MDPI and ACS Style**

Yu, Y.; Lv, P.; Liu, X.; Liu, X.
Flow-Induced Vibration Hybrid Modeling Method and Dynamic Characteristics of U-Section Rubber Outer Windshield System of High-Speed Trains. *Appl. Sci.* **2023**, *13*, 5813.
https://doi.org/10.3390/app13095813

**AMA Style**

Yu Y, Lv P, Liu X, Liu X.
Flow-Induced Vibration Hybrid Modeling Method and Dynamic Characteristics of U-Section Rubber Outer Windshield System of High-Speed Trains. *Applied Sciences*. 2023; 13(9):5813.
https://doi.org/10.3390/app13095813

**Chicago/Turabian Style**

Yu, Yizheng, Pengxiang Lv, Xiao Liu, and Xiang Liu.
2023. "Flow-Induced Vibration Hybrid Modeling Method and Dynamic Characteristics of U-Section Rubber Outer Windshield System of High-Speed Trains" *Applied Sciences* 13, no. 9: 5813.
https://doi.org/10.3390/app13095813