# Dynamic Characteristics of Urban Rail Train in Multivehicle Marshaling under Traction Conditions

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Dynamic Modeling

## 3. Model Validation

## 4. Analysis and Discussion

#### 4.1. Analysis of Safety and Critical Velocity

#### 4.1.1. Security Analysis

#### 4.1.2. Nonlinear Critical Speed Analysis

#### 4.2. Vibration and Comfort Analysis of Urban Rail Trains

#### 4.2.1. Vibration Response Analysis

^{2}. T1 is the smallest, with a variation range of 0.011~0.365 m/s

^{2}. In terms of lateral acceleration, there is no obvious difference between the carriages, and the overall variation range is 0.022~0.472 m/s

^{2}.

^{2}and 0.239~1.512 m/s

^{2}, respectively, and the lateral acceleration ranges are 0.083~1.062 m/s

^{2}and 0.086~1.042 m/s

^{2}, respectively. The vibration acceleration of the four motor cars is greater than that of the two trailers because the traction motor will generate large vibration during operation, and the gear mechanism adopts a helical gear which will generate large vertical vibration and lateral vibration during operation. The vertical vibration acceleration has a significant effect.

^{2}and 0.15~11.84 m/s

^{2}, respectively, and the ranges of the vertical and lateral accelerations of the trailer are 0.13~3.47 m/s

^{2}and 0.11~3.01 m/s

^{2}, respectively. The vibration acceleration of the axle box is significantly greater than that of the bogie frame, and the vibration acceleration of the bogie frame is significantly greater than that of the car body, indicating that the primary suspension and the secondary suspension suppress most of the vibration.

_{s}, the double frequency is denoted by 2f

_{s}, and the triple frequency is denoted by 3f

_{s}. The frequency multiplication response wheel-to-body octave response gradually decreases because of the gradual reduction in the effect of the nonlinear term generated by the nonlinear forces during the Fourier transform. The traction transmission system has a significant impact on the low-frequency vibration of the bogie frame, and the meshing frequency of the gearbox will be significantly transmitted to the bogie frame and the wheelset.

#### 4.2.2. Stability Analysis

^{2}; $f$ is the vibration frequency, Hz; and $F\left(f\right)$ is a weighting factor that considers the sensitivity of the human body to various frequencies of vibration. Here, ${W}_{\mathrm{s}}$ ≤ 2.5 is excellent grade, ${W}_{\mathrm{s}}$ ≤ 2.75 is good grade and ${W}_{\mathrm{s}}$ ≤ 3.0 is a pass grade.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Marshaling urban rail train dynamics model with the traction transmission system: (

**a**) Topology model of the urban rail train; (

**b**) The modeling flow chart of the marshaling urban rail train; (

**c**) The marshaling urban rail train dynamics model.

**Figure 2.**Sensor layout in the field test. (

**a**) Layout of sensors on axle box, and the red circle indicates that the axle box; (

**b**) Layout of sensors on gearbox, and the red circle indicates that the gearbox.

**Figure 3.**Comparison of the experimental data and simulation data in the vertical direction: the time series (

**a**) on the axle box and (

**b**) gearbox.

**Figure 5.**Wheel–rail force of urban rail train in operation: (

**a**) wheel–rail vertical force; and (

**b**) wheel–rail lateral force.

**Figure 9.**Nonlinear critical speed of each car of the marshaling urban rail train. Each curve represents the convergence rate of transverse movement of each carriage respectively.

**Figure 10.**Vibration acceleration of each component of the marshaling urban rail train: (

**a**) vertical acceleration of car body; (

**b**) lateral acceleration of car body; (

**c**) vertical acceleration of bogie frame; (

**d**) lateral acceleration of bogie frame; (

**e**) vertical acceleration of wheelset; and (

**f**) lateral acceleration of wheelset.

**Figure 11.**Time–frequency diagram of the vertical vibration acceleration of the motor car and trailer: (

**a**) car body of the motor car; (

**b**) car body of the trailer; (

**c**) bogie frame of the motor car; (

**d**) bogie frame of the trailer; (

**e**) wheelset of the motor car; and (

**f**) wheelset of the trailer.

**Figure 12.**Time–frequency diagram of the lateral vibration acceleration of the motor car and trailer: (

**a**) car body of the motor car; (

**b**) car body of the trailer; (

**c**) bogie frame of the motor car; (

**d**) bogie frame of the trailer; (

**e**) wheelset of the motor car; and (

**f**) wheelset of the trailer.

**Figure 13.**Sperling index of the marshaling urban rail train: (

**a**) vertical sperling index; and (

**b**) lateral sperling index.

Specification | Value |
---|---|

Car body mass (kg) | 4.08 × 10^{4} |

Bogie frame mass (kg) | 3188 |

Wheelset mass (kg) | 1640 |

Axle box mass (kg) | 85.367 |

Gearwheel mass (kg) | 53.15 |

Pinion mass (kg) | 5.15 |

Gearbox mass (kg) | 149.75 |

Rotor mass (kg) | 178..36 |

Motor mass (kg) | 422.82 |

Rotational inertia of car body x/y/z (t·m^{2}) | 75.06/2277.4/2277.4 |

Rotational inertia of bogie frame x/y/z (kg·m^{2}) | 2040/2710/3460 |

Rotational inertia of wheelset x/y/z (kg·m^{2}) | 725/100/725 |

Rotational inertia of axle box x/y/z (kg·m^{2}) | 1.455/2.448/2.011 |

Rotational inertia of gearwheel mass x/y/z (kg·m^{2}) | 4.55/4.85/4.555 |

Rotational inertia of pinion x/y/z (kg·m^{2}) | 0.006/0.007/0.006 |

Rotational inertia of gearbox x/y/z (kg·m^{2}) | 4.22/10.45/8.56 |

Rotational inertia of rotor x/y/z (kg·m^{2}) | 24.5/1.9/24.5 |

Rotational inertia of motor x/y/z (kg·m^{2}) | 77.5/24.7/75.2 |

Stiffness of primary suspension x/y/z (N/m) | 9.2 × 10^{6}/8 × 10^{6}/1.5 × 10^{6} |

Stiffness of secondary suspension x/y/z (N/m) | 2.06 × 10^{5}/2.06 × 10^{5}/4.41 × 10^{5} |

Stiffness between motor and bogie frame x/y/z (N/m) | 3 × 10^{7}/1 × 10^{7}/3 × 10^{7} |

Stiffness between gearbox and bogie frame x/y/z (N/m) | 3 × 10^{6}/3 × 10^{6}/3 × 10^{6} |

Damping coefficient of primary suspension x/y/z (N·s/m) | 5560/5560/1800 |

Vertical damping coefficient of secondary suspension (N·s/m) | 6 × 10^{4} |

Damping coefficient between motor and bogie frame x/y/z (N·s/m) | 1 × 10^{3}/1 × 10^{3}/1 × 10^{3} |

Damping coefficient between gearbox and bogie frame x/y/z (N·s/m) | 3 × 10^{5}/2 × 10^{5}/3 × 10^{5} |

Specification | Value |
---|---|

Tooth number of pinion/gear | 16/107 |

Modification coefficient of pinion/gear (mm) | 0.31449/−0.07614 |

Face width of pinion/gear (mm) | 70/70 |

Module (mm) | 85.367 |

Pressure angle (°) | 20 |

Helix angle (°) | 17 |

Poisson ratio | 0.3 |

Young modulus (GN/m) | 206 |

Damping coefficient (kN·s/m) | 5 |

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## Share and Cite

**MDPI and ACS Style**

Zhang, Y.; Yang, J.; Wang, J.; Zhao, Y.
Dynamic Characteristics of Urban Rail Train in Multivehicle Marshaling under Traction Conditions. *Appl. Sci.* **2023**, *13*, 3022.
https://doi.org/10.3390/app13053022

**AMA Style**

Zhang Y, Yang J, Wang J, Zhao Y.
Dynamic Characteristics of Urban Rail Train in Multivehicle Marshaling under Traction Conditions. *Applied Sciences*. 2023; 13(5):3022.
https://doi.org/10.3390/app13053022

**Chicago/Turabian Style**

Zhang, Yichao, Jianwei Yang, Jinhai Wang, and Yue Zhao.
2023. "Dynamic Characteristics of Urban Rail Train in Multivehicle Marshaling under Traction Conditions" *Applied Sciences* 13, no. 5: 3022.
https://doi.org/10.3390/app13053022