# Study of the Vibration-Energy Properties of the CRTS-III Track Based on the Power Flow Method

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## Abstract

**:**

## 1. Introduction

## 2. Model Description

#### 2.1. Vehicle Model

**x**,

**y**, and

**z**denote the longitudinal, lateral, and vertical directions, respectively.

**M**,

**k**, and

**q**denote the mass matrix, the constraint matrix, and the general force matrix, respectively; y, $\dot{y}$, and $\ddot{y}$ refer to the displacement, the velocity, and the acceleration vector, respectively; t represents time.

#### 2.2. Track Model

_{r}(t); $\overrightarrow{s}(\overrightarrow{y},t)$ refers to the small deformation of the flexible track, which can be expressed by Equation (4) in the low mode approximatively.

_{i}(t) denotes the vibration type of the elastic body; $\overrightarrow{{s}_{i}}(\overrightarrow{y},t)$ represents the modality of vibration. And the dynamic responses of the track can be depicted by Equations (3) and (4).

#### 2.3. Wheel/Rail Interaction Model

#### 2.3.1. Normal Contact

^{−8}in this study. Moreover, ΔZ(t) can be expressed by Equation (6):

_{wi}(t) is the vertical displacement of the ith wheel; Z

_{ri}(x,t) represents the vertical displacement of the one side rail at the corresponding contact position; ξ(x) is the vertical irregularity. It is worth noting that when ΔZ(t) is less than zero, the wheel/rail normal force is regarded as zero.

#### 2.3.2. Tangent Contact

_{x}, lateral creep rate ξ

_{y}, and spin creep rate ξ

_{spin}utilized in that theory are presented in Equation (7):

_{w1}, V

_{w2}, and Ω

_{w3}are the longitudinal line velocity, the lateral line velocity, and the angular velocity of the wheel at the wheel/rail contact position, respectively; V

_{r1}, V

_{r2}, and Ω

_{r3}refer to the longitudinal line velocity, the lateral line velocity, and the angular velocity of the rail at the wheel/rail contact position, respectively.

_{11}denotes the longitudinal creep coefficient; f

_{22}represents the lateral creep coefficient; f

_{23}and f

_{32}are the lateral/spin creep coefficient; f

_{33}refers to the spin creep coefficient. Furthermore, the creep coefficients can be obtained by Equation (9):

_{ij}represents the Kalker coefficient, which can be found in Kalker [21].

#### 2.4. Model Verification

#### 2.5. Power Flow Method

_{0}and V

_{0}represent the amplitudes of the force and velocity, respectively.

_{1}. The power flow of the composite concrete and the connected nodes of the rubber damping pads is summed up as the input power flow of the composite concrete, which is represented as P

_{2}. Moreover, the power flow of the base slab and the connected nodes of the rubber damping pads is summed up as the input power flow of the base slab, which is referred to as P

_{3}. Finally, the power flow is transferred from the base slab to the subgrade, which is denoted as P

_{out}. Further, the vertical transfer of the power flow in the track structure is shown in Figure 6.

_{s}(k) is the total power flow; P

_{0}refers to the reference power flow, whose value is set as 1 × 10

^{−6}N·m/s.

#### 2.6. Evaluationindexes of Vibration Energy

_{a}is the AVEL of one structure; K refers to the number of the frequency points investigated.

_{l}(k) represents the power flow of the lower structure connected by spring-damper elements in the track structure; P

_{u}(k) denotes the power flow of the upper structure connected by spring-damper elements in the track structure. The TRPF depicts the rate of the energy transferring from the upper structure to the lower one. Further, the higher the TRPF is, the more energy transfers.

## 3. Results and Discussion

#### 3.1. Power Flow of the Track Structure under Irregularities

#### 3.2. The Influence of the Stiffness of Fasteners on the Power Flow

#### 3.3. The Influence of the Stiffness of Rubber Damping Pads on the Power Flow

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The illustration of the analysis of the vibration-energy properties of the track structure.

**Figure 5.**The time-domain and frequency-domain of the nodes’ force and velocity. (

**a**) and (

**b**) are the time-domain results; (

**c**) and (

**d**) are the frequency-domain results.

**Figure 8.**The transfer rate of the power flow (TRPFs) (

**a**) from the rail to the composite slab and (

**b**) from the composite concrete to the base slab.

**Figure 11.**The TRPFs (

**a**) from the rail to the composite slab and (

**b**) from the composite slab to the base slab.

**Figure 14.**The TRPFs (

**a**) from the rail to the composite slab and (

**b**) from the composite slab to the base slab.

Parameters | Unit | Value |
---|---|---|

Mass of car body | kg | 33,200 |

Moment of inertia of car body frame about the x-axis/y-axis/z-axis | kg·m^{2} | 1.076 × 10^{5}/1.571 × 10^{6}/1.402 × 10^{6} |

Mass of bogie | kg | 2600 |

Moment of inertia of bogie frame about the x-axis/y-axis/z-axis | kg·m^{2} | 2106/1424/2600 |

Mass of wheelset | kg | 1970 |

Moment of inertia of wheelset about the x-axis/y-axis/z-axis | kg·m^{2} | 623/78/623 |

Longitudinal/lateral/vertical stiffness of primary suspension | kN/m | 980/980/1176 |

Longitudinal/lateral/vertical damping of primary suspension | kN·s/m | 0/0/19.6 |

Longitudinal/lateral/vertical stiffness of secondary suspension | kN/m | 178.4/193.1/193.1 |

Longitudinal/lateral/vertical damping of secondary suspension | kN·s/m | 58.8/58.8/98 |

Axle distance | m | 2.5 |

Bogie distance | m | 17.5 |

Rolling radius | m | 0.43 |

Parameters | Unit | Value |
---|---|---|

Density of rail | kg/m^{3} | 7850 |

Elastic modulus of rail | N/m^{2} | 2.059 × 10^{11} |

Vertical/lateral stiffness of fastener | N/m | 2.5 × 10^{7}/5 × 10^{7} |

Vertical/lateral damping of fastener | N·s/m | 2.6 × 10^{4}/1.24 × 10^{4} |

Density of slab/self-compacting concrete/base slab | kg/m^{3} | 2500/2500/2500 |

Elastic modulus of slab | N/m^{2} | 3.65 × 10^{10} |

Elastic modulus of self-compacting concrete | N/m^{2} | 3.25 × 10^{10} |

Elastic modulus of base slab | N/m^{2} | 3.25 × 10^{10} |

Surface stiffness of rubber damping pad | MPa/m | 400 |

Surface damping of rubber damping pad | Pa·s/m | 3.43 × 10^{4} |

Density of subgrade | kg/m^{3} | 1750 |

Elastic modulus of subgrade | N/m^{2} | 1.4 × 10^{8} |

Dynamic Responses | Wheel-Rail Vertical Force (kN) | Rail Vertical Displacement (mm) | Base Slab Vertical Acceleration (g) |
---|---|---|---|

Lou et al. [22] | 102.5 | 0.80 | 0.23 |

This paper | 106.6 | 0.83 | 0.21 |

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**MDPI and ACS Style**

Jiang, H.; Gao, L. Study of the Vibration-Energy Properties of the CRTS-III Track Based on the Power Flow Method. *Symmetry* **2020**, *12*, 69.
https://doi.org/10.3390/sym12010069

**AMA Style**

Jiang H, Gao L. Study of the Vibration-Energy Properties of the CRTS-III Track Based on the Power Flow Method. *Symmetry*. 2020; 12(1):69.
https://doi.org/10.3390/sym12010069

**Chicago/Turabian Style**

Jiang, Hanwen, and Liang Gao. 2020. "Study of the Vibration-Energy Properties of the CRTS-III Track Based on the Power Flow Method" *Symmetry* 12, no. 1: 69.
https://doi.org/10.3390/sym12010069