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

^{*}

## 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

- Tang, L.; Kong, X.; Li, S.; Ling, X.; Ye, Y.; Tian, S. A preliminary investigation of vibration mitigation technique for the high-speed railway in seasonally frozen regions. Soil Dyn. Earthq. Eng.
**2019**, 127, 105841. [Google Scholar] [CrossRef] - Rådeström, S.; Ülker-Kaustell, M.; Andersson, A.; Tell, V.; Karoumi, R. Application of fluid viscous dampers to mitigate vibrations of high-speed railway bridges. Int. J. Rail Transp.
**2017**, 5, 47–62. [Google Scholar] [CrossRef] - Ferreira, P.A.; López-Pita, A. Numerical modelling of high speed train/track system for the reduction of vibration levels and maintenance needs of railway tracks. Constr. Build. Mater.
**2015**, 79, 14–21. [Google Scholar] [CrossRef] - Olivier, B.; Connolly, D.P.; Alves Costa, P.; Kouroussis, G. The effect of embankment on high speed rail ground vibrations. Int. J. Rail Transp.
**2016**, 4, 229–246. [Google Scholar] [CrossRef] - Liu, L.; Song, R.; Zhou, Y.; Qin, J. Noise and Vibration Mitigation Performance of Damping Pad under CRTS-III Ballastless Track in High Speed Rail Viaduct. KSCE J. Civ. Eng.
**2019**, 23, 3525–3534. [Google Scholar] [CrossRef] - Yu, Z.; Xie, Y.; Shan, Z.; Li, X. Fatigue performance of CRTS III slab ballastless track structure un-der High-speed train load based on concrete fatigue damage constitu-tive law. J. Adv. Concr. Technol.
**2018**, 16, 233–249. [Google Scholar] [CrossRef][Green Version] - Xin, T.; Gao, L. Reducing slab track vibration into bridge using elastic materials in high speed railway. J. Sound Vib.
**2011**, 330, 2237–2248. [Google Scholar] [CrossRef][Green Version] - Zhao, C.; Wang, P. Experimental study on the vibration damping performance of rubber absorbers for ballastless tracks on viaduct. China Railw. Sci.
**2013**, 34, 8–13. [Google Scholar] - Ren, J.J.; Zhao, H.W.; Li, X.; Xu, K. Analysis of harmonic response of CRTS III prefabricated slab track with anti-vibration structure. J. Railw. Eng. Soc.
**2016**, 33, 44–50. [Google Scholar] - Xin, T.; Zhang, Q.; Gao, L.; Zhao, L.; Qu, J. Dynamic effect and structure optimization of damping layers of CRTS III slab ballastless track for high speed railway. China Railw. Sci.
**2016**, 37, 8–13. [Google Scholar] - Zhao, C.; Ping, W. Effect of elastic rubber mats on the reduction of vibration and noise in high-speed elevated railway systems. Proc. Inst. Mech. Eng. Part F J. Rail Rapid Trans.
**2018**, 232, 1837–1851. [Google Scholar] [CrossRef] - Goyder, H.G.D.; White, R.G. Vibrational power flow from machines into built-up structures, part I: Introduction and approximate analyses of beam and plate-like foundations. J. Sound Vib.
**1980**, 68, 59–75. [Google Scholar] [CrossRef] - Goyder, H.G.D.; White, R.G. Vibrational power flow from machines into built-up structures, part II: Wave propagation and power flow in beam-stiffened plates. J. Sound Vib.
**1980**, 68, 77–96. [Google Scholar] [CrossRef] - Goyder, H.G.D.; White, R.G. Vibrational power flow from machines into built-up structures, part III: Power flow through isolation systems. J. Sound Vib.
**1980**, 68, 97–117. [Google Scholar] [CrossRef] - Petersson, B.; Plunt, J. On effective mobilities in the prediction of structure-borne sound transmission between a source structure and a receiving structure, part I: Theoretical background and basic experimental studies. J. Sound Vib.
**1982**, 82, 517–529. [Google Scholar] [CrossRef] - Hussein, M.F.M.; Hunt, H.E.M. A power flow method for evaluating vibration from underground railways. J. Sound Vib.
**2006**, 293, 667–679. [Google Scholar] [CrossRef] - Li, Q.; Wu, D.J. Analysis of the dominant vibration frequencies of rail bridges for structure-borne noise using a power flow method. J. Sound Vib.
**2013**, 332, 4153–4163. [Google Scholar] [CrossRef] - Eberhard, P.; Schiehlen, W. Computational Dynamics of Multibody Systems: History, Formalisms, and Applications. J. Comput. Nonlin Dyn.
**2005**, 1, 3–12. [Google Scholar] [CrossRef] - GUYAN, R.J. Reduction of stiffness and mass matrices. AIAA J.
**1965**, 3, 380. [Google Scholar] [CrossRef] - Hertz, H. Verhandlungen des Vereins zur Beförderung des Gewerbefleisses; Duncker: Wien, Austria, 1882; pp. 174–196. [Google Scholar]
- Kalker, J.J. A fast algorithm for the simplified theory of rolling contact. Veh. Syst. Dyn.
**1982**, 11, 1–13. [Google Scholar] [CrossRef] - Lou, P.; Gong, K.; Zhao, C.; Xu, Q.; Luo, R.K. Dynamic Responses of Vehicle-CRTS III Slab Track System and Vehicle Running Safety Subjected to Uniform Seismic Excitation. Shock Vib.
**2019**, 2019, 12. [Google Scholar] [CrossRef][Green Version] - Li, W.L.; Lavrich, P. Prediction of power flows through machine vibration isolators. J. Sound Vib.
**1999**, 224, 757–774. [Google Scholar] [CrossRef]

**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 |

© 2020 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/).

## Share and Cite

**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