# Research on Geometric Parameters Optimization of Fixed Frog Based on Particle Swarm Optimization Algorithm

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Wheel/Rail Contact Analysis in the Fixed Frog Area

## 3. Dynamics Modeling of Vehicle/Fixed Frog Interaction

#### 3.1. Vehicle Dynamics Modeling

- $M$ is the inertia matrix;
- $C$ is the damping matrix;
- $K$ is the stiffness matrix;
- $\left\{F\right\}$ is the external force matrix, ignoring the track irregularity, which is mainly composed of 3 parts:$$F={F}_{wr}+{F}_{IG}+{F}_{c}$$

- ${F}_{wr}$ is the force transmitted by the rail to the wheelset, including normal force and tangential force (creep force);
- ${F}_{IG}$ is gravity, inertia force, and Coriolis force;
- ${F}_{c}$ is the zero position non-equilibrium additional force due to curve line conditions.

#### 3.2. Rail Vibration System Model

#### 3.3. Real-Time Calculation for the Wheel/Rail Interaction

## 4. Geometric Parameter Optimization Design for the Fixed Frog Area

#### 4.1. PSO Algorithm

#### 4.2. Optimization Design Method for the Wing Rail Lifting Values

- (1)
- Set the basic parameters of particle swarm optimization algorithm and initialize the population, and randomly generate N groups of wing rail lifting values according to the constraints.
- (2)
- A complete set of frog profiles is generated according to each lifting value and the profile of each key section, and the vehicle dynamics are calculated.
- (3)
- If the dynamic calculation results conform to safety conditions, the individual optimal solution and group optimal solution shall be updated according to the quality of the objective function value, and then judge whether the current iteration times have reached the maximum iterations. If the maximum iterations have been met, the optimization results shall be output.
- (4)
- If the dynamic calculation results do not meet the safety requirements or the current iterations number does not reach the maximum iterations, obtain a new group N lifting values according to Formula (14) and (15), and repeat the above calculation steps until the optimal lifting values are obtained.

#### 4.3. Optimization Design Method for the Nose Rail Height

## 5. Conclusions

- (1)
- If the wheelset is close to the nose rail with a large lateral displacement, there is a risk that the wheel will climb onto the nose rail. In order to reduce the vertical impact of wheelsets passing through the fixed frog area and improve the safety and stability of vehicles passing through the turnout center, the geometric parameters of the fixed frog should be set reasonably.
- (2)
- After wing rail lifting values optimization, the maximum vertical displacement of the wheelset is reduced by 34.6%. On the stock rail side, the maximum wheel/rail lateral force is reduced by 12.8%, and the maximum wheel/rail vertical force is reduced by 10%. On the frog side, the maximum wheel/rail lateral force is reduced by 38%, and the maximum wheel/rail lateral force is reduced by 12.5%.
- (3)
- After nose rail height values optimization, the maximum vertical displacement of the wheelset is reduced by 19.5%. On the stock rail side, the maximum wheel/rail lateral force is reduced by 15.4%, and the maximum wheel/rail vertical force is reduced by 3.5%. On the frog side, the maximum wheel/rail lateral force is reduced by 18.5%, and the maximum wheel/rail lateral force is reduced by 9.4%.

## 6. Future Work

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Wang, P. High Speed Railway Turnout Design Theory and Practice, 1st ed.; Southwest Jiaotong University Press: Chengdu, China, 2011; pp. 12–13. [Google Scholar]
- Ma, H.; Niu, Y.; Zou, X.; Zhang, J.; Yu, M. Dynamic and static contact performance analysis of wagon and fixed frog in heavy haul railway. Sci. Technol. Eng.
**2020**, 20, 14229–14233. [Google Scholar] - Chen, D. Research on Optimization Design of Rail Grinding Profile in Turnout Switch Area; Tongji University: Shanghai, China, 2019. [Google Scholar]
- Ren, Z.; Zhai, W. Vertical dynamic simulation calculation of railway vehicle passing frog. J. Southwest Jiaotong Univ.
**1997**, 1997, 46–52. [Google Scholar] - Zhai, W.; Ren, Z. Study on vertical interaction between speed increasing train and turnout. J. China Railw. Soc.
**1998**, 1998, 34–39. [Google Scholar] - Wang, P. Spatial coupling vibration model of wheel rail system in turnout area and its application. J. Southwest Jiaotong Univ.
**1998**, 1998, 52–57. [Google Scholar] - Wang, P.; Liu, X.; Kou, Z. Discussion on the longitudinal distribution law of turnout vertical stiffness along the line. J. Southwest Jiaotong Univ.
**1999**, 1999, 18–22. [Google Scholar] - Lagos, R.F.; Alonso, A.; Vinolas, J.; Pérez, X. Rail vehicle passing through a turnout: Analysis of different turnout designs and wheel profiles. Proc. Inst. Mech. Eng. Part F J. Rail Rapid Transit
**2012**, 226, 587–602. [Google Scholar] [CrossRef] - Markine, V.L.; Shevitsov, I.Y. An Experimental study on crossing nose damage of railway turnouts in the Netherlands. In Proceedings of the Fourteenth International Conference on Civil, Structural and Environmental Engineering Computing, Stirlingshire, UK, 3–6 September 2013. [Google Scholar]
- Markine, V.L.; Steenbergen, M.J.M.M.; Shevtsov, I.Y. Combatting RCF on switch points by tuning elastic track properties. Wear
**2011**, 271, 158–167. [Google Scholar] [CrossRef] - Grossoni, I.; Bezin, Y.; Neves, S. Optimisation of support stiffness at railway crossings. Veh. Syst. Dyn.
**2017**, 56, 1072–1096. [Google Scholar] [CrossRef] [Green Version] - Anderson, C.; Dahlberg, T. Wheel/rail impacts at a railway turnout crossing. Proc. Inst. Mech. Eng. Part F J. Rail Rapid Transit
**1998**, 212, 123–134. [Google Scholar] [CrossRef] - Anderson, C. Modelling and Simulation of Train-Track Interaction Including Wear Prediction; Chalmers University of Technology: Gothenburg, Sweden, 2003. [Google Scholar]
- Blanco-Saura, A.E.; Velarte-Gonzalez, J.L.; Ribes-Llario, F.; Real-Herráiz, J.I. Study of the dynamic vehicle-track interaction in a railway turnout. Multibody Syst. Dyn.
**2018**, 43, 21–36. [Google Scholar] [CrossRef] - Palsson, B.A. Optimisation of railway crossing geometry considering a representative set of wheel profiles. Veh. Syst. Dyn.
**2015**, 53, 274–301. [Google Scholar] [CrossRef] - Wan, C.; Markine, V.L.; Shevitsov, I.Y. Improvement of vehicle-turnout interaction by optimizing the shape of crossing nose. Veh. Syst. Dyn.
**2014**, 52, 1517–1540. [Google Scholar] [CrossRef] - Cao, Y.; Wang, P.; Zhao, W. Design method for rigid frog based on wheel/rail contact parameters. J. Southwest Jiaotong Univ.
**2012**, 47, 605–610. [Google Scholar] - Cao, Y.; Wang, P. Optimization of nose depth for rigid frog. J. Southwest Jiaotong Univ.
**2015**, 50, 1067–1073. [Google Scholar] - Xu, J.; Wang, P. Optimization design method for rigid frog based on wheel/rail profile type. China Railw. Sci.
**2014**, 35, 1–6. [Google Scholar] - Zhang, P.; Zhu, X.; Lei, X.; Xiao, J. Influence of wing rail lifting value on dynamic characteristics of high-speed train crossing the turnout. J. Railw. Sci. Eng.
**2019**, 16, 2903–2912. [Google Scholar] - Shen, G. Railway Vehicle System Dynamics, 1st ed.; China Railway Press: Beijing, China, 2014; pp. 45–49. [Google Scholar]
- Zhu, Q.; Dai, W.; Tan, X.; Li, C.; Xie, D. Multi-objective optimization control strategy of traction inverter based on particle swarm algorithm. J. Tongji Univ. Nat. Sci.
**2020**, 48, 287–295. [Google Scholar] - Compilation Group of the Public Works Bureau of the Ministry of Railways. Railway Public Works Technical Manual Turnout; China Railway Press: Beijing, China, 2012. [Google Scholar]
- Wang, W. Research on Particle Swarm Optimization Algorithm and Its Application; Southwest Jiaotong University: Chengdu, China, 2012. [Google Scholar]

**Figure 1.**Wheel/rail contact calculation results of key sections: (

**a**) Lateral coordinates of wheel/rail contact point; (

**b**) Vertical coordinates of wheel/rail contact point; (

**c**) Rolling circle radius.

Rigid Body | Lateral Motion | Vertical Motion | Rolling Motion | Pitch Motion | Yaw Motion | / |
---|---|---|---|---|---|---|

Car body | Y_{c} | Z_{c} | Φ_{c} | Ѳ_{c} | Ψ_{c} | / |

Frames | Y_{bn} | Z_{bn} | Φ_{bn} | Ѳ_{bn} | Ψ_{bn} | n = 1, 2 |

Wheelsets | Y_{wi} | Z_{wi} | Φ_{wi} | Ѳ_{bi} | Ψ_{wi} | i = 1, 2, 3, 4 |

Parameter | Value |
---|---|

Car body mass (kg) | 41,910 |

Frame mass (kg) | 4060 |

Wheelset mass (kg) | 1670 |

Gauge (m) | 1.435 |

Nominal rolling circle radius (m) | 0.42 |

Section Position | Optimized Value/mm |
---|---|

B | 1.07 |

C | 2.39 |

D | 3.81 |

E | 4.95 |

F | 4.98 |

G | 4.99 |

H | 5 |

I | 5 |

J | 5 |

Section Position | Optimized Value/mm |
---|---|

B | 3.22 |

C | 1.98 |

D | 1.65 |

E | 1.49 |

F | 1.23 |

G | 0.72 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 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 (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Zhang, R.; Shen, G.; Wang, X.
Research on Geometric Parameters Optimization of Fixed Frog Based on Particle Swarm Optimization Algorithm. *Appl. Sci.* **2022**, *12*, 11549.
https://doi.org/10.3390/app122211549

**AMA Style**

Zhang R, Shen G, Wang X.
Research on Geometric Parameters Optimization of Fixed Frog Based on Particle Swarm Optimization Algorithm. *Applied Sciences*. 2022; 12(22):11549.
https://doi.org/10.3390/app122211549

**Chicago/Turabian Style**

Zhang, Rang, Gang Shen, and Xujiang Wang.
2022. "Research on Geometric Parameters Optimization of Fixed Frog Based on Particle Swarm Optimization Algorithm" *Applied Sciences* 12, no. 22: 11549.
https://doi.org/10.3390/app122211549