# Optimization of the Auxiliary-Beam System in Railway Bridge Vibration Mitigation Using FEM Simulation and Genetic Algorithms

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## Abstract

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## 1. Introduction

## 2. Problem Definition

^{4}), according to the data provided in a national regulation technical note [20]. The material chosen for the auxiliary beam was steel. This element is also simply supported at both ends. Both elements were connected by one viscous damper, as described before.

## 3. Finite Element Model

#### 3.1. Validation of the Bridge Model

#### 3.2. Validation of the AB System

## 4. Genetic Algorithm

#### Validation of the GA

## 5. Results and Discussion

#### 5.1. Dependency on the Weighting Factors

#### 5.2. Dynamic Response

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

AB | Auxiliary Beam |

FE | Finite Element |

FRF | Frequency Response Function |

TMD | Tuned Mass Damper |

SUS | Stochastic Universal Sampling |

MSB | More Significant Bit |

LSB | Less Significant Bit |

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**Figure 1.**Scheme of an Auxiliary Beam (AB) system. The bridge deck (upper beam) is simply supported at both ends by the bridge pillars. The AB (lower beam) is also simply supported at the pillars, and connected to the bridge deck by a viscous damper.

**Figure 2.**Maximum acceleration values as a function of the train speed, computed with the Finite Element (FE) model for the bridge with no vibration mitigation system coupled. Each line (colour) corresponds to a different European train model, according to the legend.

**Figure 3.**Maximum values for displacement (

**left**) and acceleration (

**right**) as a function of the speed of the train for the validation case provided in [20]. The reference values are plotted as asterisks.

**Figure 5.**Fitness functions for the individual objectives, added mass (

**left**) and acceleration reduction (

**right**).

**Figure 6.**Results for 20 generations of individuals with weighting factors of 10% for mass and 90% for inertia. Each individual was located according to the parameter set ($\theta ,\gamma $) it represents, and the fitness value is shown as a colour going from blue (less fitted) to red (more fitted).

**Figure 7.**Influence of the weighting factor of the fitness function in maximum acceleration and added mass values (

**left**). Relation between added mass and maximum acceleration (

**right**). Increasing added mass (reduced weight factor for mass penalty) leads to decreasing acceleration. The rate of acceleration reduction is lower for bigger masses.

**Figure 8.**Dynamic response of the bridge for different configurations. The acceleration over time is plotted for the bridge without a damping system, and three cases shown in Table 2.

$\mathit{\gamma}$ | $\mathit{\theta}$ | Max. Acel. [m/s^{2}] | Mass [10^{3} kg] | Fitness [m/s^{2}] | |
---|---|---|---|---|---|

Individual 1 | $5.207\times {10}^{-3}$ | 0.0494 | 9.0541 | 33.299 | 68.601 |

Individual 2 | $5.215\times {10}^{-3}$ | 0.0494 | 9.0525 | 33.325 | 68.601 |

Individual 3 | $4.684\times {10}^{-3}$ | 0.0494 | 9.1651 | 31.582 | 68.581 |

**Table 2.**Parameter set values of the best individual found by the genetic algorithm for different mass weighting factors.

Mass Weighting Factor | $\mathit{\gamma}$ | $\mathit{\theta}$ |
---|---|---|

2.5% | $10.580\times {10}^{-3}$ | 0.0555 |

5% | $8.187\times {10}^{-3}$ | 0.0531 |

10% | $5.207\times {10}^{-3}$ | 0.0494 |

20% | $1.777\times {10}^{-3}$ | 0.0326 |

30% | $1.135\times {10}^{-3}$ | 0.1569 |

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**MDPI and ACS Style**

Baldonedo, J.; López-Campos, J.A.; López, M.; Casarejos, E.; Fernández, J.R.
Optimization of the Auxiliary-Beam System in Railway Bridge Vibration Mitigation Using FEM Simulation and Genetic Algorithms. *Symmetry* **2019**, *11*, 1089.
https://doi.org/10.3390/sym11091089

**AMA Style**

Baldonedo J, López-Campos JA, López M, Casarejos E, Fernández JR.
Optimization of the Auxiliary-Beam System in Railway Bridge Vibration Mitigation Using FEM Simulation and Genetic Algorithms. *Symmetry*. 2019; 11(9):1089.
https://doi.org/10.3390/sym11091089

**Chicago/Turabian Style**

Baldonedo, Jacobo, José A. López-Campos, Marcos López, Enrique Casarejos, and José R. Fernández.
2019. "Optimization of the Auxiliary-Beam System in Railway Bridge Vibration Mitigation Using FEM Simulation and Genetic Algorithms" *Symmetry* 11, no. 9: 1089.
https://doi.org/10.3390/sym11091089