# Evaluation of TMD Performance in Footbridges Using Human Walking Probabilistic Models

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## Abstract

**:**

## 1. Introduction

## 2. Human Walking Models

#### 2.1. Force Model

#### 2.2. Biodynamic Models

## 3. Walking Variability

#### 3.1. Inter-Subject Variability

#### 3.2. Intra-Subject Variability

## 4. Dynamic Equations of the Coupled Human-Structure-TMD System

## 5. Case Study

#### 5.1. Structure and Numerical Model Description

#### 5.2. TMD Design Parameters

#### 5.3. TMD Efficiency Assessment

#### 5.3.1. Load Cases

#### 5.3.2. TMD Efficiency with Intra-Subject Variability

#### 5.3.3. Sensitivity of the TMD Parameters

#### 5.3.4. Sensitivity on the Intra-Subject Variability

#### 5.3.5. The TMD Efficiency with Combined Intra- and Inter-Subject Variability

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Mass-spring-damper systems: (

**a**) primary system; (

**b**) coupled structure-TMD system; and (

**c**) coupled pedestrian-structure-TMD system.

**Figure 3.**The analyzed footbridge. (

**a**) External view, (

**b**) internal view, and (

**c**) 3D finite element model (FEM).

**Figure 5.**Acceleration amplitudes for the deterministic response (

**a**) FM, (

**b**) BM1, and (

**c**) BM2, and for the RW-intra response (

**d**) FM, (

**e**) BM1, and (

**f**) BM2.

**Figure 6.**TMD efficiency for the deterministic and RW-intra responses. (

**a**) FM, (

**b**) BM1, and (

**c**) BM2.

**Figure 7.**The TMD efficiency for the sensitivity of the TMD parameters,${f}_{r}=95\%$. (

**a**) FM, (

**b**) BM1, and (

**c**) BM2.

**Figure 8.**The TMD efficiency for the sensitivity of RW-intra responses, ${\mu}_{t}=0.005$ and $CoV=50\%.$ (

**a**) FM, (

**b**) BM1, and (

**c**) BM2

Variable | Description | Probability Distribution/Value |
---|---|---|

${f}_{p}$ | Pacing frequency | $N\left(1.87\mathrm{Hz},CoV=10\%\right)$ |

${L}_{s}$ | Step length | $N\left(0.71\mathrm{m},CoV=10\%\right)$ |

${\varnothing}_{1}$ | 1st harmonic phase angle | 0 |

${\varnothing}_{2}$ | 2nd harmonic phase angle | $UN\left(-\pi ,\pi \right)$ |

${\varnothing}_{3}$ | 3rd harmonic phase angle | Assumed equal to ${\varnothing}_{2}$ |

${\alpha}_{1}$ | 1st harmonic DLF | $N\left({\mu}_{{\alpha}_{1}},CoV=16\%\right)$ |

${\alpha}_{2}$ | 2nd harmonic DLF | $N\left(0.07,CoV=25\%\right)$ |

${\alpha}_{3}$ | 3rd harmonic DLF | $N\left(0.05,CoV=25\%\right)$ |

TMD | Footbridge 1st Vertical Bending Mode | ||||
---|---|---|---|---|---|

Variable | Description | Value | Variable | Description | Value |

${\xi}_{T}$ | TMD damping ratio | $5\%$ | ${\xi}_{s}$ | Damping ratio | 0.23% |

${f}_{r}$ | Frequency ratio | 97% | ${f}_{s}$ | Modal frequency | 1.85 Hz |

${m}_{r}$ | Mass ratio | 1% | ${m}_{s}$ | Modal mass | 39,490 kg |

${u}_{T}$ | TMD position | Mid-span |

Scenarios | Section 5.3.2 | Section 5.3.3 | Section 5.3.4 | Section 5.3.5 | ||||
---|---|---|---|---|---|---|---|---|

Intra | Det. | Intra | Det. | Intra | Det. | Intra + Inter | Inter | |

Pedestrians scenarios | 80 | 80 | 80 | 80 | 80 | 80 | 80 | 80 |

Walking models | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 |

Runs to converge | 100 | 1 | 100 | 1 | 100 | 1 | 800 | 800 |

Controlled/uncontrolled | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |

Total of simulations | 48,000 | 480 | 48,000 | 480 | 48,000 | 480 | 384,000 | 384,000 |

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**MDPI and ACS Style**

Rezende, F.; Brunet, O., Jr.; Diniz Varela, W.; Pereira, A.; Carvalho, E.
Evaluation of TMD Performance in Footbridges Using Human Walking Probabilistic Models. *Vibration* **2021**, *4*, 323-340.
https://doi.org/10.3390/vibration4020021

**AMA Style**

Rezende F, Brunet O Jr., Diniz Varela W, Pereira A, Carvalho E.
Evaluation of TMD Performance in Footbridges Using Human Walking Probabilistic Models. *Vibration*. 2021; 4(2):323-340.
https://doi.org/10.3390/vibration4020021

**Chicago/Turabian Style**

Rezende, Filipe, Otávio Brunet, Jr., Wendell Diniz Varela, André Pereira, and Eliane Carvalho.
2021. "Evaluation of TMD Performance in Footbridges Using Human Walking Probabilistic Models" *Vibration* 4, no. 2: 323-340.
https://doi.org/10.3390/vibration4020021