# Design of a Semiactive TMD for Lightweight Pedestrian Structures Considering Human–Structure–Actuator Interaction

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## Abstract

**:**

## 1. Introduction

## 2. System Modeling

#### 2.1. HSI Model

#### 2.2. Passive TMD

#### 2.3. Semiactive TMD

#### Semiactive Control Law

## 3. Optimum Design Procedure

#### 3.1. HSI Model

#### 3.2. Performance Indexes and the Optimization Problem

- -
- Normalized Peak Acceleration:$${J}_{1,P}={\sum}_{i=1}^{N}\left(\frac{max|{\ddot{x}}_{s}\left(t\right)|}{max|{\stackrel{\u02c7}{\ddot{x}}}_{s}\left(t\right)|}\right),$$
- -
- Normalized 1s-RMS Acceleration:$${J}_{1,RMS}={\sum}_{i=1}^{N}\left(\frac{RMS\left({\ddot{x}}_{s}\left(t\right)\right)}{RMS\left({\stackrel{\u02c7}{\ddot{x}}}_{s}\left(t\right)\right)}\right),$$
- -
- Inertial Mass of the Control Device:$${J}_{2}={m}_{t},$$
- -
- Saturation force:$${J}_{3}={F}_{sat}.$$

## 4. Application of the Proposed Design Methodology

#### 4.1. HSI Model

#### 4.2. Optimum Design

#### 4.3. Discussion of Results

## 5. Application of the Proposed Design Methodology under Realistic Conditions

#### 5.1. MR Modeling

#### 5.2. Implementation of the Control Law

#### 5.2.1. Low-Pass Filter

#### 5.2.2. Integrator Filter

#### 5.2.3. Deactivation Rule

#### 5.3. Optimum Design under Realistic Conditions

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

FRP | Fiber-Reinforced Polymers |

TMD | Tuned Mass Damper |

STMD | Semiactive Tuned Mass Damper |

HSI | Human–Structure Interaction |

MSDA | Mass-Spring-Damper-Actuator |

MTMD | Multiple Tuned Mass Damper |

MR | Magnetorheological |

DLF | Dynamic Load Factors |

GLF | Generated Load Factors |

SDOF | Single Degree of Freedom |

CDF | Cumulative Distribution Function |

## References

- Wei, X.; Russell, J.; Ẑivanovic, S.; Mottram, J.T. Measured Dynamic Properties for FRP Footbridges and their Critical Comparison against Structures made of Conventional Construction Materials. Compos. Struct.
**2019**, 223, 110956. [Google Scholar] - Gallegos-Calderón, C.; Naranjo-Pérez, J.; Díaz, I.M.; Goicolea, J.M. Identification of a Human-Structure Interaction Model on an Ultra-Lightweight FRP Footbridge. Appl. Sci.
**2021**, 11, 6654. [Google Scholar] [CrossRef] - Díaz, I.M.; Gallegos-Calderón, C.; Ramírez Senent, J.; Renedo, C.M.C. Interaction Phenomena to Be Accounted for Human-Induced Vibration Control of Lightweight Structures. Front. Built Environ.
**2021**, 7, 658529. [Google Scholar] [CrossRef] - Yang, F.; Sedaghati, R.; Esmailzadeh, E. Vibration Suppression of Structures using Tuned Mass Damper Technology: A State of the Art Review. Civ. Eng. J.
**2020**, 6. [Google Scholar] [CrossRef] - Ramini, F.; Aghayari, R.; Samali, B. Application of Tuned Mass Dampers for Structural Vibration Control: A State of the Art Review. Civ. Eng. J.
**2020**, 6. [Google Scholar] [CrossRef] - Soria, J.M.; Díaz, I.M.; García-Palacios, J.H. Vibration Control of a Time-Varying Modal-Parameter Footbridge: Study of Semi-Active Implementable Strategies. Smart Struct. Syst.
**2017**, 20, 525–537. [Google Scholar] - Demetriou, D.; Nikitas, N.; Tsavdaridis, K.D. Semi Active Tuned Mass Damper of Buildings: A Simple control Option. Am. J. Eng. Appl. Sci.
**2015**, 8, 620–632. [Google Scholar] [CrossRef] - Van Nimmen, K.; Verbeke, P.; Lombaert, G.; De Roeck, G.; Van den Broeck, P. Numerical and Experimental Evaluation of the Dynamic Performance of a Footbridge with Tuned Mass Dampers. J. Bridge Eng.
**2016**, 21, C4016001. [Google Scholar] [CrossRef] - Caetano, E.; Cunha, Á.; Moutinho, C.; Magalhães, F. Studies for Controlling Human-Induce Vibration of the Pedro e Inês Footbridge, Portugal. Part 2: Implementation of Tuned Mass Dampers. Eng. Struct.
**2010**, 32, 1082–1091. [Google Scholar] [CrossRef] - Weber, F.; Maślanka, M. Precise Stiffness and Damping Emulation with MR Dampers and its Application to Semi-Active Tuned Mass Dampers of Wolgograd Bridge. Smart Mater. Struct.
**2014**, 23, 015019. [Google Scholar] [CrossRef] - Koo, J.H.; Ahmadian, M.; Setareh, M.; Murray, T.M. In Search of Suitable Control Methods for Semi-Active Tuned Vibration Absorbers. J. Vib. Control
**2004**, 10, 163–174. [Google Scholar] [CrossRef] - Moutinho, C. Testing a Simple Control Law to Reduce Broadband Frequency Harmonic Vibrations using Semi-Active Tuned Mass Dampers. Smart Mater. Struct.
**2015**, 24, 055007. [Google Scholar] [CrossRef] - Zhang, D.; Pan, P.; Zeng, Y.; Guo, Y. A Novel Robust Optimum Control Algorithm and Its Application to Semi Active Controlled Base Isolated Structures. Bull. Earthq. Eng.
**2020**, 18, 2431–2460. [Google Scholar] [CrossRef] - Gu, X.; Yu, Y.; Li, Y.; Li, J.; Askari, M.; Samali, B. Experimental Study of Semi-Active Magnetorheological Elastomer Base Isolation System using Optimal Neuro Fuzzy Logic Control. Mech. Syst. Signal Process.
**2019**, 119, 380–398. [Google Scholar] [CrossRef] - Wang, L.; Nagarajaiah, S.; Shi, W.; Zhou, Y. Semi-Active Control of Walking-Induced Vibrations in Bridges using Adaptive Tuned Mass Damper considering Human-Structure-Interaction. Eng. Struct.
**2021**, 244, 112743. [Google Scholar] [CrossRef] - Maślanka, M. Optimised Semi-Active Tuned Mass Damper with Acceleration and Relative Motion Feedbacks. Mech. Syst. Signal Process.
**2019**, 130, 707–731. [Google Scholar] [CrossRef] - Pinkaew, T.; Fujino, Y. Effectiveness of Semi-Active Tuned Mass Dampers under Harmonic Excitation. Eng. Struct.
**2001**, 23, 850–856. [Google Scholar] [CrossRef] - Barrera-Vargas, C.A.; Díaz, I.M.; Soria, J.M.; García-Palacios, J.H. Enhancing Friction Pendulum Isolation Systems Using Passive and Semi-Active Dampers. Appl. Sci.
**2020**, 10, 5621. [Google Scholar] [CrossRef] - Ferreira, F.; Moutinho, C.; Cunha, Á.; Caetano, E. Use of Semi-Active Tuned Mass Dampers to Control Footbridges Subjected to Synchronous Lateral Excitation. J. Sound Vib.
**2019**, 446, 176–194. [Google Scholar] [CrossRef] - Setareh, M.; Ritchey, J.K.; Koo, J.-H.; Admadian, M. Semiactive Tuned Mass Damper for Floor Vibration Control. J. Struct. Eng.
**2007**, 133, 242–250. [Google Scholar] [CrossRef] - Weber, F.; Distl, H.; Fischer, S.; Braun, C. MR Damper Controlled Vibration Absorber for Enhanced Mitigation of Harmonic Vibrations. Actuators
**2016**, 5, 27. [Google Scholar] [CrossRef] - Díaz, I.M.; Reynolds, P. On-off Nonlinear Active Control of Floor Vibrations. Mech. Syst. Signal Process.
**2010**, 24, 1711–1726. [Google Scholar] [CrossRef] - Dougill, J.W.; Wright, J.R.; Parkhouse, J.G.; Harrison, R.E. Human Structure Interaction during Rhythmic Bobbing. Struct. Eng.
**2006**, 84, 32–39. [Google Scholar] - ACMA. Pre Standard for Load and Resistance Factor Design (LRFD) of Pultruded Fiber Reinforced Polymer (FRP) Structure; American Society of Civil Engineers ASCE: Reston, VA, USA, 2010. [Google Scholar]
- Shahabpoor, E.; Pavic, P.; Racic, V. Interaction between Walking Humans and Structure in Vertical Direction: A Literature Review. Shock Vib.
**2016**, 2016, 3430285. [Google Scholar] [CrossRef] [Green Version] - Zhang, M.; Georgakis, C.T.; Chen, J. Biomechanically Excited SMD Model of a Walking Pedestrian. J. Bridge Eng.
**2016**, 21, C4016003. [Google Scholar] [CrossRef] - Ahmadi, E.; Caprani, C.; Ẑivanovic, S.; Heidarpour, A. Experimental Validation of Moving Spring-Mass-Damper Model for Human-Structure Interaction in the Presence of Vertical Vibrations. Structures
**2021**, 29, 1274–1285. [Google Scholar] [CrossRef] - European Commission, Directorate-General for Research and Innovation; Feldmann, M.; Heinemeyer, C.; Butz, C. Advanced Load Models for Synchronous Pedestrian Excitation and Optimised Design Guidelines for Steel Footbridges; Publications Office: Luxembourg, 2009. [Google Scholar]
- Blasco, X.; Herrero, J.M.; Sanchis, J.; Martínez, M. A New Graphical Visualization of N-Dimensional Pareto Front for Decision-Making in Multiobjective Optimization. Inf. Sci.
**2008**, 128, 3908–3924. [Google Scholar] [CrossRef] - Aguirre, N. Magnetorheological Dampers: Modeling and Control Design for Civil Engineering Structure; Universitat Politècnica de Catalunya: Barcelona, Spain, 2011. [Google Scholar]
- Barrera-Vargas, C.A.; Díaz, I.M.; García-Palacios, J.H.; Soria, J.M. Semi-active Tuned Mass Damper. Magnetorheological Damper Identification and Performance Evaluation. In Proceedings of the 2nd Conference on Structural Dynamics, Gijón, Spain, 22–23 July 2021; pp. 172–182. [Google Scholar]
- Soria, J.M.; Díaz, I.M.; García-Palacios, J.H. Further steps towards the tuning of inertial controllers for broadband-frequency-varying structures. Struct Control Health Monit.
**2020**, 27, e2461. [Google Scholar] [CrossRef] - Díaz, I.M.; Reynolds, P. Robust Saturated Control of Human-Induced Floor Vibrations via a Proof-Mass Actuator. Smart Mater. Struct.
**2009**, 18, 125024. [Google Scholar] [CrossRef] [Green Version]

**Figure 1.**(

**a**) Free body diagram for a SDOF structure. (

**b**) Block diagram of the HSI model. (

**c**) Block diagram considering a moving pedestrian.

**Figure 4.**(

**a**,

**b**) show the phase control logic for upward motion with respect to an equilibrium state. (

**c**,

**d**) show the phase control logic for downward motion with respect to an equilibrium state.

**Figure 7.**Pareto front obtained for the optimum design of the (

**a**) TMD and (

**b**) STMD. The selected optimum solution is marked in red.

**Figure 8.**Dynamic response of the bare and the controlled structure: (

**a**) acceleration at midspan, (

**b**) CDF of the 1s-running RMS value of all the samples, and (

**c**) CDF of the acceleration of all the samples.

**Figure 10.**Block diagram of the HSI and STMD including all elements of the implementation and noisy signals.

**Figure 11.**Dynamic response of the footbridge with each control device: (

**a**) Acceleration at midspan; (

**b**) CDF of the 1-s-running RMS value of all the samples; (

**c**) CDF of the acceleration of all the samples.

**Figure 12.**Vertical reaction force measured in an instrumented treadmill: (

**a**) Time history; (

**b**) Fast Fourier Transform.

**Figure 13.**Objective function ${\varphi}_{1}$ of the optimum systems under different footbridge frequencies.

Structure | Value | Units | |
---|---|---|---|

${m}_{s}$ | Mass | $[286.00$ to $495.80]$ | kg |

${f}_{s}$ | Frequency | $[4.60$ to $10.40]$ | Hz |

${\zeta}_{s}$ | Damping ratio | $[0.86$ to $1.83]$ | % |

${c}_{s}$ | Damping coefficient | $[279.40$ to $931.10]$ | kg/s |

${k}_{s}$ | Stiffness | $[3.54$ to $17.05]\times {10}^{5}$ | N/m |

Human | Value | Units | |

${m}_{h}$ | Mass | $[50.41$ to $76.43]$ | kg |

${f}_{h}$ | Frequency | ${f}_{s}/4$ | Hz |

${\zeta}_{h}$ | Damping ratio | $[10$ to $40]$ | % |

${c}_{h}$ | Damping coefficient | $[164.38$ to $643.50]$ | kg/s |

${k}_{h}$ | Stiffness | $[3.52$ to $17.18]\times {10}^{3}$ | N/m |

Design Variable | Lower Bound | Upper Bound | |
---|---|---|---|

${m}_{t}$ | Mass (kg) | 10 | 45 |

${f}_{t}$ | Frequency (Hz) | 1.00 | 10.00 |

${\zeta}_{t}$ | Damping ratio (%) | 1 | 50 |

${F}_{sat}$ | Saturation force (N) | 10 | 5000 |

TMD | Value | ||
---|---|---|---|

${m}_{t}$ | Mass | 33.36 | kg |

${f}_{t}$ | Frequency | 9.95 | Hz |

${\zeta}_{t}$ | Damping ratio | 1.03 | % |

${c}_{t}$ | Damping coefficient | 42.96 | kg/s |

${k}_{t}$ | Stiffness | $1.30\times {10}^{5}$ | N/m |

${\varphi}_{1}$ | Objective function | $0.48$ | - |

STMD | Value | ||

${m}_{t}$ | Mass | 24.50 | kg |

${f}_{t}$ | Frequency | 6.15 | Hz |

${c}_{min}$ | Normal functioning | 37.91 | kg/s |

${c}_{max}$ | Blocking functioning | 1895.7 | kg/s |

${k}_{t}$ | Stiffness | $3.66\times {10}^{4}$ | N/m |

${F}_{sat}$ | Saturation force | $1010.24$ | N |

${\varphi}_{1}$ | Objective function | $0.40$ | - |

STMD MR | Value | ||
---|---|---|---|

${m}_{t}$ | Mass | 24.38 | kg |

${f}_{t}$ | Frequency | 6.00 | Hz |

${k}_{t}$ | Stiffness | $3.47\times {10}^{4}$ | N/m |

${F}_{sat}$ | Saturation force | $88.27$ | N |

${\varphi}_{1}$ | Objective function | $0.48$ | - |

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**MDPI and ACS Style**

Barrera-Vargas, C.A.; Naranjo-Pérez, J.; Díaz, I.M.; García-Palacios, J.H.
Design of a Semiactive TMD for Lightweight Pedestrian Structures Considering Human–Structure–Actuator Interaction. *Actuators* **2022**, *11*, 101.
https://doi.org/10.3390/act11040101

**AMA Style**

Barrera-Vargas CA, Naranjo-Pérez J, Díaz IM, García-Palacios JH.
Design of a Semiactive TMD for Lightweight Pedestrian Structures Considering Human–Structure–Actuator Interaction. *Actuators*. 2022; 11(4):101.
https://doi.org/10.3390/act11040101

**Chicago/Turabian Style**

Barrera-Vargas, Christian A., Javier Naranjo-Pérez, Iván M. Díaz, and Jaime H. García-Palacios.
2022. "Design of a Semiactive TMD for Lightweight Pedestrian Structures Considering Human–Structure–Actuator Interaction" *Actuators* 11, no. 4: 101.
https://doi.org/10.3390/act11040101