# On-Line Modal Parameter Identification Applied to Linear and Nonlinear Vibration Absorbers

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. Linear Vibration Absorber: Tuned Mass Damper

#### 2.2. Nonlinear Vibration Absorber: Autoparametric System

## 3. A Flexible Structure with n Degrees of Freedom

## 4. Time-Domain and On-Line Algebraic Identification of the Harmonic Excitation

#### Analysis Based on the Hilbert Transformation

## 5. Experimental Results

#### 5.1. Non-Linearity Analysis

#### 5.2. Application of a Linear Absorber

#### 5.3. Application of a Nonlinear Absorber

#### 5.4. Nonlinearity Analysis

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

FRF | Frequency response function |

RFP | Rational fractional polynomial |

PV | Principal Cauchy Value |

TMD | Tuned mass damper |

dof | degree of freedom |

${\omega}_{i}$ | Natural frequency of the ith mode or resonance |

${\zeta}_{i}$ | Damping ratio of the ith mode or resonance |

s | Complex variable $s=j\omega $ |

## References

- Connor, J.; Laflamme, S. Structural Motion Engineering; Springer: Cham, Switzerland, 2014. [Google Scholar]
- Enríquez-Zárate, J.; Abundis-Fong, H.F.; Velazquez, R.; Gutierrez, S. Passive vibration control in a civil structure: Experimental results. Meas. Control
**2019**, 52, 938–946. [Google Scholar] [CrossRef] - Korenev, B.G.; Reznikov, L.M. Dynamic Vibration Absorber: Theory and Technical Applications; Wiley: London, UK, 1993. [Google Scholar]
- Frahm, H. A Device for Damping Vibrations of Bodies. U.S. Patent 989958, 18 April 1911. [Google Scholar]
- Ormondroyd, J.; Den Hartog, J.P. The theory of the dynamic vibration absorber. J. Appl. Mech-T ASME
**1928**, 50, 9–22. [Google Scholar] - Jangid, R.S.; Datta, T.K. Performance of multiple tuned mass dampers for torsionally coupled system. Earthq. Eng. Struct. Dyn.
**1997**, 26, 307–317. [Google Scholar] [CrossRef] - Guo, Y.Q.; Chen, W.Q. Dynamic analysis of space structures with multiple tuned mass dampers. Eng. Struct.
**2007**, 29, 3390–3403. [Google Scholar] [CrossRef] - Seung-Yong, O. Tuned mass damper asymmetric coupling system for vibration control of adjacent twin buildings. Adv. Struct. Eng.
**2019**, 23, 954–968. [Google Scholar] - Khodaie, N. Vibration control of super-tall buildings using combination of tapering method and TMD system. J. Wind Eng. Ind. Aerod.
**2020**, 196, 104031. [Google Scholar] [CrossRef] - Ibrahim, R.A. Recent advances in nonlinear passive vibration isolators. J. Sound Vib.
**2008**, 314, 371–452. [Google Scholar] [CrossRef] - Feudo, S.L.; Touzé, C.; Boisson, J.; Cumunel, G. Nonlinear magnetic vibration absorber for passive control of a multi-storey structure. J. Sound Vib.
**2019**, 438, 33–53. [Google Scholar] [CrossRef] [Green Version] - Ibrahim, R.A.; Heo, H. Autoparametric vibration of coupled beams under random support motion. J. Vib. Acoust.
**1986**, 1084, 421–426. [Google Scholar] [CrossRef] - Dahlberg, T. On optimal use of the mass of a dynamic vibration absorber. J. Sound Vib.
**1989**, 1323, 518–522. [Google Scholar] [CrossRef] - Cuvalci, O.; Ertas, A.; Ekwaro-Osire, S. Nonlinear vibration absorber for a system under sinusoidal and random excitation: Experiments. J. Sound Vib.
**2002**, 249, 701–718. [Google Scholar] [CrossRef] - Hui, C.K.; Ng, C.F. Autoparametric vibration absorber effect to reducethe first symmetric mode vibration of a curved beam/panel. J. Sound Vib.
**2011**, 330, 4551–4573. [Google Scholar] [CrossRef] - Abundis-Fong, H.F.; Enríquez-Zárate, J.; Cabrera-Amado, A.; Silva-Navarro, G. Optimum design of a nonlinear vibration absorber coupled to a resonant oscillator: A case study. Shock Vib.
**2018**, 2018, 2107607. [Google Scholar] [CrossRef] [Green Version] - Ting, T.; Yan, Z.; Zou, Y.; Zhang, W. Optimal dual-functional design for a piezoelectric autoparametric vibration absorber. Mech. Syst. Signal Prcess.
**2019**, 123, 513–532. [Google Scholar] - Beltran-Carbajal, F.; Silva-Navarro, G. Adaptive-like vibration control in mechanical systems with unknown paramenters and signals. Asian J. Control
**2013**, 15, 1613–1626. [Google Scholar] [CrossRef] - Fliess, M.; Sira-Ramirez, H. An algebraic framework for linear identification. ESAIM Control Optim. Calc. Var.
**2003**, 9, 151–168. [Google Scholar] [CrossRef] - Fliess, M.; Join, C. Model-free control. Int. J. Control
**2013**, 86, 2228–2252. [Google Scholar] [CrossRef] [Green Version] - Mboup, M.; Join, C.; Fliess, M. Numerical differentiation with annihilators in noisy environment. Numer. Algorithms
**2009**, 50, 439–467. [Google Scholar] [CrossRef] [Green Version] - Enriquez-Zarate, J.; Abundis-Fong, H.F.; Silva-Navarro, G. Passive vibration control in a building-like structure using a tuned-mass-damper and an autoparametric cantilever beam absorber. In Active and Passive Smart Structures and Integrated Systems 2015; International Society for Optics and Photonics: Washington, DC, USA, 2015; Volume 9431. [Google Scholar]
- Rao, S.S. Mechanical Vibrations, 5th ed.; Prentice Hall: Upper Saddle River, NJ, USA, 2011. [Google Scholar]
- Cartmell, M.O. Introduction to Linear, Parametric and Nonlinear Vibrations; Chapman and Hall: London, UK, 1990. [Google Scholar]
- Tondl, A.; Ruijgrok, T.; Verhulst, F.; Nabergoj, R. Autoparametric Resonance in Mechanical Systems; Cambridge University Press: Cambridge, MA, USA, 2000. [Google Scholar]
- Nayfeh, A.H.; Mook, D.T. Nonlinear Oscillations; John Wiley & Sons: New York, NY, USA, 1979. [Google Scholar]
- Silva-Navarro, G.; Abundis-Fong, H.F. Evaluation of Autoparametric Vibration Absorbers on N-Story Building-Like Structures. In Nonlinear Dynamics, Volume 1; Conference Proceedings of the Society for Experimental Mechanics Series; Kerschen, G., Ed.; Springer: Cham, Switzerland, 2017; pp. 177–184. [Google Scholar]
- Heylen, W.; Lammens, S.; Sas, P. Modal Analysis, Theory and Testing; Katholieke Universiteit Leuven: Leuven, Belgium, 2003. [Google Scholar]
- Jimin, H.; Fu, Z.-F. Modal Analysis; Chapter 6 Modal Analysis of a Damped MDoF System; Butterworth-Heinemann: Oxford, UK, 2001. [Google Scholar]
- Mikusiński, P. Operational Calculus, 2nd ed.; PWN & Pergamon: Warsaw, Poland, 1983; Volume 1. [Google Scholar]
- Silva-Navarro, G.; Beltrán-Carbajal, F.; Trujillo-Franco, L.G. Adaptive-Like Vibration Control in a Three-Story Building-Like Structure with a PZT Stack Actuator. In Topics in Modal Analysis, Volume 10; Conference Proceedings of the Society for Experimental Mechanics Series; Mains, M., Ed.; Springer: Cham, Switzerland, 2015; pp. 123–131. [Google Scholar]
- Beltran-Carbajal, F.; Silva-Navarro, G. On the algebraic parameter identification of vibrating mechanical systems. Int. J. Mech. Sci.
**2015**, 92, 178–186. [Google Scholar] [CrossRef] - Feldman, M. Hilbert Transform Applications in Mechanical Vibration; John Wiley and Sons, Ltd.: Chichester, UK, 2011. [Google Scholar]
- Richardson, M.H.; Formenti, D.L. Parameter estimation from frequency response measurements using rational fraction polynomials. In Proceedings of the 1st IMAC Conference, Orlando, FL, USA, 8–10 November 1982. [Google Scholar]
- Kelly, L.G. Curve fitting and data smoothing. In Handbook of Numerical Methods and Application; Addison-Wesley: New York, NY, USA, 1967; Chapter 5. [Google Scholar]

**Figure 1.**Frequency response functions of the flexible mechanical system with a passive dynamic vibration absorber. Primary system (

**top**), Tuned Mass Damper (

**bottom**).

**Figure 5.**Experimental FRF of the six-story building-like structure in blue line and its corresponding Hilbert transformation in dotted black line.

**Figure 6.**Dynamic response of the primary system without pendulum absorber (Tuned Mass Damper case).

**Figure 9.**On-line estimation of the frequency of the excitation force using displacement measurements of the sixth floor or degree of freedom.

**Figure 11.**Experimental FRF of the three-story building-like structure with autoparametric pendulum absorber.

**Figure 12.**Experimental FRF of the three-story building-like structure with autoparametric absorber in blue line and its corresponding Hilbert transformation in dotted black line.

**Figure 13.**On-line estimation of the frequency of the excitation force using displacement measurements of the third floor or degree of freedom with pendulum absorber (nonlinear-type).

**Figure 14.**Graph of the total energy in the main system with and without nonlinear pendulum absorber.

Mode | Frequency [Hz] | Damping Ratio % |
---|---|---|

1 | 1.148 | 0.15 |

2 | 3.39 | 0.39 |

3 | 5.44 | 0.18 |

4 | 7.16 | 0.19 |

5 | 8.53 | 0.18 |

6 | 9.34 | 0.17 |

Coefficient | Value |
---|---|

${\widehat{a}}_{11}$ | 1.736 |

${\widehat{a}}_{10}$ | 254.8 |

${\widehat{a}}_{9}$ | 345 |

${\widehat{a}}_{8}$ | $2.408\times {10}^{4}$ |

${\widehat{a}}_{7}$ | $2.417\times {10}^{4}$ |

${\widehat{a}}_{6}$ | $1.03534\times {10}^{6}$ |

${\widehat{a}}_{5}$ | $7.1273\times {10}^{5}$ |

${\widehat{a}}_{4}$ | $1.97956\times {10}^{7}$ |

${\widehat{a}}_{3}$ | $8.331231\times {10}^{6}$ |

${\widehat{a}}_{2}$ | $1.357123\times {10}^{8}$ |

${\widehat{a}}_{1}$ | $2.72123\times {10}^{7}$ |

${\widehat{a}}_{0}$ | $1.45834\times {10}^{8}$ |

Coefficient | Value |
---|---|

${\widehat{a}}_{5}$ | 4.6 |

${\widehat{a}}_{4}$ | 4417 |

${\widehat{a}}_{3}$ | $1.4134\times {10}^{4}$ |

${\widehat{a}}_{2}$ | $3.95435\times {10}^{6}$ |

${\widehat{a}}_{1}$ | $6.64657\times {10}^{6}$ |

${\widehat{a}}_{0}$ | $3.4632423\times {10}^{8}$ |

**Table 4.**Modal parameters of the three-story building-like structure with nonlinear pendulum absorber.

Mode | Frequency [Hz] | Damping Ratio % |
---|---|---|

1 | 1.03 | 0.11 |

2 | 5.26 | 0.75 |

3 | 9.03 | 0.34 |

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

© 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**

Trujillo-Franco, L.G.; Silva-Navarro, G.; Beltran-Carbajal, F.; Campos-Mercado, E.; Abundis-Fong, H.F.
On-Line Modal Parameter Identification Applied to Linear and Nonlinear Vibration Absorbers. *Actuators* **2020**, *9*, 119.
https://doi.org/10.3390/act9040119

**AMA Style**

Trujillo-Franco LG, Silva-Navarro G, Beltran-Carbajal F, Campos-Mercado E, Abundis-Fong HF.
On-Line Modal Parameter Identification Applied to Linear and Nonlinear Vibration Absorbers. *Actuators*. 2020; 9(4):119.
https://doi.org/10.3390/act9040119

**Chicago/Turabian Style**

Trujillo-Franco, Luis Gerardo, Gerardo Silva-Navarro, Francisco Beltran-Carbajal, Eduardo Campos-Mercado, and Hugo Francisco Abundis-Fong.
2020. "On-Line Modal Parameter Identification Applied to Linear and Nonlinear Vibration Absorbers" *Actuators* 9, no. 4: 119.
https://doi.org/10.3390/act9040119