# Online Estimation Techniques for Natural and Excitation Frequencies on MDOF Vibrating Mechanical Systems

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## Abstract

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## 1. Introduction

- A novel algebraic technique for online estimation in time-domain of multiple natural frequencies for an important class of forced and uncertain lumped-parameter vibrating mechanical systems of n degrees of freedom is proposed.
- Closed-form algebraic formulas to simultaneously estimate online multiple frequencies of possible exogenous excitation forces affecting a vibrating system are obtained.
- Compared with other parameter identification methods, the introduced multifrequency estimation approach is performed in time-domain, algebraically and online, into a small interval of time. System parameters of mass, stiffness and damping matrices are assumed to be completely unknown.
- Measurements of available output signals, associated with some degree of freedom, are only required. Then, multifrequency estimation can be carried out using measurements of either position, velocity or acceleration.

## 2. Description of the MDOF Forced Vibrating Mechanical System

## 3. Algebraic Estimation of Natural and Forcing Frequencies

## 4. Numerical and Experimental Validation

#### 4.1. Numerical Results: 3 DOF Mass-Spring System

#### 4.2. Experimental Results: A Cantilever Euler–Bernoulli Beam

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Online algebraic estimation of the three natural frequencies for the 3 DOF vibrating mechanical system (18).

**Figure 2.**Online algebraic estimation of the three excitation frequencies for the 3 DOF vibrating mechanical system (18).

**Figure 3.**Experimental setup of the cantilever beam submitted to forces with multiple excitation frequencies.

**Figure 4.**Experimental Frequency Response Functions (FRF) of the cantilever beam for the first three mode-shapes.

**Figure 5.**Schematic diagram of the signal generation and data acquisition system for online algebraic identification.

**Figure 6.**Online algebraic estimation of the first three natural frequencies of the distributed-parameter system.

**Figure 7.**Online algebraic estimation of exogenous excitation frequencies of the distributed-parameter system.

Mass [kg] | Viscous Damping [Ns/m] | Stiffness [N/m] |
---|---|---|

${m}_{1}=5.0$ | ${c}_{1}\approx 0$ | ${k}_{1}=1200$ N/m |

${m}_{2}=3.0$ | ${c}_{2}\approx 0$ | ${k}_{2}=1000$ N/m |

${m}_{3}=2.5$ | ${c}_{3}\approx 0$ | ${k}_{3}=750$ N/m |

− | − | ${k}_{4}=400$ N/m |

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

Density | 2710 [kg/m${}^{3}$] |

Thickness | 0.00158 [m] |

Length | 1.0160 [m] |

Width | 0.0381 [m] |

Young’s modulus | 68.9 $\times {10}^{9}$ [Pa] |

Area of cross section | 6.048375 $\times {10}^{-5}$ [m${}^{2}$] |

Moment of inertia | 1.27 $\times {10}^{-11}$ [m${}^{4}$] |

Polar moment of inertia | 0.059 [kg·m${}^{2}$] |

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

1 | $0.79$ | $1.18$ |

2 | $6.63$ | $1.23$ |

3 | $19.97$ | $3.1$ |

Offline FEM | Online Algebraic Method | |||||
---|---|---|---|---|---|---|

Frequencies | Actual | Estimation | Error | Estimation | Error | |

Natural | [Hz] | [Hz] | [%] | [Hz] | [%] | |

${\omega}_{1}$ | $0.79$ | 0.8412 | 6.481 | 0.699 | −11. 43 | |

${\omega}_{2}$ | $6.63$ | 6.2710 | −5.415 | 6.626 | −0.06 | |

${\omega}_{3}$ | $19.97$ | 18.780 | −5.959 | 20.06 | 0.45 | |

Excitation | [Hz] | [Hz] | [%] | [Hz] | [%] | |

${\Omega}_{1}$ | $0.318$ | − | − | 0.3202 | 0.692 | |

${\Omega}_{2}$ | $7.96$ | − | − | 7.576 | −4.824 | |

${\Omega}_{3}$ | $15.91$ | − | − | 15.430 | −3.017 |

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**MDPI and ACS Style**

Silva-Navarro, G.; Beltran-Carbajal, F.; Trujillo-Franco, L.G.; Peza-Solis, J.F.; Garcia-Perez, O.A.
Online Estimation Techniques for Natural and Excitation Frequencies on MDOF Vibrating Mechanical Systems. *Actuators* **2021**, *10*, 41.
https://doi.org/10.3390/act10030041

**AMA Style**

Silva-Navarro G, Beltran-Carbajal F, Trujillo-Franco LG, Peza-Solis JF, Garcia-Perez OA.
Online Estimation Techniques for Natural and Excitation Frequencies on MDOF Vibrating Mechanical Systems. *Actuators*. 2021; 10(3):41.
https://doi.org/10.3390/act10030041

**Chicago/Turabian Style**

Silva-Navarro, Gerardo, Francisco Beltran-Carbajal, Luis Gerardo Trujillo-Franco, Juan Fernando Peza-Solis, and Oscar A. Garcia-Perez.
2021. "Online Estimation Techniques for Natural and Excitation Frequencies on MDOF Vibrating Mechanical Systems" *Actuators* 10, no. 3: 41.
https://doi.org/10.3390/act10030041