# Quantification of Active Structural Path for Vibration Reduction Control of Plate Structure under Sinusoidal Excitation

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## Abstract

**:**

## 1. Introduction

## 2. Modeling of Mounting System

#### 2.1. 9-DOF Modeling

#### 2.2. Calculation of Actuator Amplitude and Phase

## 3. Validation Using the Numerical Simulation

#### 3.1. Simulation Overview

**A**,

**B**, and

**C**, the

**G**,

**H**, and

**C**matrices wre constructed for the estimates. Because the state space equations are defined in continuous time, they were transformed into discrete time matrices through bilinear transformation. The observer gain

**L**was selected appropriately, and the agreement between the estimated and actual output was confirmed, as shown in Figure 5. Therefore, the validity of the model was verified, and it was applied to the feedback control system.

#### 3.2. Control Using the Calculated Amplitude and Phase

#### 3.3. Control Using the NLMS Algorithm

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 6.**Result of the vibration of the source related to each active path; the blue and red lines represent the vibration of the source before and after applying the control, respectively.

**Figure 7.**Result of the vibration of the receiver related to each active path; the blue and red lines represent the vibration of the receiver before and after applying the control, respectively.

**Figure 8.**Schematic representation of the simulation using the NLMS (Normalized Least Mean Squares) algorithm.

**Figure 9.**Result of the vibration of the source related to each active path using the NLMS algorithm; the blue and red lines represent the vibration of the source before and after applying the control, respectively.

**Figure 10.**Result of the vibration of the receiver related to each active path using the NLMS algorithm; the blue and red lines represent the vibration of the receiver before and after applying the control, respectively.

Variable | Values | Units | Variable | Values | Units |
---|---|---|---|---|---|

${m}_{s}={m}_{r}$ | $1.081$ | $\mathrm{kg}$ | ${k}_{m5}^{\ast}$ | $0.53(1+i0.256)$ | $\mathrm{kN}\hspace{0.17em}{\mathrm{mm}}^{-1}$ |

${m}_{st1}={m}_{st3}$ | $0.067$ | $\mathrm{kg}$ | ${k}_{m6}^{\ast}$ | $0.61(1+i0.300)$ | $\mathrm{kN}\hspace{0.17em}{\mathrm{mm}}^{-1}$ |

${m}_{st2}$ | $0.075$ | $\mathrm{kg}$ | ${k}_{brn\hspace{0.17em}(n=1,2)}^{\ast}$ | $0.42(1+i0.300)$ | $\mathrm{kN}\hspace{0.17em}{\mathrm{mm}}^{-1}$ |

${I}_{s,y}={I}_{r,y}$ | $16.172$ | $\mathrm{g}\hspace{0.17em}{\mathrm{m}}^{2}$ | ${l}_{sn\hspace{0.17em}(n=1,2)}$ | $100$ | $\mathrm{mm}$ |

${I}_{sf,x}={I}_{rf,x}$ | $5.940$ | $\mathrm{g}\hspace{0.17em}{\mathrm{m}}^{2}$ | ${l}_{rn\hspace{0.17em}(n=1,2)}$ | $140$ | $\mathrm{mm}$ |

${k}_{m1}^{\ast}$ | $5.46(1+i0.034)$ | $\mathrm{kN}\hspace{0.17em}{\mathrm{mm}}^{-1}$ | ${l}_{fn\hspace{0.17em}(n=1,2)}$ | $80$ | $\mathrm{mm}$ |

${k}_{m2}^{\ast}$ | $5.46(1+i0.034)$ | $\mathrm{kN}\hspace{0.17em}{\mathrm{mm}}^{-1}$ | ${l}_{rfn\hspace{0.17em}(n=1~4)}$ | $100$ | $\mathrm{mm}$ |

${k}_{m3}^{\ast}$ | $2.48(1+i0.036)$ | $\mathrm{kN}\hspace{0.17em}{\mathrm{mm}}^{-1}$ | $d$ | $50$ | $\mathrm{mm}$ |

${k}_{m4}^{\ast}$ | $0.61(1+i0.300)$ | $\mathrm{kN}\hspace{0.17em}{\mathrm{mm}}^{-1}$ | ${d}_{2}$ | $10$ | $\mathrm{mm}$ |

Source | ${a}_{st1,g1}$ | ${a}_{st2,g1}$ | ${a}_{st3,g1}$ |

Original | $0.533$ | $1.514$ | $1.095$ |

All actuators turned on | $2.934\left(450.47\%\uparrow \right)$ | $2.896\left(91.28\%\uparrow \right)$ | $0.019\left(98.19\%\downarrow \right)$ |

Receiver | ${a}_{st1,g2}$ | ${a}_{st2,g2}$ | ${a}_{st3,g2}$ |

Original | $0.2382$ | $0.37$ | $0.6122$ |

All actuators turned on | $0.1021\left(57.13\%\downarrow \right)$ | $0.165\left(55.18\%\downarrow \right)$ | $0.384\left(37.21\%\downarrow \right)$ |

Source | ${a}_{st1,g1}$ | ${a}_{st2,g1}$ | ${a}_{st3,g1}$ |

Original | $0.533$ | $1.514$ | $1.095$ |

Control using the NLMS | $1.096\left(105.62\%\uparrow \right)$ | $1.171\left(22.66\%\downarrow \right)$ | $0.374\left(65.84\%\downarrow \right)$ |

Receiver | ${a}_{st1,g2}$ | ${a}_{st2,g2}$ | ${a}_{st3,g2}$ |

Original | $0.2382$ | $0.37$ | $0.6122$ |

Control using the NLMS | $0.214\left(10.08\%\downarrow \right)$ | $0.352\left(4.86\%\downarrow \right)$ | $0.059\left(90.29\%\downarrow \right)$ |

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**MDPI and ACS Style**

Hong, D.; Kim, B.
Quantification of Active Structural Path for Vibration Reduction Control of Plate Structure under Sinusoidal Excitation. *Appl. Sci.* **2019**, *9*, 711.
https://doi.org/10.3390/app9040711

**AMA Style**

Hong D, Kim B.
Quantification of Active Structural Path for Vibration Reduction Control of Plate Structure under Sinusoidal Excitation. *Applied Sciences*. 2019; 9(4):711.
https://doi.org/10.3390/app9040711

**Chicago/Turabian Style**

Hong, Dongwoo, and Byeongil Kim.
2019. "Quantification of Active Structural Path for Vibration Reduction Control of Plate Structure under Sinusoidal Excitation" *Applied Sciences* 9, no. 4: 711.
https://doi.org/10.3390/app9040711