# Vibration Suppression of a Flexible Beam Structure Coupled with Liquid Sloshing via ADP Control Based on FBG Strain Measurement

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## Abstract

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## 1. Introduction

- (1)
- Compared with complex dynamic models, the Euler–Bernoulli beam model and spring-mass-damper equivalent model provide a simpler and more convenient way to construct the flexible-sloshing coupling dynamic model. The development of a strain-based vibration dynamic model facilitates the full utilization of FBG sensors’ information.
- (2)
- Compared with control methods that require motion parameters, the control method proposed in this paper can effectively suppress elastic vibration and sloshing even when only partial strain information is applied. Furthermore, the utilization of FBG strain information allows for direct measurement, eliminating the need for estimation of vibration parameters. This controller’s advantages in practicality make it highly suitable for engineering applications.

## 2. Preliminaries and System Descriptions

#### 2.1. Flexible-Sloshing Coupling Dynamic Model

**Assumption 1.**

**Assumption 2.**

**Remark 1**

#### 2.2. Strain-Based Vibration Dynamic Model

**Remark 2.**

## 3. Control Strategy Design

#### 3.1. Design of ADP Control Method Based on Strain Information of FBG

**Remark 3**

#### 3.2. Stability Analysis

**Assumption 3.**

**Lemma 1**

**Theorem 1.**

**Proof of Theorem 1.**

## 4. Numerical Simulations

## 5. Conclusions

- (1)
- The usages of the Euler–Bernoulli beam model and the spring-mass-damper equivalent model provide a simpler and more convenient way for constructing the flexible-sloshing coupling dynamic model.
- (2)
- The development of a strain-based vibration dynamic model facilitates the full utilization of FBG sensors’ strain information.
- (3)
- This controller can effectively suppress elastic vibration and sloshing with only partial strain information, eliminating the process of estimating the vibration motion parameters. The control strategy indicates important implications in engineering applications.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Flexible-sloshing coupling system including spring-mass-damper model and Euler–Bernoulli cantilever beam’s FEM model.

**Figure 4.**Displacement of the structure in (

**a**) free case; (

**b**) ADP case 1 with 10 sensors; (

**c**) ADP case 2 with 20 sensors; (

**d**) PD case.

**Figure 5.**Results of displacement. (

**a**) Displacement of the structure’s free end in free case, PD case, and ADP case 1; (

**b**) sloshing displacement in free case, PD case, and ADP case 1.

**Figure 6.**Results of displacement. (

**a**) Displacement of the structure’s free end for ADP control with 10 and 20 sensors; (

**b**) sloshing displacement for ADP control with 10 and 20 sensors.

**Figure 7.**Error results of displacement relative to free case. (

**a**) Displacement of the structure’s free end in PD case and ADP case 1; (

**b**) sloshing displacement in PD case and ADP case 1.

**Figure 8.**Error results of displacement between ADP case 1 and ADP case 2. (

**a**) Displacement of the structure’s free end; (

**b**) sloshing displacement.

Parameters | Value | |
---|---|---|

Aluminum alloy beam | Density (kg/m^{2}) | 2690 |

Elastic modulus (Pa) | 6.98e10 | |

Length (m) | 0.972 | |

Width (m) | 0.02 | |

Height (m) | 0.003 | |

Cuboid tank | Height of tank(m) | 0.1 |

Length of bottom (m) | 0.04 | |

Height of liquid level (m) | 0.04 | |

Water | Density (kg/m^{2}) | 1000 |

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**MDPI and ACS Style**

Kong, C.; Zhao, D.; Liang, B.
Vibration Suppression of a Flexible Beam Structure Coupled with Liquid Sloshing via ADP Control Based on FBG Strain Measurement. *Actuators* **2023**, *12*, 471.
https://doi.org/10.3390/act12120471

**AMA Style**

Kong C, Zhao D, Liang B.
Vibration Suppression of a Flexible Beam Structure Coupled with Liquid Sloshing via ADP Control Based on FBG Strain Measurement. *Actuators*. 2023; 12(12):471.
https://doi.org/10.3390/act12120471

**Chicago/Turabian Style**

Kong, Chunyang, Dangjun Zhao, and Buge Liang.
2023. "Vibration Suppression of a Flexible Beam Structure Coupled with Liquid Sloshing via ADP Control Based on FBG Strain Measurement" *Actuators* 12, no. 12: 471.
https://doi.org/10.3390/act12120471