# Active Control of Submerged Systems by Moving Mass

## Abstract

**:**

## 1. Introduction

^{6}m in comparison to electromagnetic waves, which can propagate around 10

^{2}m in water. Fluid–solid interaction is important in many applications such as vibration suppression particularly for bioacoustics submerged structures [1]; energy harvesting by galloping [2,3,4]; physical acoustics by piezo fan [5], cavitation [6], sloshing [7]; acoustical oceanography, modal analysis of submerged structures [8]; structures under cavitation [9]; active vibration control of submerged structures for remote control of surveillance [10]; sound cancellation of submerged systems [11]; atomic force microscope energy harvesting [7]; transduction, sonar, acoustic signal processing [12]; underwater communications systems and networks [7,8,9,10,11], among others. A liquid–solid interaction happens when a structure vibrates in a liquid [8], and must be addressed using both fundamental science and engineering. There are many studies focusing on vibration suppression of a solid structure by piezo ceramic elements (PZT) [2]. The neutrino–seawater interaction can be sensed by acoustical methods. The piezoelectric effect delivers the aptitude to utilize these materials as both actuators and sensors. Piezo ceramic elements have been extensively used for active vibration control in the Neutrino Telescope [2,12]. The expansion in dormancy comes about because the smooth movement influences basic vibrations, with the assumption that the normal frequencies of a structure in a liquid are altogether lower than those in air [9,13]. This marvel has been portrayed by presenting the idea of an additional virtual mass via a triangulation method. Active control strategies are regularly deficient to control the vibrations of structures, thus, we look for dynamic techniques to smother vibrations to improve the presentation of the frameworks of intrigue; for example, for the piezo ceramic elements mounted on flexible string lines fixed at the seabed [14]. The fluid–solid interaction affects structures through a wide range of applications and sizes, from microscale MEMS structures to larger ship structures [15]. The dynamic conductivity of plate structures is critical to numerous applications running from cars to designing ventures. In addition, submerged plates are fundamental pieces of ship building, atomic, sea, and maritime designing [16]. The vibration qualities of the unblemished plate combined with liquid medium have been thoroughly treated and very much archived in important writing. It is in this way realized that vibrations of the submerged unblemished plate are not quite the same as those in vacuum [17]. Brilliant structure innovation has led to dynamic controls to react to outer aggravations and can offer upgrades in framework execution without essentially expanding the weight [18]. One advantage of utilizing a brilliant structure is that it can adapt to changes in nature by detecting outside unsettling influences [19,20]. In addition to piezo ceramic elements, eddy-current-tuned mass damper and pounding-tuned mass damper are used to suppress vibrations in submerged pipelines [21,22].

_{33}), the electric charge on the electrodes of the transducer and the total displacement have a linear relation with voltage and force as:

## 2. Governing Equations

_{s}is the supporting mass of the AMD, m

_{a}is moving mass of the AMD, the nondimensionalized mass matrices of the solid plate are

**d**), equations of motion can then be written as

**M**, the natural frequencies and mode shapes for vibrating in vacuo (${\omega}_{\mathit{a}}$) can be obtained. The eigenvector of matrix

_{f}**U**satisfies the orthonormality condition:

_{1}V

_{2}], V with subscript 1 from Equation (31) and uncontrolled with subscript 2 from Equation (28), the decentralized multi-input multi-output (MIMO) negative acceleration feedback control (NAF) controller coupled equations are presented as

## 3. Results and Discussion

## 4. Conclusions

- Inviscid fluid modeling with fluid–solid interaction by Bernoulli forces is an effective approach to frequency response modeling of sloshing systems coupled by structure.
- The negative acceleration feedback control algorithm in modal-space is an effective approach to controlling the vibrating plate problem submerged in a vessel.
- The decentralized negative acceleration feedback control algorithm in modal-space is an effective approach to controlling the vibrating plate problem submerged in a vessel.

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

a | Dimension of the aluminum plate for draught of 50% (m) |

b | Length of the fluid tank (m) |

$B$ | vector consisting of plate displacement at the point of supporting mass |

C | capacitance |

${\mathrm{C}\mathrm{e}}_{\mathit{r}}$ | nonperiodic even Mathieu function (cosine-elliptic) |

d_{33} | piezoelectric charge coefficient |

E | Young’s modulus, Module of elasticity (GPa) |

F | force |

g | Gravity constant (ms^{−2}) |

H | Height of the fluid tank (m) |

K | stiffness |

L | Depth of the fluid tank (m) |

$\hspace{0.17em}q(\mathit{t})$ | vector consisting of generalized coordinates |

p | pressure (Pa) |

Q | electric charge on the electrodes of the transducer |

${\mathrm{s}\mathrm{e}}_{\mathit{r}}$ | periodic Mathieu function (sin-elliptic) |

t | thickness of the aluminum plate (m) |

$\hspace{0.17em}u(\mathit{t})$ | vector consisting of moving mass displacements |

V | voltage |

Greek symbols | |

$\mathsf{\Delta}$ | total displacement |

ρ | density (kg/m^{3}) |

ν | Poisson ratio of the aluminum plate |

${\chi}_{\mathit{i}}\hspace{0.17em}(\mathit{i}=1,2,\dots ,\mathit{n})$ | eigenfunction of a torsional Spring-free beam in x direction |

${\lambda}_{\mathit{i}}\hspace{0.17em}$ | eigenvalues of a free-free beam in z direction |

${\gamma}_{\mathit{i}}\hspace{0.17em}(\mathit{i}=1,2,\dots ,\mathit{n})$ | eigenfunction of a free-free beam in z direction |

Subscript | |

a | moving mass of the active mass damper |

f | base fluid |

p | aluminum plate |

s | supporting mass |

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**Figure 2.**Comparison of magnitude of frequency response function of the free plate, for theory and experiment.

**Figure 3.**Comparison of frequency response function magnitude of the partially submerged plate (H = 0.15 m), theory and experiment.

**Figure 4.**Comparison of frequency response function magnitude of the partially submerged plate (H = 0.3 m), theory and experiment.

**Figure 5.**Frequency response function of the partially submerged plate (H = 0.3 m) controlled by multi-input multi-output (MIMO) modal-space negative acceleration feedback control (NAF), (

**a**) magnitude and (

**b**) phase.

**Figure 6.**Frequency response function of the partially submerged plate (H = 0.3 m) controlled by decentralized MIMO modal-space NAF, (

**a**) magnitude and (

**b**) phase.

**Figure 7.**Impulse response by decentralized MIMO NAF controller of the partially submerged plate (H = 0.3 m).

**Table 1.**Experimental setup parameters [16].

Parameter | Value | Unit | Description |
---|---|---|---|

t | 2 | mm | Thickness of the aluminum plate |

$\nu $ | 0.33 | − | Poisson ratio of the aluminum plate |

ρ_{p} | 2700 | kg/m^{3} | Density of the aluminum plate |

E_{p} | 68.9 | GPa | Module of elasticity of the aluminum plate |

ρ_{f} | 1000 | kg/m^{3} | Density of the fluid |

a | 900 | mm | Dimension of the aluminum plate for draught of 50% |

b | 300 | mm | Dimension of the fluid tank |

H | 450 | mm | Dimension of the fluid tank |

L | 450 | mm | Dimension of the fluid tank |

10% Draught | 20% Draught | 30% Draught | 40% Draught | 50% Draught | |
---|---|---|---|---|---|

1 | 3.4919 | 3.4664 | 3.4509 | 3.4417 | 3.4360 |

2 | 8.5322 | 14.8380 | 21.2059 | 21.5038 | 21.4772 |

3 | 21.4308 | 21.5970 | 21.5427 | 27.6445 | 34.1324 |

4 | 27.3504 | 48.4160 | 60.4955 | 60.3272 | 60.2020 |

5 | 31.1654 | 60.5937 | 66.6982 | 85.4110 | 104.4240 |

6 | 54.4540 | 93.3672 | 118.9479 | 118.6553 | 118.3542 |

7 | 61.6181 | 93.7573 | 121.4258 | 151.4252 | 182.4435 |

8 | 64.5492 | 119.1964 | 188.4479 | 196.7006 | 196.2312 |

9 | 71.5548 | 127.6663 | 195.0043 | 227.2713 | 268.3955 |

10 | 93.7973 | 154.1316 | 208.4070 | 322.2323 | 377.0116 |

Numerical | Experimental | |
---|---|---|

1 | 3.4360 | 3.5068 |

2 | 21.4772 | 20.4717 |

3 | 34.1324 | 33.3710 |

4 | 60.2020 | 57.4699 |

5 | 104.4240 | 100.2170 |

6 | 118.3542 | 122.1825 |

7 | 182.4435 | 185.9980 |

8 | 196.2312 | 192.6422 |

9 | 268.3955 | 280.4793 |

10 | 377.0116 | 359.4597 |

Parameter | Description |
---|---|

(x_{p},y_{p}) | Observed point of vibration |

m_{a} | Active mass damper |

u | Displacement of moving mass |

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**MDPI and ACS Style**

Abdollahzadeh Jamalabadi, M.Y.
Active Control of Submerged Systems by Moving Mass. *Acoustics* **2021**, *3*, 42-57.
https://doi.org/10.3390/acoustics3010005

**AMA Style**

Abdollahzadeh Jamalabadi MY.
Active Control of Submerged Systems by Moving Mass. *Acoustics*. 2021; 3(1):42-57.
https://doi.org/10.3390/acoustics3010005

**Chicago/Turabian Style**

Abdollahzadeh Jamalabadi, Mohammad Yaghoub.
2021. "Active Control of Submerged Systems by Moving Mass" *Acoustics* 3, no. 1: 42-57.
https://doi.org/10.3390/acoustics3010005