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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

In this work, active vibration control of an underwater cylindrical shell structure was investigated, to suppress structural vibration and structure-borne noise in water. Finite element modeling of the submerged cylindrical shell structure was developed, and experimentally evaluated. Modal reduction was conducted to obtain the reduced system equation for the active feedback control algorithm. Three Macro Fiber Composites (MFCs) were used as actuators and sensors. One MFC was used as an exciter. The optimum control algorithm was designed based on the reduced system equations. The active control performance was then evaluated using the lab scale underwater cylindrical shell structure. Structural vibration and structure-borne noise of the underwater cylindrical shell structure were reduced significantly by activating the optimal controller associated with the MFC actuators. The results provide that active vibration control of the underwater structure is a useful means to reduce structure-borne noise in water.

In the last several decades, significant advances have been achieved in the field of smart materials and structures. One of the main applications of smart materials and structures is active vibration control to suppress undesirable structural vibration and noise. A smart structure has the capability to respond to changes in the external environment, as well as to a change of its internal environment. It incorporates smart materials that allow the change of system characteristics, such as stiffness or damping, in a controlled manner. Many types of smart materials are being developed as actuators and sensors, such as piezoelectric materials, shape memory alloys, electrorheological fluids, magnetorheological fluids, electrostrictive materials, magnetostrictive materials and electroactive polymers. In particular, piezoelectric materials are most commonly used as smart materials, owing to their quick response, wide bandwidth and easy implementation. Moreover, piezoelectric materials can be employed as both actuators and sensors, by taking advantage of direct and converse piezoelectric effects.

Crawley and de Luis provided pioneering work in this area, involving the development of the induced strain actuation mechanism [

Consequently, the main purpose of this work is to actively control the imposed vibration of an underwater cylindrical shell structure using MFC actuators, and to experimentally investigate the reduction of structure-borne noise due to the vibration control effect. Finite element modeling was developed to obtain a state space equation for the active control algorithm. The optimal control algorithm was designed and experimentally implemented to suppress structural vibration in water. It has been demonstrated that the imposed vibration of the underwater cylindrical shell structure was suppressed significantly, based on the designed optimal control algorithm. In addition, it has been measured that the structure-borne noise was effectively reduced, by actively controlling the vibration using MFC actuators.

A schematic diagram of the proposed underwater cylindrical shell structure for vibration control is shown in

A simple end-capped cylindrical shell structure is considered as the host structure, which can be considered as the simple model of a submarine. MFC actuators are bonded on the surface of the host structure, and perfect bonding is assumed between the host structure and actuators. The structure is considered in the water, with free-free boundary conditions.

Finite element modeling of the underwater cylindrical shell structure is conducted first. The underwater cylindrical shell structure is divided into three parts, namely the cylinder, MFC actuators and water. The cylinder and MFC actuators are finite, but the fluid is infinite in dimension. In mathematical modeling of the underwater cylindrical shell structure, the fluid-structure interaction must be considered. The equations of motion of the underwater cylindrical shell structure can be expressed as follows [_{S}_{S}_{S}_{S}_{S}_{I}_{I}_{S}_{I}_{,} is solved by applying the boundary element method, based on the inverse formulation of Euler's solution. Then, one can obtain:
_{F}

Now electro-mechanical coupling of the full structure and MFC actuators is considered. After the application of the variational principle and finite element discretization, the coupled finite element equations of motion can be expressed as follows:
_{uϕ}_{ϕu}_{ϕϕ}_{S}_{ϕ}

The above reduced equations of motion are coupled with each other. The mesh size must be fine to obtain accurate dynamic responses of the underwater cylindrical shell structure. However, the feedback control algorithm requires a small size of the system matrix, due to the limitation of computer performance. The size of the above system must be reduced for the active feedback control. The most commonly used method to reduce the size of the system is modal reduction. The above coupled equations of motion are first solved for un-damped free vibration. The mode shapes are extracted, and assembled as a modal matrix Φ. Then, the modal matrix is used to transform the global displacement vector

Substituting

The matrices _{c}_{i}_{i}_{i}

An optimal control algorithm is designed for active vibration control of the underwater cylindrical shell structure. The sensor noises and system disturbances are also considered for the actual implementation of the cylindrical shell structure. The control purpose is to regulate the unwanted vibrations of the cylindrical shell structure. Thus, the performance index to be minimized is chosen as follows:

In the above equation,

Here, _{G}

Since the states _{i}_{i}_{1} and _{2} are uncorrelated white noise characterized by covariance matrices _{1} and _{2}, as follows:

In the above equation,

Using the estimated states, the control input is obtained as follows:

A block diagram of the proposed LQG controller is presented in

This study investigates active vibration control of the underwater cylindrical shell structure. The geometries of the proposed cylindrical shell structure and MFC actuators are given in

Modal characteristics of the proposed underwater cylindrical shell structure were first investigated by using the commercial finite element analysis package ANSYS. The finite element mesh configuration is presented in

An experimental test was also conducted to validate the finite element modeling. The manufactured end-capped aluminum cylindrical shell structure is shown in

The fundamental mode shapes of the underwater cylindrical shell structure are presented in

Modal analysis of the finite element modeling is evaluated by the modal test. The experimental frequency response of the underwater cylindrical shell structure is presented in

Now, active vibration control performance of the underwater cylindrical shell structure is studied. The experimental apparatus for the active vibration control is presented in

The excitation signal, which is generated from a personal computer, was sent to the MFC exciter through a high voltage amplifier. Structural vibration was measured by the collocated MFC sensors, and proper control inputs were determined based on the designed LQG control algorithm in the dSPACE control system. The weighting matrices in ^{7} and

The maximum vibration reduction was 8 dB at the fourth resonant frequency, and the minimum reduction was 1.5 dB at the second resonant frequency. The vibration reductions at the 1st and 2nd resonant frequencies were smaller than those of the 3rd and 4th resonant frequencies. This is because of the locations of MFC actuators. The 3rd and 4th modes of the cylindrical shell structure can develop large strains of MFC actuators, whereas the 1st and 2nd modes do not. This reveals that optimum placements of the MFC actuators are important to improve the performance of the active vibration control of the underwater cylindrical shell structure. Vibration control performances under 3rd mode excitation are presented in the time domain in

Structure-borne noise due to active vibration control was also measured by hydrophone. The cylindrical shell structure was located at the center of the water pool, and sixteen hydrophones were radially located, as shown in

The hydrophone was located with equal spacing (45 degree) about the circumferential direction, and at 1 m and 2 m distances from the cylindrical shell structure, respectively. Stars represent the positions of the MFC actuators, and an arrow represents the position of the MFC exciter in the figure. The radiating sounds measured at hydrophones 2, 6, 10 and 14 under 3rd and 4th mode excitation are presented in

When a hydrophone is close to the exciting MFC, the radiating sound is larger than that at other positions. For 3rd mode excitation, as shown in

Active vibration control of an underwater cylindrical shell structure was investigated in this paper. Finite element modeling of the underwater cylindrical shell structure was developed, and dynamic characteristics were obtained by using the commercial finite element package, ANSYS. The fluid-structure interaction was modeled as an added fluid mass. Modal reduction of the given underwater cylindrical shell structure was conducted to obtain the reduced state space equations, for the design of an active feedback control algorithm. A LQG controller was designed, based on the reduced system matrix. Three MFCs were used as actuators and sensors, and one MFC was used as an exciter. Vibration control of the proposed system was evaluated by lab scale experiments. Structural vibration of the underwater cylindrical shell structure was suppressed by the designed optimal control algorithm associated with the MFC actuators. Structure-borne noise was also reduced by the active vibration control. It is concluded that active vibration control of the underwater cylindrical shell structure is useful for the reduction of structural vibration, and of structure-borne noise as well.

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2011-0021720) and by research grant from the Underwater Vehicle Research Center of Agency for Defense Development and Defense Acquisition Program Administration, Korea. These financial supports are gratefully acknowledged.

Schematic diagram of the proposed underwater cylindrical shell structure.

Block diagram of LQG control algorithm.

Geometry of the end-capped cylindrical shell structure with surface bonded MFC actuators.

Finite element mesh configuration.

Photographs of the manufactured cylindrical shell structure. (

Experimental apparatus for the modal test of the underwater cylindrical shell structure.

Fundamental mode shapes of underwater cylindrical shell structure. (

Frequency response of the underwater cylindrical shell structure.

Experimental setup for active vibration control.

Frequency responses of the underwater cylindrical shell structure with and without active control.

Actuator input voltages and control response under 3rd mode excitation.

Positions of hydrophones.

Measured radiating sound under 3rd mode excitation. (

Measured radiating sound under 4th mode excitation (

Material properties of the aluminum, MFC and water.

Young's modulus ( |
68 [GPa] | Density ( |
2,698 [kg/m^{3}] |

Poisson ratio ( |
0.32 | ||

Young's modulus 1 direction (_{1}) |
30.34 [GPa] | Young's modulus 2 direction (_{2}) |
15.86 [GPa] |

Shear modulus (_{12}) |
5.52 [GPa] | Density ( |
7,750 [kg/m^{3}] |

Poisson ratio (_{12}) |
0.31 | Poisson ratio (_{21}) |
0.16 |

Piezoelectric Constant (_{11}) |
400 [pC/N] | Piezoelectric Constant (_{12}) |
−170 [pC/N] |

Permittivity (_{11} / _{0}) |
830 [C/m^{2}] |
Permittivity (_{22} / _{0}) |
916 [C/m^{2}] |

_{2}O) | |||

Density ( |
1,000 [kg/m^{3}] |
Speed of Sound | 1,500 [m/s^{2}] |

Comparison of natural frequencies of the cylindrical shell structure in the air, and underwater.

1st | 587.4 | 193.7 | 67 |

2nd | 617.5 | 215.9 | 65 |

3rd | 830.7 | 351.7 | 58 |

4th | 1,109.8 | 617.9 | 44 |

Natural frequencies of the underwater cylindrical shell obtained by FEA and experiment.

1st | 193.7 | 208 |

2nd | 215.9 | 251 |

3rd | 351.7 | 381 |

4th | 617.9 | 577 |