^{1}

^{2}

^{3}

^{4}

^{1}

^{1}

^{2}

^{3}

^{4}

The characteristics of vibrational power flow in an infinite laminated composite cylindrical shell filled with fluid excited by a circumferential line cosine harmonic force are investigated using wave propagation approach. The harmonic motions of the shell and the fluid filled in the shell are described by Love shell theory and acoustic wave equation, respectively. Under the driving force, the vibrational power flow input into the coupled system and the transmission of the power flow carried by different internal forces (moments) of the shell in the axial direction are established. Numerical computations are implemented to investigate the vibrational power flow input and its propagation. It is found that characteristics of the vibrational power flow vary with different circumferential mode orders and frequencies, and the presence of fluid in the shell significantly affects the vibration of the shell structure. Additionally, parametric investigations are carried out to study the effects of the fiber orientation, modulus ratio

Laminated composite cylindrical shell commonly applied in structural designs is an important element of submerged and floating structures due to its excellent mechanical characteristics of high specific strength-to-weight ratio, good fatigue resistance, and ease of fabrication [

Numerous researches have been conducted on the vibration characteristics of the composite circular cylindrical shell, and most are focused on vacant shell. For example, Lam et al. [

A few investigations have been made on the vibration characteristics of the circular cylindrical shell filled with fluids. Zhang et al. [

The wave propagation approach is very valuable in the vibrational analysis of thin cylindrical shell. Xu and Zhang [

From the above, we can draw the conclusion that much attention has been paid to the free vibration and forced vibration characteristics of the laminated cylindrical shells. Few of them have investigated the vibrational power characteristics of infinite laminated cylindrical shells. In this paper, the characteristics of vibrational power flow in a laminated composite cylindrical shell filled with fluid excited by a circumferential line cosine harmonic force are investigated with wave propagation approach. The harmonic motions of the shell and the fluid filled in the shell are described by Love shell theory and Helmholtz equation, respectively. Vibrational power flow input into the coupled system and its propagation along the shell axial direction are both studied. Additionally, investigations are carried out to study the effects of the fiber orientation, modulus ratio

The composite cylindrical shell can be defined through thickness

(a) Coordinate system and circumferential modal shape. (b) Harmonic line force F applied on thin laminated composite cylindrical shell.

According to Love theory, the motion equations of the cylindrical shell can be expressed through the force components

According to Hooke’s law, the stress components

According to Love’s approximation theory,

Associated with (

For the thin laminated composite cylindrical shell, the modulus components can be obtained from the engineering elasticity constants. The reduced transformed stiffness _{12} is the shear modulus.

The following spatial displacement field of the cylindrical shell can be expressed in the form of wave propagation:

The fluid in the cylindrical shell is supposed to be incompressible and inviscid which should satisfy the acoustic wave equation. The motion equation of the fluid can be expressed as _{n}() is the Bessel function of circumferential mode order

In the vibrational analysis, the radical displacement for the cylindrical shell and filled fluid should be the same at the interface between them to ensure contacting with each other. So the coupled boundary at the interface should meet the following:

The shell wall is excited by a harmonic line force _{0} represents the amplitude of harmonic line force. The cosine function

The solutions of the equations are

Through making the inverse Fourier transform of (

When the shell is excited by an external harmonic force

Here, the asterisk represents the complex conjugate, and

And the nondimensional input power flow can be expressed as

When the shell wall is subjected to the radial exciting force, the vibrational input power flow will be transmitted in the axial direction from the exciting location, along with the generated vibration waves propagating. The displacement components

The total vibrational power flow can be written as

Once the driving force inputs power flow into the shell-fluid coupled system, the four kinds of power flow will be transmitted in the axial direction from the exciting location. Due to symmetry of the transmission, only half of the input power will be considered in the positive direction of the shell. From the viewpoint of energy, four kinds of power flow can be characterized by the ratios of the power flow carried by different shell internal forces (moment) to the total power in the shell wall, namely,

In this paper, a simple numerical method [

To check the accuracy of the uncoupled vibration (the case of ^{2}, _{1}=0.25, and ^{3}. The comparisons are presented for the geometric ratios

Comparison of the frequency parameters for a three-layered, cross-ply [0°/90°/0°] cylindrical shell with SS boundary conditions (

| n | Lam[ | Zhang[ | Present |
---|---|---|---|---|

10 | 1 | 0.083908 | 0.083908 | 0.083908 |

2 | 0.030009 | 0.030008 | 0.030009 | |

3 | 0.015193 | 0.015191 | 0.015193 | |

4 | 0.012176 | 0.012174 | 0.012176 | |

5 | 0.015231 | 0.015230 | 0.015231 | |

6 | 0.021179 | 0.021178 | 0.021179 | |

| ||||

20 | 1 | 0.023590 | 0.023589 | 0.023590 |

2 | 0.007904 | 0.007903 | 0.007904 | |

3 | 0.005869 | 0.005868 | 0.005869 | |

4 | 0.009020 | 0.009019 | 0.009020 | |

5 | 0.014236 | 0.014235 | 0.014236 | |

6 | 0.020801 | 0.020800 | 0.020801 |

To check the accuracy of the coupled vibration (the case of ^{3}, Poisson’s ratio ^{3}. The modal vector of the coupled vibration is defined as mode of the axial wavenumber

Comparison of frequency for a C-C cylindrical shell (

Frequency (HZ) | |||||
---|---|---|---|---|---|

Order | Modal vector (m, n) | FEM/BEM(2000) | FEM/BEM(2800) | Reference [ | present |

1 | 1,2 | 4.91 | 4.89 | 4.93 | 4.92 |

2 | 1,3 | 9.13 | 9.00 | 8.94 | 8.91 |

3 | 2,3 | 10.8 | 10.64 | 10.64 | 10.61 |

4 | 2,2 | 11.19 | 11.12 | 11.48 | 11.18 |

5 | 3,3 | 14.79 | 14.55 | 14.66 | 14.32 |

6 | 1,4 | 18.99 | 18.55 | 18.26 | 18.37 |

7 | 2,4 | 19.46 | 19.00 | 18.73 | 19.22 |

8 | 3,4 | 20.7 | 20.21 | 19.96 | 20.18 |

Some numerical computations are conducted to investigate the vibration analysis of the coupled system, including the power flow input into the shell wall and its power flow transmission in the axial direction. Besides, the influence factors such as the fiber orientation, modulus ratio ^{2}, _{1}=0.25, ^{3}, and ^{3}. The magnitude of radial force is supposed to be

The nondimensional input flow

The comparison of nondimensional input power flow into the shell filled and unfilled with water. Black dashed line: in vacuo; red solid line: water-filled.

The power flow carried by different internal forces (moment)

Power transmitted by different shell forces (moments) for shell filled and unfilled with water: (a)

The power flow

Power transmitted by the shell filled and unfilled with water: (a)

In order to illuminate the influence of layer orientation of fiber on the vibration analysis, the fluid-filled cylindrical shell with cross-ply [90°/0°/90°] is chosen to compare with the original model. The input flow

The comparison of nondimensional input power flow into the shell with different cross-ply: black dashed line: [90°/0°/90°]; red solid line: [0°/90°/0°].

The power flow transmitted by internal forces (moments) of the shell with cross-ply [90°/0°/90°] against the axial distance

Power transmitted by different shell forces (moments) with cross-ply [90°/0°/90°]: (a)

The power flow

The comparison of power transmitted by the shell with different cross-ply: (a)

In order to investigate the influence of modulus ratio

The comparison of nondimensional input power flow into the shell with

The power flow transmitted by internal forces (moments) of the thicker shell against the axial distance

Power transmitted by different shell forces (moments) with

The power flow

The comparison of power transmitted by the shell with

In order to investigate the influence of shell thickness-to-radius on the vibration analysis of the coupled system, the shell thickness increases to _{s}, twice that in the original model. The input flow against the driving frequency for fluid-filled cylindrical thicker shell is plotted in Figure

The comparison of nondimensional input power flow into the shell with

The power flow transmitted by internal forces (moments) of the thicker shell against the axial distance

Power transmitted by different shell forces (moments) with

The power flow

The comparison of power transmitted by the shell with

The paper aims to investigate the characteristics of vibrational power flow in an infinite laminated composite cylindrical shell filled with fluid by using the wave propagation approach. The vibrational power flow input into the coupled system and its propagation are established under a circumferential line cosine harmonic force. Numerical computations are implemented to study the effects of the fiber orientation, modulus ratio

The authors declare that there are no conflicts of interest regarding the publication of this paper.

The present work is supported by the National Natural Science Foundation of China (Grants nos. 51609089, 51579110, and 51079059), the China Postdoctoral Science Foundation (Grant no. 2016M592338), the National High Technology Research and Development Program of China (863 Program, Grant no. 2012AA112601), the Offshore Flexible Pipe Project from Ministry of Industry and Information Technology, and the Fund Project Independent Innovation Research Fund of Huazhong University of Science and Technology (Grant no. 2015TS004).