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The effect of structural vibration on the propagation of acoustic pressure waves through a cantilevered 3-D laminated beam-plate enclosure is investigated analytically. For this problem, a set of well-posed partial differential equations governing the vibroacoustic wave interaction phenomenon are formulated and matched for the various vibrating boundary surfaces. By employing integral transforms, a closed form analytical expression is computed suitable for vibroacoustic modeling, design analysis, and general aerospace defensive applications. The closed-form expression takes the form of a kernel of polynomials for acoustic pressure waves showing the influence of linear interface pressure variation across the axes of vibrating boundary surfaces. Simulated results demonstrate how the mode shapes and the associated natural frequencies can be easily computed. It is shown in this paper that acoustic pressure waves propagation are dynamically stable through laminated enclosures with progressive decrement in interfacial pressure distribution under the influence of high excitation frequencies irrespective of whether the induced flow is subsonic, sonic , supersonic, or hypersonic. Hence, in practice, dynamic stability of hypersonic aircrafts or jet airplanes can be further enhanced by replacing their noise transmission systems with laminated enclosures.

The control of vibration and noise propagation from industrial plants, aircraft engines and noise generating machines has remained an active research area for several decades. Nonetheless, limited literature exist in the area of noise-structure dynamic interaction modeling. However, within the context of analytical and experimental studies in acoustic -structure dynamics, a number of investigations have been reported in [

In particular, Fang et al. [

For some cases, with simplified assumptions, closed-form solutions are possible usually with rigorous mathematical intrigues and manipulations. One of such methods involves the computation of a transfer function from the frequency response data. Even at that, the limit of these identification techniques is that, the system identification algorithms work satisfactorily only for low-order systems and systems with separated nodes.

However, for 3-D acoustic enclosures, with vibrating boundary surfaces, these identification techniques are fronted with difficulty, especially with two-input one-output systems. Nevertheless, investigations of vibroacoustic pressure waves propagation through a 3-D acoustic enclosure having vibrating laminated boundary surfaces has not been widely reported in literature. Within the context of transverse vibrations of laminated structures, with non-uniform interfacial pressure distribution, some interesting results have been reported recently in [

In Section

The problem here is to examine analytically, the effect of the nature of load, frequency variation and the pressure gradient on the acoustic pressure waves propagation.

A general theory of the energy dissipation properties of press-fit joints in the presence of Coulomb friction as originally developed by Goodman and Klumpp provides the basis for the physics of the problem. As illustrated in Figure

there is continuity of stress distributions at the interface to sufficiently hold the equivalent layers together both in the pre- and post-slip conditions,

a stick elastic slip with presence of Coulomb friction occurs at the interface of the sandwich vibrating boundary to dissipate energy and does not remain constant as a function of some other variables such as spatial distance, time or velocity.

Problem geometry of 3-D composite structure.

2-D view of the composite structure geometry through _{3}

2-D view of the composite structure geometry through _{4}

The formulated vibroacoustic wave equation is given by the relation_{1,}_{2}, _{3}, _{4} are derived in the appendix. In the meantime, the 2-D views of the structural loadings through _{3}, _{4} are illustrated in Figures (

In this investigation we will simplify the solution of (_{1 } while the boundary effects on the pressure waves propagation are restricted to the following domains (_{2}, _{3}, _{4}).

Under this circumstance, (_{2}_{3}_{4} from the following relations, viz,

By invoking the Fourier inversion, the solution of (

The above equation can be simplified via the following relations, viz,

In acoustics, sound intensity

In this paper, we shall limit analysis to intensity variation along the spatial variable

In view of the above relations, the magnitude of the sound intensity can be evaluated from (

In this paper, vibration and noise propagation control emanating from complex engineering systems such as industrial power plants, aircraft engines, space propulsive devices and machine enclosures is investigated. The acoustic-structure configuration of interest is the one in which an acoustic disturbance is prompted by one of the vibrating boundaries of the enclosure such as in aircraft cabin noise transmission and systems for outer space exploration. The modeling techniques employed for this study derives from recent advances made in the mechanics of sandwich structures, with non-uniform interfacial pressure distribution. The acoustic-structure dynamic interaction problem is simplified by assuming zero initial conditions prior to the excitation of the upper boundary surface. In the formulated problem, the upper layer of the sandwich elastic plates is subjected to harmonic excitation force as illustrated in Figure

Acoustic natural frequency profile for the case

Acoustic natural frequency profile for the case

Acoustic natural frequency profile for the case

Acoustic natural frequency profile for the case

Acoustic natural frequency profile for the case

In each zone, the general pattern of the frequency profiles showed that the results are dependent on the geometry of the 3-D enclosure and ambient sound velocity. With respect to the effect of modal parameters, Figure

Parameters for simulation of results.

Definition | Symbol | Value |
---|---|---|

Applied force amplitude | 1 MN | |

Poisson ratio | 0.33 | |

Coefficient of friction | 0.14 | |

Interfacial pressure | P | |

3-D: - Length | 0.4 m–1.2 m | |

-Width | 0.3 m | |

-Thickness | 0.8 m | |

Thickness of laminated boundary surface on | 0.005 m | |

Excitation frequency | 1 HZ | |

Modulus of rigidity of materials | ^{-2} | |

Velocity of Sound through the 3-D enclosure | 343 ms^{-1} | |

Density of material | 2810 kg m^{-3} | |

Density of air | 1.13 kg m^{-3} |

In the subsonic zone, the natural frequency profiles are inversely proportional to the axial length of the enclosure and decrease monotonically to a constant value irrespective of the axial length as we approach sonic flow. However, in the supersonic zone, the natural frequencies are further attenuated in reversed order to their respective critical values before increasing monotonically in the hypersonic zone.

With respect to the effect of higher modal parameters on the natural frequency profiles, as demonstrated in Figures

With respect to the profiles of vibroacoustic pressure, Figures

Vibroacoustic pressure profile for the case

Vibroacoustic pressure profile for the case

Vibroacoustic pressure profile for the case

Vibroacoustic pressure profile for the case

Vibroacoustic pressure profile for the case

Vibroacoustic pressure profile for the case

Vibroacoustic pressure profile for the case

Vibroacoustic pressure profile for the case

Vibroacoustic pressure profile for the case

Vibroacoustic pressure profile for the case

The pictures in Figures

With respect to the effect of low excitation frequency through the boundary surface, the interfacial pressure gradients do not seem to have any strong effect on the acoustic pressure profiles. In fact we note in the lower range of Mach number, that the acoustic pressure waves rise initially before dropping at Mach 2 and increased marginally. However, as we move beyond Mach 16, the effects of the pressure gradients are noticeable.

On the other hand, Figures

As can be observed, the pressure profiles are increasing with time. In particular, we note that pressure profile magnitude with lower boundary excitation frequency are significantly higher compare with the effect of higher boundary excitation frequency. On the other hand, Figure

In this paper, explicit closed form solutions for the vibroacoustic characteristic frequencies, pressure waves and sound intensity or transmission quality through a 3-D enclosure with a vibrating laminated boundary surface is investigated. The vibroacoustic properties are shown to be dependent on, interfacial pressure gradients and boundary surfaces excitation frequencies. The results presented in this study can be positively exploited for the design of modern airplanes, aero-elastic structures and propulsive devices for the launching of space systems.

For Case _{1},

For Case _{2},

For Case _{3} (mirror reflection of _{1}),

For Case _{4} (mirror reflection of _{2}),

For Case _{1},

For Case _{2},

For Case _{3},

For Case _{4},

For case _{1},

For case _{2,}

For case _{3},

For case _{4},

Length of sandwich laminates

Width of sandwich laminates

Height of sandwich laminates

Ambient speed of sound

Speed of sound through the 3-D enclosure

Differential operator

Modulus of rigidity of the laminate

Applied end force amplitude

Space coordinate along the length of the laminate

Space coordinate along the width of the laminate

Space coordinate along the height of the laminate

Sound Intensity

Dynamic response

Dynamic response in Laplace transform plane

Dynamic response in Fourier transform plane

Dynamic response in Fourier-Laplace transform plane

Coefficient of friction at the interface of sandwich composite elastic beam

Ambient air density

Time coordinate

Clamping pressure at the interface

Shear stress at the upper half of the laminates interface

Shear stress at the lower half of the laminates interface