^{1}

^{1}

^{2}

^{3}

^{4}

^{1}

^{2}

^{3}

^{4}

Vibration of structures due to external sound is one of the main causes of interior noise in cavities like automobile, aircraft, and rotorcraft, which disturb the comfort of passengers. Accurate modelling of such phenomena is required in eigenfrequency analysis and in designing an active noise control system to reduce the interior noise. In this paper, the effect of periodic noise travelling into a rectangular enclosure is investigated with finite element method (FEM) using COMSOL Multiphysics software. The periodic acoustic wave is generated by a point source outside the enclosure and propagated through the enclosure wall and excites an aluminium flexible panel clamped onto the enclosure. The behaviour of the transmission of sound into the cavity is investigated by computing the modal characteristics and the natural frequencies of the cavity. The simulation results are compared with previous analytical and experimental works for validation and an acceptable match between them were obtained.

Modelling of sound propagation in an enclosure is of considerable importance in the design and analysis of an active noise control system. Reduction of noise in aircrafts, automobiles and house appliances is important due to their annoying effects on human. Many of these applications can be modelled by a 3D cavity with a flexible boundary condition on one of its sides. An approach to this modelling is to consider an enclosure with rigid boundary conditions [

In modelling the effect of coupling between the flexible plate and the enclosure, both simply supported and clamped boundaries have been used, but several studies used only simply supported [

Computational techniques have been employed to solve the vibro-acoustic problems thanks to the rapid advancement of computing power. Finite element and boundary element methods are two examples of computational techniques, which can be used to study the characteristics of sound radiation from a box–type structure. The boundary element method provides a versatile means of solving acoustic radiation problem in arbitrary shaped regions, but in order to be used efficiently the elements must be smaller than a fraction of the acoustic wavelength. Therefore, in problems with three-dimensional geometry, modelling can be performed on a desktop computer just for frequencies up to a few tens or at most hundreds of Hz [

In this paper a reliable finite element model for the analysis of the vibro-acoustic behaviour of a rectangular enclosure is developed. It is assumed that the sound is transmitted through the flexible panel that is attached to the enclosure with clamped boundary conditions. The modal analysis of such an enclosure was simulated and the results are compared with the analytical and experimental results obtained in related study [

Producing a sound propagation pattern in an enclosure due to multiple reflections is quite involved. In fact, the response of an acoustic-structure system in the fully coupled case combining the acoustics and structures are quite different from the response of the uncoupled case [

Figure

Dimension of a rectangular enclosure.

The Helmholtz equation, which describes a harmonic wave equation propagating in medium while neglecting dissipation, is represented as

Substituting (

The natural frequencies of the acoustical system are obtained by assuming that the boundaries of the enclosure are hard, hence the pressure gradients on all boundaries are set to zero:

The solution of (

Figure

Here,

Equation (

In the case of a coupled system, the effect of the flexible plate on the sound field inside the enclosure as well as the effect of sound field on the flexible plate must be considered together. In coupling the flexible plate to the enclosure, acceleration of the plate was considered as a source of sound. The pressures at rigid boundaries are zero, but at the top of the enclosure where the flexible plate was placed, it is equal to the acceleration of the plate. Therefore, the homogenous Helmholtz equation (

As previously mentioned, the combined modal analysis of the coupled system represents the sound propagation of the system with different excitation frequencies. In this work, sound wave is the source of pressure on the plate. The sound waves were generated by a point source. The governing equation for a point source with the power

In the experimental studies reported in [

Experimental model setting and arrangement of piezoelectric, (a) 3D view of enclosure system with microphones, (b) 2D schematic of plate with piezoelectric (c) zone view of symmetric piezoelectric on the plate.

The enclosure was excited by airborne pressure generated by a loudspeaker placed at some distance above of enclosure, and three microphones inside the enclosure were to measure the pressure level inside the enclosure. The location of the microphones inside the enclosure is shown in Figure

In this section, sound travel into the medium and its interaction with a solid medium are modelled using a finite element software COMSOL Multiphysics. This modelling procedure requires two modules: one for simulating the acoustic medium and the other for flexible plate. After proper selection of the modules, the three dimensional model with the same dimensions described in Section

The “acoustic” module of COMSOL was used to model the enclosure, which utilised partial differential equations based on time harmonic and frequency domain analysis. The boundaries and properties of the medium were set to be hard sound boundary and air characteristics respectively. The dimensions were specified according to the work of [

The point of origin of the enclosure was placed at (

A few mode shapes of the rectangular enclosure (a) mode number (1,0,0) at 281.12 Hz, (b) mode number (0,0,1) at 338.98 Hz, (c) mode number (0,1,0) at 379.64 Hz, (d) mode number (1,0,1) at 446.27 Hz.

The “Structural Mechanics” module was used to perform the modal analysis of the plate. By adjusting the parameters of the equations, the mode shapes of the solid stress and strain of the plate is plotted in Figure

A few mode shapes of the rectangular plate (a) mode number (1,1) at 44.28 Hz, (b) mode number (1,2) at 75.66 Hz, (c) mode number (2,1) at 103.19 Hz, (d) mode number (2,2) at 127.05 Hz.

All edges are selected to be fixed with clamped boundary condition. The shapes of the mesh elements are second order triangular with 18730 mesh elements. Eigenfrequency solver and analysis are selected to solve the model to give the first 17 eigenmodes around 100 Hz. According to this solver, time for solving the model is 17.156 seconds and the number of degree of freedom is 16010. By replacing the parameters mentioned above for subdomains and edges, the model is solved and the mode shapes of the plate are shown in Figure

The complete coupled system has been simulated and is presented in this section. Two modules (Acoustic and Structural Mechanics) are used in this simulation. The pressure wave source was represented by a point source outside the cavity. A sphere with a reasonably large diameter outside the enclosure was used to envelope the air-filled acoustic domain. On the outer spherical perimeter of the air domain, radiation condition with the spherical wave was used. This boundary condition allows a spherical wave to travel out of the system, giving only minimal reflections for the non-spherical components of the wave. The radiation boundary condition is useful when the surroundings are only a continuation of the domain [

A few mode shapes of the coupled system (a) first mode at 59.8042 Hz, (b) second mode at 88.878 Hz, (c) third mode at 125.061Hz, (d) fourth mode at 149.398 Hz.

The natural frequencies of the system without any coupling between the flexible plate and enclosure were computed from the FEM model. These results are shown in Table

Comparison eigenfrequencies between finite element model with analytical results in [

Mode | Analytical | Finite element (COMSOL) | Error % | ||

Panel | Enclosure | Panel | Enclosure | ||

(1,1) | 41.6 | — | 44.28 | — | 6.0524 |

(1,2) | 73.7 | — | 75.66 | — | 2.5905 |

(2,1) | 95.0 | — | 103.19 | — | 7.9368 |

(2,2) | 124.6 | — | 127.05 | — | 1.9284 |

(1,0,0) | — | 281.3 | — | 281.12 | 0.0640 |

(0,0,1) | — | 337.6 | — | 338.98 | 0.4071 |

(0,1,0) | — | 375.1 | — | 379.64 | 1.1959 |

(1,0,1) | — | 439.5 | — | 446.27 | 1.5170 |

(1,1,0) | — | 468.9 | — | 479.60 | 2.2310 |

(1,0,0) | — | 504.7 | — | 511.83 | 1.3930 |

(1,1,1) | — | 577.8 | — | 578.96 | 0.2004 |

Error % | 2.3197 |

The next step is to calculate the resonant frequencies of the coupled system. The results were compared with both the analytical and experimental results reported in [

Comparison eigenfrequency analysis between simulations, analytical and experimental results.

mode | Finite element | Analytical | Experiment | FEM | FEM |

modelling (FEM) | error 1% | error 2% | |||

using COMSOL | versus | versus | |||

software | analytical | experimenta | |||

1 | 55.11 | 40.9 | 50.1 | 25.7848 | 9.0909 |

2 | 82.74 | 72.4 | 74 | 12.4970 | 10.5632 |

3 | 116.2 | 93.3 | 91.5 | 19.7074 | 21.2565 |

4 | 138.03 | 123.1 | 119.5 | 10.8165 | 13.4246 |

5 | 147.01 | 124.1 | 124.5 | 15.5840 | 15.3119 |

6 | 196.81 | 174.7 | 172 | 11.2342 | 12.6061 |

7 | 216.39 | 194.4 | 192 | 10.1622 | 11.2713 |

8 | 227.59 | 203.3 | 194.05 | 10.6727 | 14.7370 |

9 | 243.87 | 241.8 | 235 | 0.8488 | 3.6372 |

10 | 283.28 | 251.2 | 245 | 11.3245 | 13.5131 |

11 | 289.25 | 283.3 | 275.5 | 2.0570 | 4.7537 |

12 | 293.49 | 319.5 | 313 | 8.8623 | 6.6476 |

13 | 323.72 | 342.3 | 338 | 5.7395 | 4.4112 |

14 | 342.03 | 359 | 352 | 4.9616 | 2.9149 |

15 | 370.58 | 380.5 | 374 | 2.6769 | 0.9229 |

16 | 382.35 | 405.1 | 395.5 | 5.9500 | 3.4393 |

Average of errors % | — | 9.929963 | 9.281338 |

In order to improve the accuracy of the model, the mesh element size must be smaller than 0.002 of wavelength (L), where

This would satisfy the rule of ten to twelve degree of freedom per wavelength. However, this cannot be achieved with our current computing facility (CPU Intel Pentium Dual, 1.60 GHz, Ram, and 2.93 GB). Better computing facility shall be employed in future work.

Note that the analytical results of eigenfrequency analysis of plate and couple condition are taken from Al-bassiouni’s method [

Modelling and analysis of the sound-structure interaction of a rectangular enclosure has been addressed in this work. The enclosure has five rigid walls with one aluminium plate as a flexible wall. The effect of a point source noise placed at a distance outside the enclosure on the interior of the enclosure was investigated. The incident noise induced vibrations of the plate, and this in turn generated fluctuation of acoustic pressure inside the enclosure. The modal analysis of the plate, enclosure, and the coupling effects between them were performed and the behaviour of the sound field inside the enclosure was inferred from the shapes of modes at different frequencies for the coupled system. The eigenfrequency analysis showed that the results obtained from the finite element model was reliable with 90% accuracy when compared with the analytical and experimental results published in related. Therefore, the finite element model can be used further in the studies of active noise control.

Dimensions of the enclosure

Complex sound pressure amplitude

Wave number

Angular frequency

Speed of sound

Eigenvalue

Density of air

Outward-pointing unit normal vector

Modes number

Bending rigidity

Displacement of flexible plate in x and y direction

Poisson’s ratio

Young’s modulus

Mass density

Bessel function

Constant.