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Modeling and Eigenfrequency Analysis of Sound-Structure Interaction in a Rectangular Enclosure with Finite Element Method

Modeling and Eigenfrequency Analysis of Sound-Structure Interaction in a Rectangular Enclosure... Hindawi Publishing Corporation Advances in Acoustics and Vibration Volume 2009, Article ID 371297, 9 pages doi:10.1155/2009/371297 Research Article Modeling and Eigenfrequency Analysis of Sound-Structure Interaction in a Rectangular Enclosure with Finite Element Method 1 1 2 3 Samira Mohamady, Raja Kamil Raja Ahmad, Allahyar Montazeri, Rizal Zahari, and Nawal Aswan Abdul Jalil Departement of Electrical and Electronic Engineering, Universiti Putra Malaysia, 43400 UPM SERDANG, Selangor, Malaysia Faculty of Electrical Engineering, Iran University of Science and Technology, Narmak 16846-13114, Tehran, Iran Department of Aerospace Engineering, Universiti Putra Malaysia, 43400 UPM SERDANG, Selangor, Malaysia Departement of Mechanical and Manufacturing Engineering, Universiti Putra Malaysia, 43400 UPM SERDANG, Selangor, Malaysia Correspondence should be addressed to Samira Mohamady, arimasim@ieee.org Received 26 July 2009; Accepted 20 November 2009 Recommended by Massimo Viscardi 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. Copyright © 2009 Samira Mohamady et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction field inside an enclosure has been developed. Asymptotic modal analysis technique has been proposed [8]toanalyze Modelling of sound propagation in an enclosure is of such problems and has been shown to have advantages over considerable importance in the design and analysis of an traditional methods used for solving dynamic problems with active noise control system. Reduction of noise in aircrafts, a large number of modes. Furthermore, a mechanics-based automobiles and house appliances is important due to their analytical model has also been developed [9] to address the annoying effects on human. Many of these applications interactions between a panel and the sound field inside a can be modelled by a 3D cavity with a flexible boundary rectangular enclosure. In this work, a rectangular enclosure with a flexible panel with piezoelectric actuators attached to condition on one of its sides. An approach to this modelling is to consider an enclosure with rigid boundary conditions it w modelled. The studies in [10] for an irregular enclosure [1, 2] and extends to the case where one of its boundaries is with two flexible panels are extended here. Finite element considered as flexible [3, 4]. Excitation of the flexible plate by models have been constructed to study similar problems sound source will cause vibrations on the plate and induces [11, 12]. In several studies, the geometry of the enclosure noise inside the cavity. Due to the coupling between struc- was considered to be non rectangular, but the same modal tural vibrations and acoustical field, these systems are termed analysis strategy was used to study the behaviour of the vibro-acoustic systems. Several analytical and experimental sound travelling within it [13]. studies have been conducted to study the behaviour of these In modelling the effect of coupling between the flexible vibro-acoustic systems. In the earliest studies in [5–7]a plate and the enclosure, both simply supported and clamped comprehensive modal based theoretical framework of sound boundaries have been used, but several studies used only 2 Advances in Acoustics and Vibration z z Flexible plate L L zc Thickness of zc plate = h L L xc xc x x L L yc yc y y (a) (b) Figure 1: Dimension of a rectangular enclosure. simply supported [4, 14, 15] because the analytical derivation of an acoustic-structure system in the fully coupled case of the model for the coupled system is less complex. There- combining the acoustics and structures are quite different fore, finite element technique is more useful in modelling from the response of the uncoupled case [3, 4]. Therefore, clamped boundaries. in modelling the sound in closed spaces, modal analysis of Computational techniques have been employed to solve the enclosure must be performed. In this section, the natural the vibro-acoustic problems thanks to the rapid advance- frequencies of the plate and the enclosure in the case of an ment of computing power. Finite element and boundary uncouple condition are examined initially followed by the element methods are two examples of computational tech- case of coupled or clamped condition. niques, which can be used to study the characteristics of sound radiation from a box–type structure. The boundary 2.1. Enclosure with Rigid Walls. Figure 1(a) shows an enclo- element method provides a versatile means of solving sure with one of its edges located at the origin 0 of a acoustic radiation problem in arbitrary shaped regions, but Cartesian coordinate system. Here L , L and L are the xc yc zc in order to be used efficiently the elements must be smaller length, width and height respectively. than a fraction of the acoustic wavelength. Therefore, in The Helmholtz equation, which describes a harmonic problems with three-dimensional geometry, modelling can wave equation propagating in medium while neglecting be performed on a desktop computer just for frequencies up dissipation, is represented as to a few tens or at most hundreds of Hz [16]. In this paper a reliable finite element model for the 2 2 ∇ + k p = 0, (1) analysis of the vibro-acoustic behaviour of a rectangular enclosure is developed. It is assumed that the sound is where p is complex sound pressure amplitude and k is the transmitted through the flexible panel that is attached to the wave number which is related to angular frequency ω and enclosure with clamped boundary conditions. The modal speed of sound c by analysis of such an enclosure was simulated and the results are compared with the analytical and experimental results ω k = . (2) obtainedinrelated study[3]. An error analysis of the obtained resonant frequencies was used to validate the devel- oped model. The finite element modelling was performed Substituting (2) into (1) while introducing air density ρ [17] gives a homogeneous Helmholtz equation: using COMSOL Multiphysics software that provides exclu- sive structural and acoustical modules as well as the ability of connecting them together to develop a structural-acoustics 1 ω p ∇· − ∇p − = 0. (3) system necessary for further studies in this area. The rest of ρ ρ c 0 0 this paper is organized as follows; in Section 2 the theories behind the developed model and fundamental physics of the In addition, the eigenvalue λ is related to the eigenfrequency system are presented. This is followed by the derivation of by [17] the governing equations of the plate, enclosure and coupled system. In Section 3, the modal properties of the plate, λ = i2πf = iω. (4) enclosure and coupled vibro-acoustic system are simulated using COMSOL software. Finally comparisons between finite Substituting (4) into (3) and extending to three dimensions element model with the analytical and experimental results the enclosure can be written as obtain in [3]are presented. 2 2 2 2 ∂ p ∂ p ∂ p λ p + + + = 0. (5) 2 2 2 2 ∂x ∂y ∂z ρ c 2. Modelling of Sound in an Enclosure s Producing a sound propagation pattern in an enclosure due The natural frequencies of the acoustical system are to multiple reflections is quite involved. In fact, the response obtained by assuming that the boundaries of the enclosure Advances in Acoustics and Vibration 3 32 1 PZT PZT acuator Mic. 2 C Plate PVDF sensor Mic. 3 yp Mic. 1 xc PZT (a) (b) (c) Figure 2: 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. are hard, hence the pressure gradients on all boundaries are Here, A and B are used to satisfy the orthogonality n n set to zero: conditions and could be determined by using orthogonal characteristics of these vibration modes [18]. In (11), J p | , p | , p | , x x=0 x x=L y y=0 and I are the Bessel function [18] and the wave number is (6) 4 2 computed from k = ω ρ/D. The expressions (cos) and (sin) p | , p | , p | . y Z z y=L z=L y z=0 z at the right hand side of (11) mean replacing either one of The solution of (3) with boundary conditions introduced them in this equation appropriately. in (6)isgiven as [2] Equation (11) is the general solution for the vibration modes of the solid rectangular plate. Here, J (kx)J (ky) n−m m n πy n πx y n πz x z and I (kx)I (ky) will be used to construct the free n−m m p = A cos cos cos , nxnynz L L L x y z vibration solution of a rectangular thin plate with different (7) edge conditions [18]. In this work, fully clamped boundary with n , n , n = 0, 1, 2,... , x y z conditions is assumed and the edge lengths are defined as a and b. Therefore, the boundary conditions were defined as where n , n and n are the modes number. Using the x y z derivation in [2], the eigenfrequencies of the enclosure can W| = 0, W| = 0, x=0 x=a be further written as ⎡ ⎤ 1/2 W| = 0, W| = 0, y=0 y=a 2 2 c n y n s x z ⎣ ⎦ f = + + . (8) n n n x y z ∂W ∂W 2 L L L (12) x y z | = 0, | = 0, x=0 x=a ∂x ∂x ∂W ∂W 2.2. Flexible Plate. Figure 1(b) shows the position of the | = 0, | = 0. y=0 y=a flexible plate with a constant thickness h. The free harmonic ∂y ∂y vibration partial differential equation of plate, [18]can be Solving (9) with the clamped boundary conditions yield the written as vibration modes [18]: 4 2 D∇ W − ω ρh = 0, (9) W x, y (n,m) where W(x,y), ∇ and D are the displacement of flexible plate in x and y direction, biharmonic differential operator = A {J (kx) + J [k(a − x)]} J ky + J k b − y n n−m n−m m m and bending rigidity respectively. The bending rigidity [18] +B {I (kx) + I [k(a − x)]}) n n−m n−m is given by mπ nπ Eh × I ky + I k b − y cos cos , m m D = , (10) 2 2 12(1 − v ) (13) where E, v, ω,and ρ are the young’s modulus, Poisson’s where m and n are even numbers. ratio, natural frequency and mass density respectively. The nth vibration mode of the rectangular plate in (9)can be 2.3. Model of the Coupled System. In the case of a coupled written in a compact form as [18] system, the effect of the flexible plate on the sound field W x, y inside the enclosure as well as the effect of sound field on the flexible plate must be considered together. In coupling +∞ mπ cos the flexible plate to the enclosure, acceleration of the plate = A J (kx)J ky + B I (kx)I ky . n n−m m n n−m m sin was considered as a source of sound. The pressures at rigid m=−∞ boundaries are zero, but at the top of the enclosure where (11) yc zc xp h h pzt p 4 Advances in Acoustics and Vibration 0.2Max:2.215 0.2Max:2.215 2.215 2.215 0 0 2.154 2.154 −0.2 −0.2 2.093 2.093 0.2 0.2 2.033 2.033 1.972 1.972 0 0 1.912 1.912 1.851 1.851 −0.2 −0.2 1.791 1.791 0.2 0.2 1.73 1.73 0 0 −0.2 −0.2 1.67 1.67 Min: 1.67 Min: 1.67 (a) (b) 0.2 Max: 2.215 0.2 Max: 2.215 2.215 2.215 0 0 2.154 2.154 −0.2 −0.2 2.093 2.093 0.2 0.2 2.033 2.033 1.972 1.972 0 0 1.912 1.912 1.851 1.851 −0.2 −0.2 1.791 1.791 0.2 0.2 1.73 1.73 0 0 −0.2 −0.2 1.67 1.67 Min: 1.67 Min: 1.67 (c) (d) Figure 3: 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 flexible plate was placed, it is equal to the acceleration the point R = R in an infinite homogeneous space is of the plate. Therefore, the homogenous Helmholtz equation 1 π Pc (3)becomes [19] s (3) ∇· − ∇p = 2 δ (R − R ), (15) ρ ρ 0 0 ∇p · n = a , (14) a n 0 (3) where δ (R) is the Dirac delta function in three dimensional space [19]. where, n is the outward-pointing unit normal vector seen from inside the acoustics domain, and a is the normal acceleration of the plate. Acoustic pressure was also coupled 3. Experimental Work in Related Study to the flexible plate as a boundary load pressure in the direction of the normal vectors. In the experimental studies reported in [3],arectangular As previously mentioned, the combined modal analysis cube with five rigid acrylic sheets was constructed and an of the coupled system represents the sound propagation aluminium panel was clamped on of it. The Thickness of of the system with different excitation frequencies. In this acrylic sheets and aluminium sheets used were 25.4 mm work, sound wave is the source of pressure on the plate. The and 1.588 mm, respectively. The rectangular cube with the sound waves were generated by a point source. The governing dimension of L = 0.6096 [m], L = 0.4572 [m], L = xc yc zc equation for a point source with the power P, and located at 0.508 [m] are shown in Figure 2(a), and the plate has the Advances in Acoustics and Vibration 5 Max: 0.402 Max: 0.402 0.4 0.4 0.35 0.35 0.3 0.3 0.25 0.25 0.2 0.2 0.15 0.15 0.1 0.1 0.05 0.05 Min: −7.282e − 6 Min: −7.282e − 6 (a) (b) Max: 0.402 Max: 0.402 0.4 0.4 0.35 0.35 0.3 0.3 0.25 0.25 0.2 0.2 0.15 0.15 0.1 0.1 0.05 0.05 0 0 Min: −7.282e − 6 Min: −7.282e − 6 (c) (d) Figure 4: 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. following dimension: L = 660.4, L = 508.0which is Each module provides a wide range of equations, which xp yp illustrated in Figure 2(b). was needed in specifying subdomains, boundaries, edges and The enclosure was excited by airborne pressure generated points. The theories and equations behind this model are by a loudspeaker placed at some distance above of enclosure, based on the governing equations in Section 2. and three microphones inside the enclosure were to measure the pressure level inside the enclosure. The location of the 4.1. Modelling the Enclosure. The “acoustic” module of microphones inside the enclosure is shown in Figure 2(b). COMSOL was used to model the enclosure, which utilised These microphones are used to sense all modes of enclosure partial differential equations based on time harmonic and using personal computer and dSPACE interface. Nine piezo- frequency domain analysis. The boundaries and properties electric (PZT-5H) elements were patched symmetrically onto of the medium were set to be hard sound boundary and air the plate depicted in Figure 2(c).These actuators are used to characteristics respectively. The dimensions were specified measure the plate response. according to the work of [3], which is repeated here: L = 0.6096 [m], L = 0.4572 [m], xc yc 4. Modelling with COMSOL Multiphysics (16) [ ] L = 0.508 m . zc In this section, sound travel into the medium and its interaction with a solid medium are modelled using a finite The point of origin of the enclosure was placed at element software COMSOL Multiphysics. This modelling (−0.3048, −0.2286, −0.254). Using eigenfrequency analysis procedure requires two modules: one for simulating the of the model, the first 4 eigenfrequencies with modal shapes acoustic medium and the other for flexible plate. After proper are shown in Figure 3. Due to the hard boundaries used selection of the modules, the three dimensional model with in this analysis, the material of the body of enclosure can the same dimensions described in Section 3 was sketched. be neglected. The shapes of mesh elements were selected 6 Advances in Acoustics and Vibration 0.2Max:2.215 0.2Max:2.215 Max: 3.862 Max: 3.862 2.215 2.215 0 0 2.154 2.154 −0.2 3.5 −0.2 3.5 2.093 2.093 3 3 0.2 0.2 2.033 2.033 2.5 2.5 1.972 1.972 2 2 0 0 1.912 1.912 1.5 1.5 1.851 1.851 −0.2 −0.2 1 1 1.791 1.791 0.2 0.2 0.5 0.5 1.73 1.73 0 0 −0.2 −0.2 1.67 1.67 0 0 Min: 1.67 Min: −2.272 Min: 1.67 Min: −2.272 (a) (b) 0.2Max:2.215 0.2Max:2.215 Max: 3.862 Max: 3.862 2.215 2.215 0 0 2.154 2.154 3.5 3.5 −0.2 −0.2 2.093 2.093 3 3 0.2 0.2 2.033 2.033 2.5 2.5 1.972 1.972 2 2 0 0 1.912 1.912 1.5 1.5 1.851 1.851 −0.2 −0.2 1 1 1.791 1.791 0.2 0.2 0.5 0.5 1.73 1.73 0 0 −0.2 −0.2 1.67 0 1.67 0 Min: 1.67 Min: −2.272 Min: 1.67 Min: −2.272 (c) (d) Figure 5: 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. Table 1: Comparison eigenfrequencies between finite element to be tetrahedralinnormalsize. Theenclosure model model with analytical results in [3] for uncoupled system. consists of 10727 mesh elements. Using the eigenfrequency analysis of the model yield the modal shapes with the Analytical Finite element (COMSOL) first four eingenfrequencies depicted in Figure 3.Solution Mode Error % Panel Enclosure Panel Enclosure time and degree of freedom are 24.281 seconds and 56725, (1,1) 41.6 — 44.28 — 6.0524 respectively. (1,2) 73.7 — 75.66 — 2.5905 (2,1) 95.0 — 103.19 — 7.9368 4.2. Modelling of the Plate. The “Structural Mechanics” (2,2) 124.6 — 127.05 — 1.9284 module was used to perform the modal analysis of the plate. (1,0,0) — 281.3 — 281.12 0.0640 By adjusting the parameters of the equations, the mode (0,0,1) — 337.6 — 338.98 0.4071 shapes of the solid stress and strain of the plate is plotted (0,1,0) — 375.1 — 379.64 1.1959 in Figure 4. The dimension of the plate is larger than the (1,0,1) — 439.5 — 446.27 1.5170 enclosure, and this is the same as the experimental work of (1,1,0) — 468.9 — 479.60 2.2310 [3]. For this dimension, the coupled mode was constructed (1,0,0) — 504.7 — 511.83 1.3930 as discussed in Section 2.3 . All edges are selected to be fixed with clamped boundary (1,1,1) — 577.8 — 578.96 0.2004 condition. The shapes of the mesh elements are second Error % 2.3197 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 parameters mentioned above for subdomains and edges, the solver, time for solving the model is 17.156 seconds and model is solved and the mode shapes of the plate are shown the number of degree of freedom is 16010. By replacing the in Figure 4. Advances in Acoustics and Vibration 7 Table 2: Comparison eigenfrequency analysis between simulations, analytical and experimental results. Finite element FEM FEM modelling (FEM) error 1% error 2% mode Analytical Experiment 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 4.3. Simulation of the Coupled System. Thecompletecoupled and enclosure were computed from the FEM model. These system has been simulated and is presented in this section. results are shown in Table 1 and they are compared with Two modules (Acoustic and Structural Mechanics) are used those extracted by the analytical model in [3]. The first 11 in this simulation. The pressure wave source was represented modes of the system consists of the first four modes of the by a point source outside the cavity. A sphere with a plate and the first seven enclosure modes. The mean errors reasonably large diameter outside the enclosure was used of 2.3% indicate that the results are in good agreement with to envelope the air-filled acoustic domain. On the outer the analytical model. It is possible to improve the accuracy of spherical perimeter of the air domain, radiation condition the FEM model by decreasing the element size of the mesh with the spherical wave was used. This boundary condition applied in finite element modelling. On the other hand, the allows a spherical wave to travel out of the system, giving only memory required for this purpose will put a bound on the minimal reflections for the non-spherical components of the obtainable accuracy of the model. These limitations are more wave. The radiation boundary condition is useful when the obvious when comparing the results of couple condition, surroundings are only a continuation of the domain [19]. which requires more element sizes calculations described in Two sub modules from the acoustic tab was used to support the next section. thickness and inner dimension of cavity, namely the “Solid stress strain” for thickness and “Pressure Acoustic” for the 5.2. Model of the Coupled System. The next step is to calculate air inside it. The coupling of flexible plate vibration with the resonant frequencies of the coupled system. The results the acoustic pressure inside the enclosure and vice versa was were compared with both the analytical and experimental performed using both “Acoustic” and “Structure Mechanic” results reported in [3]. Table 2 shows the comparison of the modules. For this purpose, some variables were set to coupled system’s resonant frequencies between the analytical make the connection between these two modules. COMSOL and experimental model and those obtained using the FEM software in such an analysis will enable the combination of model. In experimental work rather than rectangular cube different acoustical and vibration phenomena in one model. and flexible plate, piezoelectric elements are patched to the The variation of some of the mode shapes are depicted in plate to actuate and control the plate for future applications. Figure 5. The mean of the absolute relative error percentage of 9.929963 is achieved due to comparison with analytical work 5. Results and Discussion and 9.281338 percent due to experimental work. The results 5.1. Model of the Uncoupled System. The natural frequencies indicate that finite element method with COMSOL software of the system without any coupling between the flexible plate is able to predict the natural frequency of the whole modes, 8 Advances in Acoustics and Vibration such that active noise control system can be designed using References this model. Results that obtained by smaller mesh element [1] A. Montazeri, J. Poshtan, and M. H. Kahaei, “Modal analysis sizes gave better results. for global control of broadband noise in a rectangular In order to improve the accuracy of the model, the mesh enclosure,” Journal of Low Frequency Noise Vibration and element size must be smaller than 0.002 of wavelength (L), Active Control, vol. 26, no. 2, pp. 91–104, 2007. where [2] A. Montazeri, J. Poshtan, and M. H. 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Modeling and Eigenfrequency Analysis of Sound-Structure Interaction in a Rectangular Enclosure with Finite Element Method

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Hindawi Publishing Corporation
Copyright
Copyright © 2009 Samira Mohamady et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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1687-6261
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1687-627X
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10.1155/2009/371297
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Hindawi Publishing Corporation Advances in Acoustics and Vibration Volume 2009, Article ID 371297, 9 pages doi:10.1155/2009/371297 Research Article Modeling and Eigenfrequency Analysis of Sound-Structure Interaction in a Rectangular Enclosure with Finite Element Method 1 1 2 3 Samira Mohamady, Raja Kamil Raja Ahmad, Allahyar Montazeri, Rizal Zahari, and Nawal Aswan Abdul Jalil Departement of Electrical and Electronic Engineering, Universiti Putra Malaysia, 43400 UPM SERDANG, Selangor, Malaysia Faculty of Electrical Engineering, Iran University of Science and Technology, Narmak 16846-13114, Tehran, Iran Department of Aerospace Engineering, Universiti Putra Malaysia, 43400 UPM SERDANG, Selangor, Malaysia Departement of Mechanical and Manufacturing Engineering, Universiti Putra Malaysia, 43400 UPM SERDANG, Selangor, Malaysia Correspondence should be addressed to Samira Mohamady, arimasim@ieee.org Received 26 July 2009; Accepted 20 November 2009 Recommended by Massimo Viscardi 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. Copyright © 2009 Samira Mohamady et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction field inside an enclosure has been developed. Asymptotic modal analysis technique has been proposed [8]toanalyze Modelling of sound propagation in an enclosure is of such problems and has been shown to have advantages over considerable importance in the design and analysis of an traditional methods used for solving dynamic problems with active noise control system. Reduction of noise in aircrafts, a large number of modes. Furthermore, a mechanics-based automobiles and house appliances is important due to their analytical model has also been developed [9] to address the annoying effects on human. Many of these applications interactions between a panel and the sound field inside a can be modelled by a 3D cavity with a flexible boundary rectangular enclosure. In this work, a rectangular enclosure with a flexible panel with piezoelectric actuators attached to condition on one of its sides. An approach to this modelling is to consider an enclosure with rigid boundary conditions it w modelled. The studies in [10] for an irregular enclosure [1, 2] and extends to the case where one of its boundaries is with two flexible panels are extended here. Finite element considered as flexible [3, 4]. Excitation of the flexible plate by models have been constructed to study similar problems sound source will cause vibrations on the plate and induces [11, 12]. In several studies, the geometry of the enclosure noise inside the cavity. Due to the coupling between struc- was considered to be non rectangular, but the same modal tural vibrations and acoustical field, these systems are termed analysis strategy was used to study the behaviour of the vibro-acoustic systems. Several analytical and experimental sound travelling within it [13]. studies have been conducted to study the behaviour of these In modelling the effect of coupling between the flexible vibro-acoustic systems. In the earliest studies in [5–7]a plate and the enclosure, both simply supported and clamped comprehensive modal based theoretical framework of sound boundaries have been used, but several studies used only 2 Advances in Acoustics and Vibration z z Flexible plate L L zc Thickness of zc plate = h L L xc xc x x L L yc yc y y (a) (b) Figure 1: Dimension of a rectangular enclosure. simply supported [4, 14, 15] because the analytical derivation of an acoustic-structure system in the fully coupled case of the model for the coupled system is less complex. There- combining the acoustics and structures are quite different fore, finite element technique is more useful in modelling from the response of the uncoupled case [3, 4]. Therefore, clamped boundaries. in modelling the sound in closed spaces, modal analysis of Computational techniques have been employed to solve the enclosure must be performed. In this section, the natural the vibro-acoustic problems thanks to the rapid advance- frequencies of the plate and the enclosure in the case of an ment of computing power. Finite element and boundary uncouple condition are examined initially followed by the element methods are two examples of computational tech- case of coupled or clamped condition. niques, which can be used to study the characteristics of sound radiation from a box–type structure. The boundary 2.1. Enclosure with Rigid Walls. Figure 1(a) shows an enclo- element method provides a versatile means of solving sure with one of its edges located at the origin 0 of a acoustic radiation problem in arbitrary shaped regions, but Cartesian coordinate system. Here L , L and L are the xc yc zc in order to be used efficiently the elements must be smaller length, width and height respectively. than a fraction of the acoustic wavelength. Therefore, in The Helmholtz equation, which describes a harmonic problems with three-dimensional geometry, modelling can wave equation propagating in medium while neglecting be performed on a desktop computer just for frequencies up dissipation, is represented as to a few tens or at most hundreds of Hz [16]. In this paper a reliable finite element model for the 2 2 ∇ + k p = 0, (1) analysis of the vibro-acoustic behaviour of a rectangular enclosure is developed. It is assumed that the sound is where p is complex sound pressure amplitude and k is the transmitted through the flexible panel that is attached to the wave number which is related to angular frequency ω and enclosure with clamped boundary conditions. The modal speed of sound c by analysis of such an enclosure was simulated and the results are compared with the analytical and experimental results ω k = . (2) obtainedinrelated study[3]. An error analysis of the obtained resonant frequencies was used to validate the devel- oped model. The finite element modelling was performed Substituting (2) into (1) while introducing air density ρ [17] gives a homogeneous Helmholtz equation: using COMSOL Multiphysics software that provides exclu- sive structural and acoustical modules as well as the ability of connecting them together to develop a structural-acoustics 1 ω p ∇· − ∇p − = 0. (3) system necessary for further studies in this area. The rest of ρ ρ c 0 0 this paper is organized as follows; in Section 2 the theories behind the developed model and fundamental physics of the In addition, the eigenvalue λ is related to the eigenfrequency system are presented. This is followed by the derivation of by [17] the governing equations of the plate, enclosure and coupled system. In Section 3, the modal properties of the plate, λ = i2πf = iω. (4) enclosure and coupled vibro-acoustic system are simulated using COMSOL software. Finally comparisons between finite Substituting (4) into (3) and extending to three dimensions element model with the analytical and experimental results the enclosure can be written as obtain in [3]are presented. 2 2 2 2 ∂ p ∂ p ∂ p λ p + + + = 0. (5) 2 2 2 2 ∂x ∂y ∂z ρ c 2. Modelling of Sound in an Enclosure s Producing a sound propagation pattern in an enclosure due The natural frequencies of the acoustical system are to multiple reflections is quite involved. In fact, the response obtained by assuming that the boundaries of the enclosure Advances in Acoustics and Vibration 3 32 1 PZT PZT acuator Mic. 2 C Plate PVDF sensor Mic. 3 yp Mic. 1 xc PZT (a) (b) (c) Figure 2: 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. are hard, hence the pressure gradients on all boundaries are Here, A and B are used to satisfy the orthogonality n n set to zero: conditions and could be determined by using orthogonal characteristics of these vibration modes [18]. In (11), J p | , p | , p | , x x=0 x x=L y y=0 and I are the Bessel function [18] and the wave number is (6) 4 2 computed from k = ω ρ/D. The expressions (cos) and (sin) p | , p | , p | . y Z z y=L z=L y z=0 z at the right hand side of (11) mean replacing either one of The solution of (3) with boundary conditions introduced them in this equation appropriately. in (6)isgiven as [2] Equation (11) is the general solution for the vibration modes of the solid rectangular plate. Here, J (kx)J (ky) n−m m n πy n πx y n πz x z and I (kx)I (ky) will be used to construct the free n−m m p = A cos cos cos , nxnynz L L L x y z vibration solution of a rectangular thin plate with different (7) edge conditions [18]. In this work, fully clamped boundary with n , n , n = 0, 1, 2,... , x y z conditions is assumed and the edge lengths are defined as a and b. Therefore, the boundary conditions were defined as where n , n and n are the modes number. Using the x y z derivation in [2], the eigenfrequencies of the enclosure can W| = 0, W| = 0, x=0 x=a be further written as ⎡ ⎤ 1/2 W| = 0, W| = 0, y=0 y=a 2 2 c n y n s x z ⎣ ⎦ f = + + . (8) n n n x y z ∂W ∂W 2 L L L (12) x y z | = 0, | = 0, x=0 x=a ∂x ∂x ∂W ∂W 2.2. Flexible Plate. Figure 1(b) shows the position of the | = 0, | = 0. y=0 y=a flexible plate with a constant thickness h. The free harmonic ∂y ∂y vibration partial differential equation of plate, [18]can be Solving (9) with the clamped boundary conditions yield the written as vibration modes [18]: 4 2 D∇ W − ω ρh = 0, (9) W x, y (n,m) where W(x,y), ∇ and D are the displacement of flexible plate in x and y direction, biharmonic differential operator = A {J (kx) + J [k(a − x)]} J ky + J k b − y n n−m n−m m m and bending rigidity respectively. The bending rigidity [18] +B {I (kx) + I [k(a − x)]}) n n−m n−m is given by mπ nπ Eh × I ky + I k b − y cos cos , m m D = , (10) 2 2 12(1 − v ) (13) where E, v, ω,and ρ are the young’s modulus, Poisson’s where m and n are even numbers. ratio, natural frequency and mass density respectively. The nth vibration mode of the rectangular plate in (9)can be 2.3. Model of the Coupled System. In the case of a coupled written in a compact form as [18] system, the effect of the flexible plate on the sound field W x, y inside the enclosure as well as the effect of sound field on the flexible plate must be considered together. In coupling +∞ mπ cos the flexible plate to the enclosure, acceleration of the plate = A J (kx)J ky + B I (kx)I ky . n n−m m n n−m m sin was considered as a source of sound. The pressures at rigid m=−∞ boundaries are zero, but at the top of the enclosure where (11) yc zc xp h h pzt p 4 Advances in Acoustics and Vibration 0.2Max:2.215 0.2Max:2.215 2.215 2.215 0 0 2.154 2.154 −0.2 −0.2 2.093 2.093 0.2 0.2 2.033 2.033 1.972 1.972 0 0 1.912 1.912 1.851 1.851 −0.2 −0.2 1.791 1.791 0.2 0.2 1.73 1.73 0 0 −0.2 −0.2 1.67 1.67 Min: 1.67 Min: 1.67 (a) (b) 0.2 Max: 2.215 0.2 Max: 2.215 2.215 2.215 0 0 2.154 2.154 −0.2 −0.2 2.093 2.093 0.2 0.2 2.033 2.033 1.972 1.972 0 0 1.912 1.912 1.851 1.851 −0.2 −0.2 1.791 1.791 0.2 0.2 1.73 1.73 0 0 −0.2 −0.2 1.67 1.67 Min: 1.67 Min: 1.67 (c) (d) Figure 3: 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 flexible plate was placed, it is equal to the acceleration the point R = R in an infinite homogeneous space is of the plate. Therefore, the homogenous Helmholtz equation 1 π Pc (3)becomes [19] s (3) ∇· − ∇p = 2 δ (R − R ), (15) ρ ρ 0 0 ∇p · n = a , (14) a n 0 (3) where δ (R) is the Dirac delta function in three dimensional space [19]. where, n is the outward-pointing unit normal vector seen from inside the acoustics domain, and a is the normal acceleration of the plate. Acoustic pressure was also coupled 3. Experimental Work in Related Study to the flexible plate as a boundary load pressure in the direction of the normal vectors. In the experimental studies reported in [3],arectangular As previously mentioned, the combined modal analysis cube with five rigid acrylic sheets was constructed and an of the coupled system represents the sound propagation aluminium panel was clamped on of it. The Thickness of of the system with different excitation frequencies. In this acrylic sheets and aluminium sheets used were 25.4 mm work, sound wave is the source of pressure on the plate. The and 1.588 mm, respectively. The rectangular cube with the sound waves were generated by a point source. The governing dimension of L = 0.6096 [m], L = 0.4572 [m], L = xc yc zc equation for a point source with the power P, and located at 0.508 [m] are shown in Figure 2(a), and the plate has the Advances in Acoustics and Vibration 5 Max: 0.402 Max: 0.402 0.4 0.4 0.35 0.35 0.3 0.3 0.25 0.25 0.2 0.2 0.15 0.15 0.1 0.1 0.05 0.05 Min: −7.282e − 6 Min: −7.282e − 6 (a) (b) Max: 0.402 Max: 0.402 0.4 0.4 0.35 0.35 0.3 0.3 0.25 0.25 0.2 0.2 0.15 0.15 0.1 0.1 0.05 0.05 0 0 Min: −7.282e − 6 Min: −7.282e − 6 (c) (d) Figure 4: 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. following dimension: L = 660.4, L = 508.0which is Each module provides a wide range of equations, which xp yp illustrated in Figure 2(b). was needed in specifying subdomains, boundaries, edges and The enclosure was excited by airborne pressure generated points. The theories and equations behind this model are by a loudspeaker placed at some distance above of enclosure, based on the governing equations in Section 2. and three microphones inside the enclosure were to measure the pressure level inside the enclosure. The location of the 4.1. Modelling the Enclosure. The “acoustic” module of microphones inside the enclosure is shown in Figure 2(b). COMSOL was used to model the enclosure, which utilised These microphones are used to sense all modes of enclosure partial differential equations based on time harmonic and using personal computer and dSPACE interface. Nine piezo- frequency domain analysis. The boundaries and properties electric (PZT-5H) elements were patched symmetrically onto of the medium were set to be hard sound boundary and air the plate depicted in Figure 2(c).These actuators are used to characteristics respectively. The dimensions were specified measure the plate response. according to the work of [3], which is repeated here: L = 0.6096 [m], L = 0.4572 [m], xc yc 4. Modelling with COMSOL Multiphysics (16) [ ] L = 0.508 m . zc In this section, sound travel into the medium and its interaction with a solid medium are modelled using a finite The point of origin of the enclosure was placed at element software COMSOL Multiphysics. This modelling (−0.3048, −0.2286, −0.254). Using eigenfrequency analysis procedure requires two modules: one for simulating the of the model, the first 4 eigenfrequencies with modal shapes acoustic medium and the other for flexible plate. After proper are shown in Figure 3. Due to the hard boundaries used selection of the modules, the three dimensional model with in this analysis, the material of the body of enclosure can the same dimensions described in Section 3 was sketched. be neglected. The shapes of mesh elements were selected 6 Advances in Acoustics and Vibration 0.2Max:2.215 0.2Max:2.215 Max: 3.862 Max: 3.862 2.215 2.215 0 0 2.154 2.154 −0.2 3.5 −0.2 3.5 2.093 2.093 3 3 0.2 0.2 2.033 2.033 2.5 2.5 1.972 1.972 2 2 0 0 1.912 1.912 1.5 1.5 1.851 1.851 −0.2 −0.2 1 1 1.791 1.791 0.2 0.2 0.5 0.5 1.73 1.73 0 0 −0.2 −0.2 1.67 1.67 0 0 Min: 1.67 Min: −2.272 Min: 1.67 Min: −2.272 (a) (b) 0.2Max:2.215 0.2Max:2.215 Max: 3.862 Max: 3.862 2.215 2.215 0 0 2.154 2.154 3.5 3.5 −0.2 −0.2 2.093 2.093 3 3 0.2 0.2 2.033 2.033 2.5 2.5 1.972 1.972 2 2 0 0 1.912 1.912 1.5 1.5 1.851 1.851 −0.2 −0.2 1 1 1.791 1.791 0.2 0.2 0.5 0.5 1.73 1.73 0 0 −0.2 −0.2 1.67 0 1.67 0 Min: 1.67 Min: −2.272 Min: 1.67 Min: −2.272 (c) (d) Figure 5: 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. Table 1: Comparison eigenfrequencies between finite element to be tetrahedralinnormalsize. Theenclosure model model with analytical results in [3] for uncoupled system. consists of 10727 mesh elements. Using the eigenfrequency analysis of the model yield the modal shapes with the Analytical Finite element (COMSOL) first four eingenfrequencies depicted in Figure 3.Solution Mode Error % Panel Enclosure Panel Enclosure time and degree of freedom are 24.281 seconds and 56725, (1,1) 41.6 — 44.28 — 6.0524 respectively. (1,2) 73.7 — 75.66 — 2.5905 (2,1) 95.0 — 103.19 — 7.9368 4.2. Modelling of the Plate. The “Structural Mechanics” (2,2) 124.6 — 127.05 — 1.9284 module was used to perform the modal analysis of the plate. (1,0,0) — 281.3 — 281.12 0.0640 By adjusting the parameters of the equations, the mode (0,0,1) — 337.6 — 338.98 0.4071 shapes of the solid stress and strain of the plate is plotted (0,1,0) — 375.1 — 379.64 1.1959 in Figure 4. The dimension of the plate is larger than the (1,0,1) — 439.5 — 446.27 1.5170 enclosure, and this is the same as the experimental work of (1,1,0) — 468.9 — 479.60 2.2310 [3]. For this dimension, the coupled mode was constructed (1,0,0) — 504.7 — 511.83 1.3930 as discussed in Section 2.3 . All edges are selected to be fixed with clamped boundary (1,1,1) — 577.8 — 578.96 0.2004 condition. The shapes of the mesh elements are second Error % 2.3197 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 parameters mentioned above for subdomains and edges, the solver, time for solving the model is 17.156 seconds and model is solved and the mode shapes of the plate are shown the number of degree of freedom is 16010. By replacing the in Figure 4. Advances in Acoustics and Vibration 7 Table 2: Comparison eigenfrequency analysis between simulations, analytical and experimental results. Finite element FEM FEM modelling (FEM) error 1% error 2% mode Analytical Experiment 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 4.3. Simulation of the Coupled System. Thecompletecoupled and enclosure were computed from the FEM model. These system has been simulated and is presented in this section. results are shown in Table 1 and they are compared with Two modules (Acoustic and Structural Mechanics) are used those extracted by the analytical model in [3]. The first 11 in this simulation. The pressure wave source was represented modes of the system consists of the first four modes of the by a point source outside the cavity. A sphere with a plate and the first seven enclosure modes. The mean errors reasonably large diameter outside the enclosure was used of 2.3% indicate that the results are in good agreement with to envelope the air-filled acoustic domain. On the outer the analytical model. It is possible to improve the accuracy of spherical perimeter of the air domain, radiation condition the FEM model by decreasing the element size of the mesh with the spherical wave was used. This boundary condition applied in finite element modelling. On the other hand, the allows a spherical wave to travel out of the system, giving only memory required for this purpose will put a bound on the minimal reflections for the non-spherical components of the obtainable accuracy of the model. These limitations are more wave. The radiation boundary condition is useful when the obvious when comparing the results of couple condition, surroundings are only a continuation of the domain [19]. which requires more element sizes calculations described in Two sub modules from the acoustic tab was used to support the next section. thickness and inner dimension of cavity, namely the “Solid stress strain” for thickness and “Pressure Acoustic” for the 5.2. Model of the Coupled System. The next step is to calculate air inside it. The coupling of flexible plate vibration with the resonant frequencies of the coupled system. The results the acoustic pressure inside the enclosure and vice versa was were compared with both the analytical and experimental performed using both “Acoustic” and “Structure Mechanic” results reported in [3]. Table 2 shows the comparison of the modules. For this purpose, some variables were set to coupled system’s resonant frequencies between the analytical make the connection between these two modules. COMSOL and experimental model and those obtained using the FEM software in such an analysis will enable the combination of model. In experimental work rather than rectangular cube different acoustical and vibration phenomena in one model. and flexible plate, piezoelectric elements are patched to the The variation of some of the mode shapes are depicted in plate to actuate and control the plate for future applications. Figure 5. The mean of the absolute relative error percentage of 9.929963 is achieved due to comparison with analytical work 5. Results and Discussion and 9.281338 percent due to experimental work. The results 5.1. Model of the Uncoupled System. The natural frequencies indicate that finite element method with COMSOL software of the system without any coupling between the flexible plate is able to predict the natural frequency of the whole modes, 8 Advances in Acoustics and Vibration such that active noise control system can be designed using References this model. Results that obtained by smaller mesh element [1] A. Montazeri, J. Poshtan, and M. H. Kahaei, “Modal analysis sizes gave better results. for global control of broadband noise in a rectangular In order to improve the accuracy of the model, the mesh enclosure,” Journal of Low Frequency Noise Vibration and element size must be smaller than 0.002 of wavelength (L), Active Control, vol. 26, no. 2, pp. 91–104, 2007. where [2] A. Montazeri, J. Poshtan, and M. H. Kahaei, “Analysis of the global reduction of broad band noise in a telephone kiosk using a MIMO modal ANC system,” International Journal of L = . (17) Engineering Science, vol. 45, pp. 679–697, 2007. [3] M. Al-Bassyiouni and B. Balachandran, “Sound transmission through a flexible panel into an enclosure: structural-acoustics This would satisfy the rule of ten to twelve degree of model,” Journal of Sound and Vibration, vol. 284, no. 1-2, pp. freedom per wavelength. However, this cannot be achieved 467–486, 2005. with our current computing facility (CPU Intel Pentium [4] B. Fang, A. G. Kelkar, S. M. Joshi, and H. R. Pota, “Mod- Dual, 1.60 GHz, Ram, and 2.93 GB). Better computing elling, system identification, and control of acoustic-structure facility shall be employed in future work. dynamics in 3-D enclosures,” Control Engineering Practice, vol. 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