Copyright 2012 Tech Science Press CMES, vol.2067, no.1, pp.1-14, 2012

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1 CMES Galley ro Only lease Return in 48 Hours Copyright 01 Tech Science ress CMES, vol.067, no.1, pp.1-14, 01 rediction High-frequency Vibro- Coupling in Anechoic Chamber Using Energy Finite Element Method and Energy Boundary Element Method Miaoxia ie 1, ueming Li 1 and Hualing Chen 1, Abstract: Energy finite element method(efem) is a promising method to solve high-frequency vibro- problem. Energy boundary element method (EBEM) is an effective way to compute high-frequency sound radiation in the unbounded medium. Vibro- coupling cavity structure in anechoic chamber includes both the interior field and unbounded exterior field. In order to predict this kind high-frequency vibro- coupling problem in anechoic chamber, an approach combined EFEM and EBEM is developed in this paper. As a numerical example, the approach is applied to solve the high-frequency vibro coupling response a cubic cavity structure excited by a point sound source in an anechoic chamber. The result shows the approach is the feasibility. Keywords: Energy finite element method; Energy boundary element method; anechoic chamber 1 Introduction A series tests for an aircraft must be performed on ground to ensure aircraft reliability before using. Testing the environment is a very important part this verification. Aircraft during flight can be considered as being in a free sound field, which can be simulated in an anechoic chamber on ground. For most anechoic chamber tests, test pieces are placed in the anechoic room, a sound source is placed at another location away from the test pieces, and the internal and external field characteristics and test piece vibration are measured. These experiments ten have to be repeated multiple times to obtain reproducible results. Alternatively, testing time and cost can be reduced significantly with a proper nu- ro 1 State Key Laboratory for Strength and Vibration Mechanical Structure, i an Jiaotong University, i an ,.R. China Institute Vibration and Noise Control, School Mechanical Engineering, i an Jiaotong University, i an ,.R. China

2 CMES Galley ro Only lease Return in 48 Hours merical simulation test program. Therefore the study anechoic chamber simulation methods is very meaningful. Finite element method (FEM) and boundary element method (BEM) [Brancati and Aliabadi (01); Brancati et al. (011)] is instead quite mature methods to deal with the vibro- problems in anechoic cham- "are" ber. However, FEM and BEM are not always efficient and accurate, especially for high frequency ranges, since structural response is extremely sensitive to material and geometry details [Bitsie (1996)]. In order to capture the structural characteristic length, a much smaller mesh size is necessary for FEM or BEM models, which results in the procedure being computationally expensive or even prohibitive. Statistical energy analysis (SEA) is an alternative efficient method which is suitable for solving high-frequency problems and has been widely used as an analysis tool in practice. But local modeling details that concern designers are usually ignored in SEA due to the fact that SEA is based on the division sub-structures. The energy finite element method (EFEM) presented by Nefske et al [Nefske and Sung (1989)] is a new tool for high-frequency vibro- coupling analysis. In this method, governing differential equations are derived in terms energy variables which are solved by applying the finite element approach. Compared with SEA, EFEM can obtain the detail vibro- response information. It is a promising effective method to high-frequency vibro- prediction. This method can compute the high-frequency field generated from structural vibration. The radiation efficiency in EFEM determines the amount power radiated in the fluid due to structural vibration and the total amount the radiated power can be computed by integrating the intensity over all the structural elements which are in contact with the fluid. However, EFEM is not able to compute the field at unbound medium since the governing difference equation for space is based on plane wave approximation, and particularly valid for highly reverberant al environments at high frequency. ro 45 CMES, vol.067, no.1, pp.1-14, 01 8 Copyright 01 Tech Science ress Energy boundary element method (EBEM), which was developed along with EFEM, is an effective way to compute high-frequency sound radiation in the unbounded medium an external structure. From EFEM analysis, the power radiated from each structural element is obtained. Since the vibration energy in the structural elements and the intensity radiating from structural elements in contact with the fluid are incoherent among different elements, the incoherent intensity on the interface between a structure and an cavity can be regarded as a boundary condition in high-frequency EFEM interior computations. Similarly, the incoherent intensities radiating from each element on the outer structure surface in the unbound medium comprise the boundary conditions for EBEM computations [Wang et al. (004)]. Therefore, the field at specific field points around the structure can be computed by EBEM.

3 CMES Galley ro Only lease Return in 48 Hours. rediction High-frequency Vibro- Coupling For the cavity structure, the vibro- response includes both the exterior field and the interior field. If the exterior field is closed by absorbent material, means that it is a bounded space, then EFEA can be used to solve the interior and exterior field simultaneously by viewing the absorbent material as a boundary condition [Vlahopoulos and Wang (008)]. However, in anechoic chamber, exterior field is in an unbounded medium, so the EBEA and EFEA must be combined to deal with this kind problem. ro 83 In 1996, an indirect boundary element method coupling structural vibration and field was presented by Bitsie [Bitsie (1996)], in which structural vibration was computed by EFEM and field was calculated with EBEM. In 004, Wang [Wang et al. (004)] presented an EBEM formulation for calculating sound radiation at high frequency from a radiator with an arbitrary shape. In 004, Choi and Dong [6-8] evaluated the exterior field a vehicle excited by force as instead a way combing EFEM and EBEM. In 006, Nicklos and hang [Vlahopoulos and Wang (008)] studied the structural vibration and interior field ex- "by using cited by an exterior source. In 007, Raveendra and Hong [Raveendra and " hang (007); Hong et al. (007)] performed an analysis interior and vibration characteristics a simplified airplane cabin and van models using hybrid EFEM/EBEM. instead "and" In this paper, we study the interior and exterior fields a cavity structure excited by an exterior source in an anechoic chamber using EFEM/EBEM. In reference [Vlahopoulos and Wang (008)], Coupling the cavity structure viinstead "In this method, the anechoic chamber is taken as an unbounded medium, and the bration with itsvibro interior transmission computed bystructure EFEM. and Theexterior input medium. Coupling system consist interioris medium, cavity the is interior field calculating and the cavitythe structure vibration is computed power needed between in EFEM obtained from energy by EFEM. Excitation imposed on the structure come from a point sound source placed in exterior medium. After this EFEA the structural surface, which generated byfrom a point sound element source,isby EBEM.which is took as the coupling analysis, the energy radiated each structural determined, source EBEA field to calculate the exterior field.each As anstructural example, a element cubic shell excited by a point Then, the interior and energy radiated from is determined sound by EFEM. Inanthis paper, the anechoic chamber is taken as "an unsource in anechoic chamber is studied to illustrate the feasibility. bounded medium, and reference [Vlahopoulos and Wang (008)] is followed through. The energy radiated from each structure element becomes the boundary condition a second EBEM, giving the exterior field generated by structural vibration. Then, the method solving vibro- coupling the whole structural- system in anechoic chamber is presented. As an example, a cubic shell excited by a point sound source in an anechoic chamber is studied to illustrate the feasibility Basic theory high-frequency structure- response As stated in introduction, an approach combined EFEM and EBEM is developed to solve the anechoic chamber high-frequency vibro- coupling problem. Then the basic theory EFEM and EBEM is introduced. as following

4 CMES Galley ro Only lease Return in 48 Hours Copyright 01 Tech Science ress.1 Basic theory EFEM CMES, vol.067, no.1, pp.1-14, 01 The governing equations for the structural- coupling a complex system are (1) wherec is the wave speed; subscripts B,L and T express the bending, longitudinal and transverse shear waves, respectively; subscript g means group speed; subscripts s and a are used to denote the structure domain and the domain; η is the damping loss factor; ηrad is the radiation damping; ω is the radian frequency the instead "average over a time periodover and over harmonic excitation; he i is the time average overaenergy period and space average a a wavelength." wavelength energy ; and hπ i is the time average over a period and space average overinstead a wavelength input power. "average input power over a time period and over a wavelength." ro 110 CgB he sb i + (ηsb + ηrad ) ω he sb i = hπ sb i (ηsb + ηrad ) ω CgL he sl i + ηsl ω he sl i = hπ sl i ηsl ω CgT he st i + ηst ω he st i = hπ st i ηst ω c a he a i + ηa ω he a i = hπ a i ηa ω The finite element method is used to numerically solve the governing equations. At the junctions between elements, the energy is discontinuous while the power flow is continuous and is represented by instead "Using Galerkin weighted Ks 0 Es Fs + Jss + Jsa = 0 Ka Ea Fa residual scheme and the Lagrangian shape function as the trial function, taking consider the compatible condition at the discontinuous junctions, Equation (1) can be written in the form matrix as" () 115 Ks and Ka are the system matrices for the structural elements and elements, Jss and Jsa are joint matrices between discontinuous junctions, {E} is the vector nodal energy, and {F} is the vector input power Basic theory EBEM Acoustic p ρv + ρc and the intensity as I = Energy is defined as e = 1 1 Re (pv ) where p is pressure, v is vibration velocity the particle, ρ is "speed sound" the medium s mass, c is the transmitted velocity the instead wave in the medium, Re is expressed as the real part the number, and * denotes instead "denotes real part a complex number pv*" conjugation.

5 CMES Galley ro Only lease Return in 48 Hours. 5 rediction High-frequency Vibro- Coupling expression The ensemble averaging operator is applied to the equation for e and I, resulting in R k ρ he i = S σ (p) 64πρ r4 + 3π ds r (3) R ρc k ~E ds = S σ (p) hii 3π r r the source strength at point p on the sturcture surface where σ (p) denotes the strength the energy source placed at point p the surface the structure model, r is stated as the distance between the field point surface S wavenumber and point p, and k is wave number. The surfacestructure S the structure is divided into n quadrilateral or triangular elements and the source strength σ j on each element is considered to be constant. Eq.(3) can be written in the discrete form i n h R ρ k ρ he i = σ j (p) S j 64π r(ξ + ds, )4 3π r(ξ, ) j=1 i n h R ρc ~Er ds hi i = σ j (p) S j 3π k r(ξ, ) j=1 (4) ro where s j is the surface the jth element and ξ is a point the jth element. Then the energy expression in Eq.(4) can be written in matrix form [K] {σ } = {p} (5) T power vector, {}n 1 = p 1 p p n is the specified boundary condition T {σ }n 1 = σ1 σ σn is an unknown variables vector, and each term Ki j the matrix [K] is solved by R R k ρc ( ~ ni ds)ds i 6= j Si S j 3π r(ξ,η) Er ~ (6) Ki j = Ai k ρc i= j Here where 16π available The source strength on each element is obtained by solving Eq.(5). Then, by employing thestrength time and frequency averaged and SubstitutingEq.(4), the source into Eq.(4), the energy energy and intensity in exterior medium can be determined intensity can be determined at any field point within the medium s exterior structure. come from vibration structure. The energy is obtained by EFEM. The The power radiated from eachenergy element avibrating structure is computed relation between power and the energy sturcture can be expressed as follow with the structural EFEM analysis, providing the energy in the structure. The power radiated from a structure immersed in an medium is calculated by using pi = Sj ηrad ωesb hds (7)

6 CMES Galley ro Only lease Return in 48 Hours. 6 Copyright 01 Tech Science ress CMES, vol.067, no.1, pp.1-14, 01 where ηrad is radiation damping, ω is the circle frequency and h is the thickness the structure. The energy esb on the nodes the structure s surface and the power radiating from each element a vibrating structure pi is expressed by = W EsB the (8) The matrix W is comprised NE rows and N columns, where each value is written as Wi j = Si i ωhds i j ηrad if (9) if i j = 1 when node j is located on the ith boundary element, and i j = 0 when node j is not on the ith boundary element. Eq.(5) can now be written as where Here [K] {σ } = [W ] {esb } 16 Input power from an exterior source In the anechoic chamber, the cavity structure is excited usually by exterior source. The input power must be got if we predict the coupling response the interior field and the vibration cavity structure by using EFEA. The input power at each structural element which is in contact with the exterior medium is provided by the information energy radiated by source and the transmission coefficient from field to structure. ro 17.3 (10) The energy any field generated by m point sources is m ρ0 k ρ0 e 0 = σ i + (11) 64π ri4 3π ri i=1 σi is the source strength the ith source. The incident power πinc on each energy finite element is determined from the exterior energy e 0 using a plane wave approximation, πinc = e 0 c0 A (1) structure where A is the area the EFEM element and c0 is the wave speed in the medium. The power transmitted to a plane due to an impinging plane wave is plate evaluated as τas = ρ0 c30 σrad ρs cb h f (13)

7 CMES Galley ro Only lease Return in 48 Hours. rediction High-frequency Vibro- Coupling 7 By using Eq. (1) and (13), the input power applied on an EFEM element due to the external energy can be written as source A cubic cavity structure model comprised six plates, as shown in Fig.1, is constructed and studied. The dimensions the cubic cavity structure are 1m x 1m x m and the thickness the plates is 0.01m. The plates all have the same physical properties: the oung s modulus is.4e11a, oisson s ratio is 0.30, Mass is 7800Kg/m3 and the hysteresis damping factor is The cubic cavity structure is immersed in air and the space enclosed by the plates is filled with air with mass 1.5Kg/m3, speed 343m/s and hysteresis damping factor The distance between the center the cubic cavity and point source is 5.5m. The location the point source in relation to the cubic cavity structure is shown in Fig.1. Fig. shows the energy finite element model for the structure. The point source strength is unit strength. The response at several frequencies (000Hz, 8000Hz 16000Hz and 3000Hz) is predicted. ro 143 Numerical example Models (14) σrad e 0 πin = τas πinc = ρ0 c40 A ρs cb h f Figure 1: Schematic diagram the point source and the structure Figure : EFEA model for the structure Results and discussion Acoustic energy is converted to the root mean square pressure with the approximation p = c0 eρ0 and expressed in terms decibel level using the pressure reference 10 5 a. In the case 000Hz, the exterior pressures generated by structural vibration excited by the point source are shown in Fig.3. The distribution interior pressure in the medium, the distribution interior pressure at the y=0.5 symmetric plane,

8 CMES Galley ro Only lease Return in 48 Hours. 8 Copyright 01 Tech Science ress CMES, vol.067, no.1, pp.1-14, Figure 3: Exterior pressure generated by structural vibration Energy Energy Energy ro Figure 4: Distribution Figure 5: Distribution Figure 6: Distribution interior pressure interior pressure energy on the plate at the y=0.5 symmetric plane 156 and the computed energy distribution on the shell are shown in Fig.4, Fig.5 and Fig.6, respectively. 157 In the case 8000Hz, the corresponding results are shown as follow. in Fig.7-Fig In the case 16000Hz, the corresponding results are shown as follow. in Fig.11-Fig In the case 3000Hz, the corresponding results are shown as follow. in Fig.15-Fig The results the approach combined EFEM and EBEM on the above model show that: 1. With the increasing frequency, the difference between maximum and minimum pressure inside the cubic cavity structure become small. The exterior pressure and the vibration energy get smaller and smaller as the frequency increase.. The variation pressure in the medium enclosed by the plate is very small as energy propagation and decay makes the pres, but the distribution trend is

9 CMES Galley ro Only lease Return in 48 Hours. 9 rediction High-frequency Vibro- Coupling Figure 7: Exterior pressure generated by structural vibration Energy Energy Energy ro E-05 9E-05 8E-05 9E-05 8E-05 7E-05 8E-05 7E-05 6E-05 7E-05 6E-05 5E-05 6E-05 5E-05 4E-05 5E-05 4E-05 3E-05 3E-05 4E-05 E-05 E-05 3E-05 E-05 Figure 8: Distribution Figure 9: Distribution Figure 10: Distribution interior pressure interior pressure energy on the plate at the y=0.5 symmetric plane sure diminish from the boundary to the centre. Explanation for this is that energy is transmitted from the shell to the centre the medium since the pressure the plate facing the point source is larger than the pressure on the plate on the far side the cubic cavity structure from the point source. 3. Energy on the four lateral plates increases from the bottom to the top the cubic cavity structure because the bottom plate is fixed. However, the energy is thelarger plate which contour is not parallel to the bottom plate; thethe energy for is closer to the load location is large. structural elements closer to the load location. 4. If a sound absorbing material is utilized, EBEA don t suit any more. In this case, EFEA is adapted, and the absorbing material is taken as boundary element in EFEA which is similar with the method in the paper Brancati, A.

10 CMES Galley ro Only lease Return in 48 Hours. 10 Copyright 01 Tech Science ress CMES, vol.067, no.1, pp.1-14, Figure 11: Exterior pressure generated by structural vibration Energy E-05 8E-05 9E-05 7E-05 8E-05 6E-05 7E-05 5E-05 6E-05 4E-05 5E-05 3E-05 4E-05 E-05 3E-05 E-05 ro Figure 1: Distribution Figure 13: Distribution Figure 14: Distribution interior pressure interior pres- energy on the plate sure at the y=0.5 symmetric plane Comparison EFEA with SEA and FEA 185 In order to prove that the hybrid method EFEA and EBEA is the more suitable method to deal with the high frequency vibro- in anechoic chamber, comparison EFEA with SEA and FEA has been done in terms accuracy and calculating time E-05 8E-05 7E-05 6E-05 5E-05 4E-05 3E-05 E-05 Energy Energy Comparison EFEA with FEA In FEA, the biggest factor affecting the computation time is the number elements, so the number elements can roughly determine the computation time. For the cuboid, according to the accuracy requirement, the required number elements is shown as table 1. It can be seen that there need more and more elements as the frequency increasing. However, in EFEA, the elastic wave propagation problem in EFEM is similar as the heat transfer problem in FEM. So, there is no need to change the mesh size with frequency. Frequency need not to be considered in EFEM mesh models. So more

11 CMES Galley ro Only lease Return in 48 Hours. 11 rediction High-frequency Vibro- Coupling Figure 15: Exterior pressure generated by structural vibration Energy 66.8 Energy Energy E E E-05 6E-05 5E E E-05 5E-05 5E-054.5E-05 4E E E-05 4E-05 4E-053.5E-05 3E E E-05 3E-05 3E-05.5E-05.5E-05.5E-05 E-05 E-05 E-051.5E E E-05 5E-06 5E-06 5E-06 ro Figure 16: Distribution Figure 17: Distribution Figure 18: Distribution interior pressure interior pres- energy on the plate sure at the y=0.5 symmetric plane Table 1: Required number elements in FEA Frequency 000Hz 4000Hz 8000Hz 16000Hz element FEA high frequency, more convenient energy finite element analysis is Comparison EFEA with SEA 199 In order to make clear the advantage EFEA over SEA, the cubic cavity structure excited by exterior source is calculated by using SEA. At 000Hz, the results are shown in Fig From the Fig.19, the vibration energy the top plate and the front plate is

12 CMES Galley ro Only lease Return in 48 Hours. Copyright 01 Tech Science ress CMES, vol.067, no.1, pp.1-14, 01 1 ro Figure 19: The distribution energy calculated by SEA J and J, respectively. 08 From the Fig.6, the energy calculated by EFEA is obtained. In order to compare the result EFEA to SEA, the energy is converted to energy by the way multiplying energy by corresponding volume. Thus, the vibration energy the top plate and the front plate calculated by EFEA is J and J, respectively. It can be seen that EFEA and SEA has approximate results, however, the distribution energy on structure can be obtained by EFEA Conclusion Considering a structure excited by a point sound source in an anechoic chamber, the high frequency vibro- response is hard to reach with conventional finite element analysis because the computational cost. EFEA based on energy conservation is effective. Continued the method in the reference [Vlahopoulos and Wang An an approach solving the anechoic chamber high-frequency vibro- (008)], coupling problem using EFEA and EBEA is presented in this paper. The anechoic chamber is taken as an unbounded medium. The exterior/interior fields the structure and the vibration energy distribution the structure within an anechoic chamber are obtained using an EFEM/EBEM developed in this paper. As an example, a cubic shell excited by a point sound source in an anechoic

13 CMES Galley ro Only lease Return in 48 Hours rediction High-frequency Vibro- Coupling 13 chamber is studied. The result actually is reasonable and indicates that it is feasible and convenient to solving vibro- coupling the whole structural- system in anechoic chamber using EFEM/EBEM proposed in this paper. It is to believe that the method described here can accurately simulate anechoic chamber high-frequency vibration- test results and provide an alternative to chamber testing structural components. Acknowledgement: This work is supported by NSFC ( , 1117) References Bitsie, F. (1996): The structural-s energy finite element method and energy boundary element method, in School Mechanical Engineering. urdue University: West Lafayette. Brancati, A. and M.H. Aliabadi (01): Boundary element simulation for local active noise control using an extended volume. Engineering analysis with boundary elements, 36: p Brancati, A., M.H. Aliabadi, and A. Milazzo (011): An improved hierarchical ACA technique for sound absorbent materials. CMES, 78(1): p Choi, K.K., et al. (004): Design Sensitivity Analysis and Optimization High Frequency Radiation roblems Using Energy Finite Element Method and Energy Boundary Element Method, in AIAA/ISSMO10th Multidisciplinary Analysis and Optimization Conference. Albany, New ork. Dong, J. (004): Design sensitivity analysis and optimization high frequency structural- problems using energy finite element method and energy boundary element method, in Mechanical engineering. The University Iowa: Iowa. Dong, J., et al. (007): Sensitivity Analysis and Optimization Using Energy Finite Element and Boundary Element Methods. AIAA Journal, 45(6): p Hong, S.., H.W. Kwon, and J.D. Kim. (007): Car Interior and Exterior Multidomain Noise Analysis using Energy Flow Analysis (EFA) Stware, NASEFAC++. in Noise and Vibration Conference and Exhibition SAE paper St. Charles, Illinois. ro Nefske, D.J. and S.H. Sung (1989): ower flow finite element analysis dynamic systems: basic theory and applications to beams. ASME Transaction, Journal Vibration, Acoustics Stress and Reliability in Design, 111(1): p Raveendra, S.T. and W. hang. (007): Vibro- Analysis Using a Hybrid Energy Finite Element /Boundary Element Method, in SAE paper St. Charles, Illinois.

14 CMES Galley ro Only lease Return in 48 Hours Copyright 01 Tech Science ress CMES, vol.067, no.1, pp.1-14, 01 Vlahopoulos, N. and A. Wang (008): Combining Energy Boundary Element with Energy Finite Element Simulations for Vehicle Airborne Noise redictions, in 008 SAE Congress. Detroit, Michigan. Wang, A., N. Vlahopoulos, and K. Wu (004): Development an energy boundary element formulation for computing high-frequency sound radiation from incoherent intensity boundary conditions. Journal Sound and Vibration, 78(1-): p ro