摘要
研究空腔内部噪声问题是一个具有极强实际意义的问题,其应用对象包括汽车驾驶室、飞机机舱和舰船船舱,而在结构-声学耦合振动分析基础上进行声场优化是一个正在发展的方向。本文在分析了国内外现有研究结构-声学耦合方法有缺点的基础上,基于间接Trefftz理论,发展了一种新的结构-声学耦合分析方法——波函数法,详细推导了板结构、声学空腔和结构-声学耦合系统的波函数模型,并针对该方法中的两个关键问题,波函数序列与多域问题,提出了有效的解决方法;在此基础上,提出了一种基于波函数法的结构-声学灵敏度计算方法;然后应用波函数法,建立了相干双激励下的结构-声学耦合系统波函数模型,从理论上研究了相干激励下的声场分布规律;最后对所提出方法的有效性和可靠性进行了实验验证。
基于间接Trefftz理论,详细推导了用波函数法求解平板与空腔耦合系统在外力作用下的三维结构-声学耦合系统的理论计算模型。论文利用任意边界条件的长方体空腔,推导并证明了新的声学波函数系列。通过一些计算实例,将该方法的数值仿真结果与精确解及有限元数值解进行对比,验证了波函数法的有效性,同时证明该方法在收敛速度、计算精度和计算时间上都有很大的优势,能将计算频率覆盖到中频区域。针对现有表征结构-声学耦合强度参数的缺陷,提出了新的耦合强度计算公式,使其更加全面和合理。
针对波函数法的多域问题,提出了一种基于边界连续性条件的域分解方法,分别建立了平板结构、声学空腔和结构-声学耦合系统的多域计算模型,从而扩展了该方法的应用范围。数值仿真结果表明,这种域分解技术是正确而有效的,对收敛速度和计算时间没有太大的影响。
基于波函数法理论,对结构-声学灵敏度进行了探讨,通过对结构与声学响应的波函数展开式直接相对于设计变量求导,便得到了结构-声学灵敏度表达式,然后对系统模型方程求导,求得灵敏度表达式中的未知参数,从而得到结构-声学灵敏度的最终计算公式。通过这种方法,分别建立了平板结构、声学空腔以及结构-声学耦合系统的灵敏度计算模型。通过实例研究,讨论了结构及声学响应相对于设计变量的灵敏度变化规律。
应用波函数法,利用平板和空腔波函数模型以及它们的耦合关系,建立了相干双激励下的结构-声学耦合系统波函数模型。并针对一个长方体实例,从理论上验证了该模型的正确性,并讨论了激励之间的相位差、板结构的阻尼以及腔体表面上的吸声材料对空腔声场分布的影响,从仿真结果可以看出:激励之间的相位差对声场分布有较大的影响;抑制板结构的振动,能有效降低其近场声压,但对于远场分布影响很小;而吸声材料的使用导致声场能量的重新分配,材料附近的声压降低,而其相对的那一边声压却有升高。
利用一个梯形体结构-声学耦合系统,以及一个实际轿车的内部声场实例,通过实验对所提出的方法进行了验证。
最后对全文进行了总结和展望。
It’s very important to study the noise problem inside a cavity. The objects studied in this field include driving cabs of automobile, passenger compartments of a plane and cabins of a ship. The optimization of sound field based on the structure-acoustic analysis is developing rapidly. Based on the analysis about the advantages and disadvantages of the existing researches on structure-acoustic problems and indirect Trefftz method, a new method, wave based method (WBM), is brought forward for structure-acoustic studies in this paper. The WBM model of plate, cavity and structure-acoustic coupled system are deduced in detail, and the two key problems of WBM, namely wave function series and muti-domain calculation, are effectively handled. Then, a new structure-acoustic sensitivity calculation method is presented based on WBM. The WBM model of structure-acoustic coupled system under two coherent excitations is established and the principle of sound field distribution is studied in theory. Lastly, the validity of the proposed method is proved by experiments.
Based on indirect Trefftz theory, the 3D structure-acoustic coupled system calculation model is deduced in detail using WBM, which is applied to palte and cavity coupled system under point force excitation. Using a cuboid cavity with random boundary conditions, deduced and proved a new wave function series. The validity of the WBM is confirmed by comparing the simulation results with the analytical and FEM results. At the same time, the great advantage of the method in convergence rate, calculation precision and calculation speed are proved and the method can extend the calculation range to mid-frequency. Focused on the flaw of the existing parameter indicating the strength of structure-acoustic coupling interaction, a new formulation is presented, which is more comprehensive and reasonable.
Based on the continuity of boundary conditions, the method of domain decomposition is presented and the multi-domain calculation model of plate, acoustic cavity and structure-acoustic coupled system are established, which extend the application range of the method. The simulation result shows that the domain decomposition technique is valid and has negletable influence on convergence rate and calculation speed.
The structure-acoustic sensitivity is discussed based on WBM. The structure-acoustic sensitivity formula is obtained by directly derivate on the structural and acoustic response expansion formula. Then, derivating on the system model gets the unknown parameter and forms the final structure-acoustic sensitivity expression. The sensitivity calculation model of plate, acoustic cavity and structure-acoustic coupled system are established. The principle of structural and acoustic response sensitivity comparing with design parameters are discussed through some examples.
The WBM model of the structure-acoustic coupled system excited by two coherent inputs is established using plate, cavity WBM model and its coupling relationship. Based on a cuboid cavity example, the validity of the model is proved in theory. The influence of phase difference between inputs, damping of plate and absorption material of cavity surfaces on sound field distribution are discussed. The simulation result shows that the phase difference between inputs influence the sound field very much; suppressing the vibration of the plate can effectively reduce nearfield sound pressure, however, the influence on farfield sound pressure is very tiny; the using of absorption material can lead to the redistribution of sound power, in other words, the pressure near the absorption material decreased and the pressure of the opposite side increased.
The proposed theory in this dissertation is tested by examples of a trapezoid structure-acoustic coupled system and the interior sound field of a real car.
Lastly, the summarization and expectation are given.
引文
[1]何作镛,赵玉芳.声学理论基础.北京:国防工业出版社,1981.
[2]邵汝椿,黄振吕.机械噪声及其控制.广州:华南理工大学出版社,1997.
[3]孙林.国内外汽车噪声法规和标准的发展.汽车工程,2000,3(22):154-158
[4]乌惠乐.汽车噪声研究的国内外现状与前景.汽车技术.1989,5:13-15
[5] R.M .Lyon. Power flow between linearly coupled oscillators. The Journal of Acoustical Society of America, 1962, 34:623-639.
[6] P.W .Smith. Response and radiation of structural modes excited by sound. The Journal of Acoustical Society of America, 1962, 34:761-779.
[7] P.W .Smith. Coupling of sound and panel vibration below the critical frequency. The Journal of Acoustical Society of America, 1964, 36:1516-1520.
[8]王彦琴,盛美萍,孙进才.统计能量分析预测飞机壁板隔声量及舱室内声场分布.声学技术,2003,22(4):219-222.
[9] Radcliffe, C.J., Huang, X.L. Putting statistics into the statistical energy analysis of automotive vehicles. Journal of Vibration and Acoustics, Transactions of the ASME, 1997, 119(4): 629-634.
[10]伍先俊,朱石坚.基于Autosea2的家电产品声品质设计.研发技术,2005,2:53-56.
[11] Cuschieri J M, Sun J C. Use of statistical energy analysis for rotating machinery. Journal of Sound and Vibration, 1994, 170(2):181-190.
[12] Mark Jay. Data Acquisition Using PC Multimedia Functions. Sound and Vibration, 2000, 20(1):16-21.
[13] Heron K. Advanced statistical energy analysis. Philosophical Transactions of the Royal Society of London, 1994, 346(A):501-510.
[14] Cuschieri J M, Sun J C. Use of statistical energy analysis for rotating machinery. Journal of Sound and Vibration, 1994, 170(2):181-190.
[15] Radcliffe, C.J.; Huang, X.L. Putting statistics into the statistical energy analysis of automotive vehicles. Journal of Vibration and Acoustics, Transactions of the ASME, 1997, 119(4): 629-634.
[16] Mace, B.R. Wave coherence, coupling power and statistical energy analysis. Journal of Sound and Vibration, 1997, 199(3): 369-380.
[17] Priya Thamburaj, J. Q. Sun. Effect of Material and Geometry on the Sound and Vibration Transmission across a Sandwich Beam. Journal of Vibration and Acoustics, 2001, 123(2):205-212.
[18] D.Cui, I.A.Craighead. Reduction in machine noise and vibration levels based on the statistical energy analysis method. Proceedings of the Institution of Mechanical Engineers, 2000, 214(E3): 147-156.
[19] L.D.Pope.Band-limited power flow into enclosures. The Journal of Acoustical Society of America, 1980, 68(3):823-826.
[20] L.D.Pope.Development and validation of preliminary analytical models for aircraft interior noise prediction. The Journal of Acoustical Society of America, 1982, 72(2): 276-291.
[21]仪垂杰.功率流理论及其在装甲车上的应用(博士学位论文).西安交通大学,1994.
[22] A.J.Pretlove.Free vibrations of a rectangular panel backed by a closed rectangular cavity. Journal of Sound and Vibration, 1965, 2:197-209.
[23] A.C raggs.The use of simple three-dimensional acoustic finite element for determining the natural modes and frequencies of complex shaped enclosure. Journal of sound and vibration, 1972, 23(4):331-339.
[24] Wolf J.A, Nefske D, Howell L.T. Structural-Acoustic Finite Element Analysis of the Automobile Passenger Compartment.Transactions of the Society of Automotive Engineer, 1976, 85:857-864.
[25] Joseph A, Wolf J.A.Modal Synthesis for Combined Structural-Acoustic Systems. J.AIAA, 1977, 5(5):743-745.
[26] Macneal R H, Citerley R, Chargin M.A Symmetric Modal Formulation of Fluid-Structure Interaction.ASME Paper 80-C2/PVP-117, 1980.
[27] Nefske D R, Wolf J A, Howell L T. Structural-Acoustic Passenger Compartment: A Review of Current Practice. Journal of sound and vibration, 1982, 80(2):247-266.
[28] D R Nefske, S H Sung. Vehicle Interior Acoustic Design Using Finite Element Methods.Int.J.of Vech.Des, 1985, 6(l):371-385.
[29] Vaicaitis R, Slazak M. Cabin Noise Control for Twin Engine General Aviation Aircraft.NASA-CR-165833, 1982.
[30]王勖成,邵敏编著.有限单元法基本原理和数值方法[M].北京:清华大学出版社,2003.
[31] A.Craggs. The transient response of a coupled plate-acoustic system using plate and acoustic finite elements. Journal of Sound and Vibration, 1971, 15:509-528.
[32] A.Craggs.An acoustic finite element approach for studying boundary fexibrility and sound transmission between irregular enclosures. Journal of Sound and Vibration, 1973, 30:343-357.
[33] M.Petyt, J.Lea, G.H.Koopmann.A finite element method for determining the acoustic modes of ir regularly shaped cavities. Journal of Sound and Vibration, 1976, 45: 495-502.
[34] J.L.Richards, S.K.Jha.A simplified finite element method for studying acoustic characteristics inside a car cavity. Journal of Sound and Vibration, 1979, 63:61-72.
[35] L.Cheng.Fluid-structural coupling of a plate ended cylindrical shell: vibration and internal sound field. Journal of Sound and Vibration, 1994, 174(5):641-654.
[36] L.E.Gilroy.Finite element methods for analyzing coupled fluid/structure systems. Proceedings of International Conference on Noise Control Engineering (Noise93), 1993.
[37]石秀勇,李国祥,胡玉平.发动机飞轮螺栓的三维有限元计算分析[J].中国机械工程, 2006,8(17):845-848.
[38] R. Courant. Variational method for solutions of problems of equilibrium and vibrations. Bull. Am. Math. Soc., 1943, 49: 1-23.
[39] M. J. Turner, R. W. Clough, H. C. Martin. Stiffness and deflection analysis of complex structures. J. Aero. Sci., 1956, 23:805-823.
[40] R. W. Clough. The finite element method in plane stress analysis. Proc. 2nd ASME Conference on Electronic Computation, Pittsburgh, Pa., Sept., 1960.
[41]姚振汉,杜庆华.边界元法应用的若干近期研究及国际新进展.清华大学学报(自然科学版),2001,41(4/5):89-93.
[42] Perrey-Debain, E.Analysis of convergence and accuracy of the DRBEM foraxisymmetric Helmholtz-type equation. Engineering Analysis with Boundary Elements. 1999, 23(8):703-7111.
[43] Golberg, M.A., Chen, C.S., Bowman, H. Some recent results and proposals for the use of radial basis functions in the BEM. Engineering Analysis with Boundary Elements. 1999, 23(4):285-296.
[44] Popov, V., Power, H. DRM-MD approach for the numerical solution of gas flow in porous media, with application to landfill. Engineering Analysis with Boundary Elements.1999, 23(2):175-188.
[45] Songping, Zhu, Satravaha, Pornchai.Solving nonlinear time-dependent diffusion equations with the dual reciprocity method in Laplace space. Engineering Analysis with Boundary Elements.1996, 18(1):19-27.
[46] Niku, S. M., Adey, R. A.Computational aspect of the dual reciprocity method for dynamics. Engineering Analysis with Boundary Elements.1996, 18(1):43-60.
[47] Jumarhon, Bartur, Amini, Sia,Chen, Ke. On the boundary element dual reciprocity method. Engineering Analysis with Boundary Elements.1997, 20(3):205-211.
[48] Zhang Z., Vlahopoulos N., Raveendra S.T, Zhang K.Y.A computational field reconstruction process based on an indirect boundary element formulation. The Journal of Acoustical Society of America, 2000, 108(5):2167-2178.
[49] Raveendra S.T. An efficient indirect boundary element technique acoustic analysis. Int.J.Num.Meth.Engng., 1999, (44):59-76.
[50] Guiggiani M et al. A general algorithm for the numerical solution of hypersingular BEM.Journal of Applied Mechanics, 1992, 59(2): 604-614.
[51] Peter R. Johnston, David Elliott. A generalisation of Telles’method for evaluating weakly singular boundary element integrals. Journal of Computational and Applied Mathematics, 2001, (131): 223–241.
[52]赵志高.结构声辐射的机理与数值方法研究(博士学位论文).华中科技大学,2005.
[53] D.Martin, M.Aliabadi. A BE hyper-singular formulation for contact problems using non-conforming discretization.Computers and Structures, 1998, (69):557-569.
[54]吴九汇.车辆内腔声固耦合分析的覆盖域方法研究(博士学位论文).西安交通大学,2001.
[55] J. H.W u,H.L.Chen, W.B.An.A Method to Predict Sound Radiation from a Plate-Ended Cylindrical Shell Excited by an External Force.Journal of Sound and Vibration,2000,237(5):793-803.
[56] Wu Jiuhui,Chen Hualing,Hu Xuanli.Method to calculate interior sound field of arbitrary-shaped closed thin shell.Chinese Journal of Acoustics,2001,20(1):81-87.
[57]范成高,陈南,张肃.基于改进Trefftz解析法的封闭空腔噪声有源控制.动力学与控制学报,2006,4(4):380-384.
[58] Trefftz E. Ein Gegensttick zum ritzschen Verfahren. Proc.2nd Int.Cong.Appl.Mech., 1926, 23:131-37.
[59] Antes, H. On a regular boundary integral equation and a modified Trefftz method in Reissner's plate theory.Engineering Anal, 1984, l (3):49-53.
[60] Hochard, Ch., Proslier, L. A simplified analysis of plate structures using Trefftz functions. Int.J.numer.Meth.Engng, 1992, 34:179-95.
[61] Kita E, Kamiya N. Trefftz method: an overview.Advances in Engineering Software, 1995, 24:3-12.
[62] Ismael Herrera, GOnzalo Alduncin.Contribution to free boundary problems using boundary elements: Trefftz approach.Computer Methods in Applied Mechanics and Engineering,1984,42(3):257-271
[63] Herrera I, Gurgeon, H.Boundary methods C-complete systems for stokes problems. Computer Methods in Applied Mechanics and Engineering, 1982, 30(5):225-241.
[64] Zielinski, A.P.,Zienkiewicz, O.C.Generalized finite element analysis with T-complete boundary solution functions.Int.J.Num.Meth.Eng.,1985,21:509-528.
[65] Zielinski, A.P., Herrera, I.Trefftz method: fitting boundary conditions. Int.J.Num.Meth. Eng., 1987, 24:871-891.
[66] Zielinski, A.P.On trial functions applied in the generalized Trefftz method.Advances in Engineering Software, 1995, 24:147-155.
[67] C.Vanmaele, D.Vandepitte, W.Desmet. An efficient wave based prediction technique for plate bending vibrations. Coput.Methods.Appl.Mech.Engrg, 2007,4 5:110-136.
[68] B.Pluymers, W.Desmet, D.Vandepitte. Application of an efficient wave-based prediction technique for the analysis of vibro-acoustic radiation problems. Journal ofcomputational and applied mathematics, 2004, 168:353-364.
[69] Cheung, Y.K., Jin, W.G., Zienkiewicz, O.C. Direct solution procedure for solution of harmonic problems using complete, non-singular, Trefftz functions.Comm. Appl. Num. Meth. 1989,5:159-169.
[70] Cheung, Y.K., Jin, W.G., and Zienkiewicz, O.C. Solution of Helmholtz equation by Trefftz method.Int. J. Num. Meth. Eng., 1991, 32:63-78.
[71] Jin, W.G., Cheung, Y.K., and Zienkiewicz, O.C. Application of the Trefftz method in plane elasticity problems. Int. J. Num. Meth. Eng., 1990, 30:1147-1161.
[72] Jin, W.G., Cheung, Y.K., and Zienkiewicz, O.C. Trefftz method for Kirchoff plate bending problems. Int. J. Num. Meth. Eng., 1993, 36:765-781.
[73] Jirousek, J., and Guex, L. The hybrid-Trefftz finite element model and its application to plate bending. Int. J. Num. Meth. Eng., 1986, 23:651-693.
[74] Johnston, R.L., and Fairweather, G.The method of fundamental solutions for problems in potential flow. Appl. Math. Modelling, 1984, 8:265-270.
[75] W. Desmet, B. Van Hal, P. Sas, D. Vandepitte.A computationally efficient prediction technique for the steady-state dynamic analysis of coupled vibro-acoustic systems. Adv. in Engg. Software, 2002, 33:527–540.
[76] L. Cremers, K.R. Fyfe, P. Sas.A variable order infinite element for multi-domain boundary element modelling of acoustic radiation and scattering.J. Appl. Acoust., 2002, 59:185–220.
[77] Desmet W. A wave based prediction technique for coupled vibro-acoustic analysis (Ph.D. dissertation). K.U. Leuven, 1998
[78] K.Matsuda, Y.Kanemitsu, S.Kijimoto. A wave-based controller design for general flexible structures. Journal of sound and vibration, 1998, 216(2):269-279.
[79] W.Desmet, B.pluymers, P.sas.Vibro-acoustic analysis procedures for the evaluation of the sound insulation characteristics of agricultural machinery cabins.Journal of Sound and Vibration, 2003, 266(3):407-441.
[80] B. Pluymers, A. Hepberger, W. Desmet. Experimental validation of the wave based prediction technique for the analysis of the coupled vibro-acoustic behavior of a 3D cavity. Computational Fluid and Solid Mechanics, 2003, 3:1483-1487.
[81] J.R. Chang, R.F. Liu.An asymmetric indirect Trefftz method for solving free-vibrationproblems.Journal of Sound and Vibration, 2004, 275:991-1008.
[82]彭伟才,何锃.WBM法分析声腔的中频响应.噪声与振动控制,2006,12(6):58-61.
[83]彭伟才,何锃.WBM法在多域声学分析中的应用.动力学与控制学报, 2006, 4(3): 278-283.
[84] Leuridan, J.Software for acoustics crossing the chasm.Proceedings Euro Noise'95, 1995:17-32.
[85] M.Dalenbring, A.Zdunek.On the use of three-dimensional h- and p-version finite elements in solvingvibration response problems.Journal of sound and vibration, 2005, 288:907-929.
[86] Saikat Dey ,Joseph J. Shirron, Luise S. Couchman.Mid-frequency structural acoustic and vibration analysis in arbitrary, curved three-dimensional domains.Computers and Structures,2001,79:617-629.
[87] R. M. GRICE.R.J.PINNINGTON.Analysis of the flexural vibration of a thin-plate box using a combination of finite element analysis and analytical impedances.Journal of Sound and Vibration, 2002, 249(3):499-527.
[88] L. Ji, B.R. Mace, R.J. Pinnington. A mode-based approach for the mid-frequency vibration analysis of coupled long- and short-wavelength structures.Journal of Sound and Vibration, 2006, 289:148-170.
[89] X. Zhao, N. Vlahopoulos.A basic hybrid finite element formulation for mid-frequency analysis of beams connected at an arbitrary angle.Journal of Sound and Vibration, 2004, 269:135-164.
[90] G. C. Hsiao, R. E. Kleinman.Applications of boundary integral equation methods in 3D electromagnetic scattering.Journal of Computational and Applied Mathematics, 1999, 104(2):89-110.
[91] G.C. Hsiao, E. Schnack, W.L. Wendland.A hybrid coupled finite-boundary element method in elasticity.Computer Methods in Applied Mechanics and Engineering, 1999,173(3):287-316.
[92] G. C. Hsiao, B. N. Khoromskij,W. L. Wendland.Preconditioning for boundary element methods in domain decomposition.Engineering Analysis with Boundary Elements, 2001, 25(4):323-338.
[93] G. Zhu, A.A. Mammoli,H. Power.A 3-D indirect boundary element method for bounded creeping flow of drops.Engineering Analysis with Boundary Elements,2006,30(10):856-868.
[94] Chen, J.T., Chen, K.H. Dual integral formulation for determining the acoustic modes of a two-dimensional cavity with a degenerate boundary. Engineering Analysis with Boundary Elements, 1998, 21(2):105-116.
[95] Harold Yan, Alan Parrett, Wayne Nack.Statistical energy analysis by finite elements for middle frequency vibration.Finite Elements in Analysis and Design, 2000, 35: 297-304.
[96]宋朝,黄其柏,徐志云,丁律辉.浮筏隔振系统仿真结果的中间频段数据处理方法.噪声与振动控制,2006,21(6):9-11.
[97]徐志云,吴崇健,付爱华.浮筏隔振效果估算方法研究.舰船科学技术, 2006, 28(2): 64-68.
[98] Zhengbong Ma, Ichiro Hagiwara.Sentivity Analysis Methods for Coupled Acoustic Structural System, Part I: Modal Sentivity. AIAA journal, 1991, 29(5):113-125.
[99] Zhengdong Ma,lchiro Hagiwara. Sensitivity Analysis Methods for Coupled Acoustic-Structural System Part II: Direct Frequency Response and Its Sentivities. AIAA journal, 1991, 29(6):56-67.
[100] Ichiro Hagiwara,Wakae Kozukue,Zhengbong Ma. The development of eigenmode sensitivity analysis methods for coupled acoustic-structural systems and their application to reduction of vehicle interior noise. Finite Elements in Analysis and Design, 1993, 14(6):235-248.
[101] Zhengdong Ma, IchiroH agiwara. Sentivity Calculation for Conducting Modal Frequency Response Analysis of coupled acoustic-structural systems. JSME InternationalJournal, 1992, 35(3):14-21.
[102] S .A.H ambric. Sensitivity Calculation for BroadBand Acoustic Radiated Noise Design Optimization Problems. Journal of Sound and Vibration, 1995, 117(1): 136-144.
[103] K.K. Choi, I. Shim, S. Wang. Design sensitivity analysis of structure-induced noise and vibration. Journal of Sound and Vibration,1997, 119(3):173-179.
[104] S .Wang.Design sensitivity analysis of noise, vibration and harshness of vehicle body structure. Mech.Struct.Mach, 1999, 27(3):317-336.
[105]王登峰.拖拉机驾驶室内部噪声响应的灵敏度分析.农业机械学报,1993, 24(1): 25-30.
[106]吴小清,王登峰,徐诚,程悦荪.壁面有吸声材料时驾驶室内噪声响应的灵敏度分析.农业机械学报, 1999, 15(3):20-24.
[107]王登峰,程悦荪,刘明树,赵荣宝.拖拉机驾驶室声固耦合动态择优声学设计.农业机械学报,1994,25(3):23-26.
[108] Gu Y X, Wang H W. The Sensitivity analysis Methods of Coupled acoustics- structure SystemsAnd Application in Interior Noise Reducing. Prow EPMESC VIII, Shanghai, 2001.
[109] Christos I, Papadopoulos. Redistribution of the low frequency acoustic modes of a room: a finite element-based optimisation method. Applied Acoustics, 2001, 62(3): 1267-1285.
[110] Jianhui Luo, Hae Chang Gea. Optimal Stiffener Design for Interior Sound Reduction Using a Topology Optimization Based Approach. Journal of Vibration and Acoustics, 2003, 125(7):267-272.
[111] Takewaki I. Optimal frequency design of tower structures via an approximation concept Computers&Structures, 1996, 58(3):445-452.
[112] Min S, Kikuchi N, Park Y C. Optimal topology design of structures under dynamicloads Structural Optimization, 1999, 17:208-218.
[113] J.H.Kanne, S.Mao.A boundary element formulation for acoustic shape sensitivity analysis. J. Acoust. Soc. Am., 1991, 90 (l):561-573.
[114] D.C. Smith, R.J. Bernhard.Computation of acoustic shape design sensitivity using a boundary element method.Trans.ASME, J. Vib. Acoust., 1992, 114:127-132.
[115] Bon-Ung Koo, Jeong-Guon Ih,Byung-Chaf Lee.Acoustic shape sensitivity analysis using the boundary intergral equation.J.Acoust.Soc.Am.,1998,104 (5): 2851-2860.
[116] Bon-UngKoo, Byung-ChafLee, Jeong-Guon Ih.A non-singular boundary intergral equation for acoustic problems.J.Sound Vib., 1996, 192:263-279.
[117] E .W. Constans, A.D. Belegundu.Design approach form inimizing sound power from vibrating shell structures.AIAA J, 1998, 36(2):134-139.
[118] S.Marburg. Efficient optimization of a noise transfer function by modification of a shell structure geometry一Part I: Theory. Struct Multidisc Optim, 2001, 24(5):51-59.
[119] M. Tinnsten, B. Esping, M. Jonsson. Optimization of acoustic response. Structural Optimization, 1999, 18(1):36-47.
[120] Semyung Wang, Jeawon Lee. Acoustic Design Sensitivity Analysis and Optimization for Reduced Exterior Noise. AIAA Journal, 2001, 39(4):1237-1245.
[121] D.C. Hodgson, M.M. Sadek.A technique for the prediction of the noise field from an arbitrary vibrating machine. Engineering Science Division, 1983, 197(8):189-197.
[122] M.M.Nigm, M.M.Sadek. Noise emitted from a drop hammer tup and the effects of tup configurations. Proc.ASME Conf.Coputer Aided Design, Chicago, USA, 1982: 323-342.
[123]赵翔.旋转式冰箱压缩机辐射噪声的计算与实验研究.航天工艺,1993, 17(4): 14-17.
[124]赵翔,谢状宁,黄幼玲.自由场结构体声辐射研究.声学学报,1994,19(1):22-31.
[125]余兴悼,方新,周禹.用FEM-BEM法预测箱型振动构件的声辐射.华中理工大学学报,1993, 21(1):163-168.
[126] Y.K agawa, R.S himoyama, T.Yamabuchi. Boundary element models of the vocal tract and radiation field and their response characteristics.j ournal of Sound and Vibration, 1992, 157(3):385-403.
[127] C.Y.R.Ceng, A.F.Seybert, T.W.Wu. A multi domain boundary element solution for silencer muffler perdiction. Journal of Sound and Vibration, 1991, 151(1):119-129.
[128] S.M.Kirkup, D.J.Henwood. Computational solution of acoustic radiation problems by Kussmaul's boundary element method. Journal of Sound and Vibration, 1992, 158(2): 295-305.
[129]葛蕴珊,王芝秋,张志华.声边界元法在内燃机机体辐射噪声预报中的应用研究.内燃机学报, 1995, 13(1):78-83.
[130] D.E.Montgomery, R.L.West, R.A.Burdisso.Acoustic radiation prediction of a compressor housing from three-dimension experimental spatial dynamics modeling.Applied Acoustics, 1996, 47(2):165-185.
[131] A. Leissa. Vibration of plates [M].Acoustical Society of America, Woodbury, NY, 1993.
[132] C. Vanmaele, W. Desmet, D. Vandepitte. On the use of the wave based method for the steady-state dynamic analysis of three-dimensional plate assemblies. Proceedings of ISMA2004, Leuven, Belgium, 2004:1002-1013.
[133] Langley, R.S. Some perspectives on wave-mode duality in SEA.IUTAM symposium on statistical energy analysis, Southampton, 1997.
[134]曹国雄.弹性矩形薄板振动.北京:中国建筑工业出版社,1983.
[135] Atalla, N., Bernhard, R.J.Review of numerical solutions for lowfrequency structural-acoustic problems.Applied Acoustics, 1994, 43:271-294.