基于能量有限元法的板耦合结构振动特性分析
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摘要
相对于传统有限元法及统计能量分析方法,能量有限元法因其在中高频的普遍适用性,以及可预示结构中某特定点的振动特性等优点,已经成为研究振动及噪声不可或缺的方法。本文以国家自然科学基金“基于功率流有限元分析的复杂柔性振动体系的智能控制研究”(No.50805088)和汽车仿真与控制国家重点实验室开放基金“汽车驾乘室振动噪声的功率流有限元分析及主动控制研究’(No.20091110)为基本课题来源,以不同耦合板结构为基本研究对象,重点讨论了能量有限元法在其结构中的应用。本文研究的主要内容及创新点主要包括如下几个方面。
详细推导了薄板结构中弯曲、面内纵波及面内剪切波的控制方程,为后续章节建立能量平衡方程及计算能量传递系数提供最必要的基础条件。
将不同耦合结构、不同激励情况下三个坐标轴方向上的振动位移表达式进行了整合,使其在计算时仅需删减相对应的项,从而使表达式具有相当的适应性。详细讨论了不同激励情况下能量传递系数随入射角度的变化趋势,以及频率对能量传递系数的影响。
讨论了三种不同波型入射情况下耦合板结构能量密度分布情况,分析了激励频率及阻尼对能量密度分布的影响,并且讨论了在较低频率情况下面内波激励时能量有限元法的适用性问题,为确定该方法针对不同波型的频率适用范围提出了大胆的推测。讨论了弯曲激励情况下门型耦合板结构中弯曲波能量密度与总能量密度的差值,得出了在中间激励板上两者相差无几,两侧支撑板中面内波影响在频率不大情况下可以忽略的结论,这为计算时忽略面内波影响提供了最直接的依据。
Energy Finite Element Method has become an essential method using to predict mechanical vibration and noise, for its advantages such as ubiquity in frequencies ranging from medium to high, and the ability that it can predict the vibration responses at any point we want, compared to the Finite Element Method and Statistical Energy Analysis. It took some different coupled-plate structures as the subjects in this paper which is supported by the project of National Natural Science Foundation of Chinese (NSFC)"The Intelligent Control Research of Complicated Flexible Vibration System Based on Power Flow Finite Element Method"(No.50805088) and Foundation of State Key Laboratory of Automobile Dynamic Simulation at Jilin University "Active control of the vibration and noise of vehicle cabs based on power flow finite element analysis"(No.20091110). It focused on the research on vibration of coupled-plate structures using Energy Finite Element Method. The major contents of this paper are listed as follows.
The flexural and in-plane wave equations of thin plate, which can provide the theory basis for building the energy balance equations and energy transfer coefficients, have been deduced. And, the energy transfer coefficients of different coupled-plate structures with different incident angle have been derived. The results showed that there was waveform conversion at coupled lines. And the energy transfer coefficients changed with the incident angle. It obtained the coupled matrices of reinforced beam-plate coupled structures and L-shaped plates based on the energy transfer coefficients. The global matrix has been assembled in this paper. And the energy density of structures was calculated with the Matlab software.
This Energy Finite Element Method was successfully applied to some practical engineering structures. Through numerical simulations, the energy density and the intensity fields of flexural and in-plane waves of the structures were obtained.
详细推导了薄板结构中弯曲、面内纵波及面内剪切波的控制方程,为后续章节建立能量平衡方程及计算能量传递系数提供最必要的基础条件。
将不同耦合结构、不同激励情况下三个坐标轴方向上的振动位移表达式进行了整合,使其在计算时仅需删减相对应的项,从而使表达式具有相当的适应性。详细讨论了不同激励情况下能量传递系数随入射角度的变化趋势,以及频率对能量传递系数的影响。
讨论了三种不同波型入射情况下耦合板结构能量密度分布情况,分析了激励频率及阻尼对能量密度分布的影响,并且讨论了在较低频率情况下面内波激励时能量有限元法的适用性问题,为确定该方法针对不同波型的频率适用范围提出了大胆的推测。讨论了弯曲激励情况下门型耦合板结构中弯曲波能量密度与总能量密度的差值,得出了在中间激励板上两者相差无几,两侧支撑板中面内波影响在频率不大情况下可以忽略的结论,这为计算时忽略面内波影响提供了最直接的依据。
Energy Finite Element Method has become an essential method using to predict mechanical vibration and noise, for its advantages such as ubiquity in frequencies ranging from medium to high, and the ability that it can predict the vibration responses at any point we want, compared to the Finite Element Method and Statistical Energy Analysis. It took some different coupled-plate structures as the subjects in this paper which is supported by the project of National Natural Science Foundation of Chinese (NSFC)"The Intelligent Control Research of Complicated Flexible Vibration System Based on Power Flow Finite Element Method"(No.50805088) and Foundation of State Key Laboratory of Automobile Dynamic Simulation at Jilin University "Active control of the vibration and noise of vehicle cabs based on power flow finite element analysis"(No.20091110). It focused on the research on vibration of coupled-plate structures using Energy Finite Element Method. The major contents of this paper are listed as follows.
The flexural and in-plane wave equations of thin plate, which can provide the theory basis for building the energy balance equations and energy transfer coefficients, have been deduced. And, the energy transfer coefficients of different coupled-plate structures with different incident angle have been derived. The results showed that there was waveform conversion at coupled lines. And the energy transfer coefficients changed with the incident angle. It obtained the coupled matrices of reinforced beam-plate coupled structures and L-shaped plates based on the energy transfer coefficients. The global matrix has been assembled in this paper. And the energy density of structures was calculated with the Matlab software.
This Energy Finite Element Method was successfully applied to some practical engineering structures. Through numerical simulations, the energy density and the intensity fields of flexural and in-plane waves of the structures were obtained.
引文
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[2]Mickol, J D, and Bernhard, R J.1986, An Investigation of Energy Transmission due to Flexural Wave Propagation in Lightweight, Built-Up Structures, Report No.0353-4 HL 86-40, Ray W. Herrick Laboratories, Purdue University, West Lafayette, Indiana
[3]Lyon R H, Mdidanik G. Power Flow between Linearly Coupled Oscillators [J]. J Acoust. Soc. Am.,1962,34(5):623-639
[4]Lyon R H, Scharton T D. Vibrational Energy Transmission in a Three Element Structure [J]. Journal of Sound and Vibration,1965,38:253-261
[5]Lyon R H. Statistical Energy Analysis of Dynamical System:Theory and Application [M]. Cambridge:MIT press,1975
[6]Craik R J M. The Prediction of Sound Transmission through Building Using Statistical Energy Analysis [J]. Journal of Sound and Vibration,1982,82(4):505-516
[7]Davies H G. Random Vibration of Distributed Systems Strongly Coupled at Discrete Points [J]. J Acoust. Soc. Am.,1973,54(2):507-515
[8]Lotz R, Crandall S H. Prediction and Measurement of the Proportionality Constant in Statistical Energy Analysis of Structures [J]. J Acoust. Soc. Am.,1973,54(2):516-524
[9]Maidanik G. Response of Ribbed Panel to Reverberant Acoustic Field [J]. J Acoust. Soc. Am.,1962,34(6):809-826
[10]Newland D E. Calculation of Power Flow between Coupled Oscillators [J]. Journal of Sound and Vibration,1966(3):262-276
[11]Newland D E. Power Flow between a Class of Coupled Oscillators [J]. J Acoust. Soc. Am., 1968,43(3):553-559
[12]Scharton T D, Lyon R H. Power Flow and Energy Sharing in Random Vibration [J]. J Acoust. Soc. Am.,1968,43(6):1332-1343
[13]Smith P W Jr. Response and Radiation of Structural Modes Excited by Sound [J]. J Acoust. Soc. Am.,1962,34(5):640-647
[14]Smith P W Jr. Statistical Models of Coupled Dynamical Systems and Transition from Weak to Strong Coupling [J].J Acoust. Soc. Am.,1979,65 (3):695-698
[15]Fahy F J, Yao D Y. Power Flow between Non-conservatively Coupled Oscillators [J]. Journal of Sound and Vibration,1987,114(1):1-11
[16]Chen G, Soong T T. Power Flow and Energy Balance between Nonconservatively Oscillators [J]. Journal of Sound and Vibration,1991,113:448-454
[17]Sun J C, etal. Power Flow and Energy Balance of Non-conservatively Coupled Structures, I:theory [J]. Journal of Sound and Vibration,1987,112(2):321-330
[18]孙进才等.统计能量分析(SEA)研究的新进展[J].自然科学进展,1998,8(2)：129-136
[19]Lase Y, Jezequel L. Analysis of A Dynamic System Based on A New Energetic Formulation. International Congress on Intensity Techniques,1990. France:145-150P
[20]Wohlenver J C, Bernhand R J. Mechanical energy flow models of rods and beams [J]. Journal of Sound and Vibration.1992,153(1):1-19P
[21]Cho P E, Bernhand R J. Coupling of continuous beam models. Proceedings of Inter-noise 92,1992.Toronto:487-492P
[22]Bouthier O M, Bernhard R J. Models of space-averaged energetics of plates [J]. AIAA Journal.1992,30(3):616-622P
[23]Bouthier O M, Bernhard R J. Simple models of energy flow in vibrating membranes [J]. Journal of Sound and Vibration.1995,182(1):129-147P
[24]Bouthier O M, Bernhard R J. Simple models of energetics of transversely vibrating plates [J]. Journal of Sound and Vibration.1995,182(1):149-164P
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[26]Moens I., Vandepitte D., Sas.p.. Experimental validation of energy flow analysis on a two dimensional frame Proceedings ISMA 21 conference, Belgium,1996
[27]Beale L S, Accorsi M L. Power flow in two and three dimensional frame structures [J]. Journal of Sound and Vibration,1995,18(4):685-702
[28]Wang X U, Koss L L. Frequency response functions for structural intensity, Part Ⅰ:Theory [J]. J Sound and Vibration,1995,185(2):229-334
[29]Wang X U, Koss L L. Frequency response functions for power transfer ratio and coherence [J]. J Sound and Vibration,1992,155(1):55-73
[30]Stiehl A L. Power flow analysis of beam members with multiple wave types [J]. Finite Element in Analysis and Design.1996,21(3):253-264
[31]Moens I, ed. Study of the applicability of the energy finite element method for plates. Proceedings Novem,2000. Frace
[32]Park D H, Hong S Y. Power flow models and analysis of in-plane waves in finite coupled thin plates. Journal of Sound and Vibration,2001,244(4):651-668
[33]Seong-Hoon Seo, Suk-Yoon Hong, Hyun-Gwon Kil. Power flow analysis of reinforced beam-plate coupled structures [J]. Journal of Sound and Vibration,2003,259(5):1109-1129
[34]Weiguo Zhang etal. High-Frequency vibration analysis of thin elastic plates under heavy fluid loading by an energy finite element formulation[J].Journal of Sound and Vibration, 2003,263(1):21-46
[35]Weiguo Zhang etal. An energy finite element formulation for high-frequency vibration analysis of externally fluid-loaded cylindrical shells with periodic circumferential stiffeners subjected to ax-symmetric excitation[J].Journal of Sound and Vibration,2005, 282:679-700
[36]Ju-Bum Han etal. Energy flow model for thin plate considering fluid loading with mean flow[J]. Journal of Sound and Vibration,2012,331:5326-5346.
[37]Weiguo Zhang etal. An alternative energy finite element formulation based on incoherent orthogonal waves and its validation for marine structures [J].Finite Element in Analysis and Design,2002,38:1095-1113.
[38]A. Wang, N. Vlahopoulos etal. Energy Finite Element Analysis of the NASA Aluminum Test-Bed Cylinder. SAE Paper No.2005-01-2372,2005 SAE Noise and Vibration Conference, Traverse City, MI
[39]G. Zhang, and N. Vlahopoulos. Validation of an EFEA formulation for computing the vibrational response of complex structures. SAE Paper No.2007-01-2324, SAE 2007 Transactions Journal of Passenger Cars-Mechanical Systems
[40]N. Vlahopoulos etal. Structure-Acoustic Simulations of Naval Vehicles using an Energy Finite Element Method. MAST Americas,2010, June 2010, Crystal City, VA
[41]K.K. Choi, and J. Dong. Parametric Design Sensitivity Analysis of High Frequency Structural-Acousitc Problemes Using Energy Finite Element Method. ASME 2003 Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Chicago, Illinois, USA,2003, DETC2003/DAC-48753
[42]N.H. Kim eatl. Energy Flow Analysis and design Sensitivity of Structural Problems at High-Frequencies [J]. Journal of Sound and Vibration.2004,269:213-250
[43]Weiguo Zhang etal. High Frequency Vibro-Acoustic Analysis Using Energy Finite Element Method. SAE 2009 world congress, Detroit, Michigan, USA, SAE International, 2009-01-0771
[44]Jun Dong and Kyung K. Choi etal. Sensitivity Analysis and Optimization Using Energy Finite Element and Boundary Element Methods [J]. AIAA Journal,2007,45:1187-1198
[45]Miaoxia Xie etal. Prediction of High-frequency Vibro-acoustic Coupling in Anechoic Chamber Using Energy Finite Element Method and Energy Boundary Element Method [J]. Computer Modeling in Engineering & Sciences,2012,85:65-78
[46]W. Zhang etal. Sound Package Analysis using Energy Finite Element Method, Noise and Vibration. Conference and Exhibition, St. Charles, Illinois, USA, SAE International, 2007-01-2309
[47]Sung-Min Lee. Energy Finite Element Method for High Frequency Vibration Analysis of Composite Rotorcraft Structures [D]. Ph.D. Thesis, University of Michigan, USA,2010.
[48]Seyed-Javid Moravaeji. Modeling of Trim Panels in the Energy Finite Element Analysis[D].Ph.D. Thesis, University of Michigan, USA,2008
[49]J.-H. Song etal. Vibrational energy flow analysis of penetration beam-plate coupled structures [J]. Journal of Mechanical Science and Technology 2011,25 (3):567-576.
[50]Yong-Ho Park, and Suk-Yoon Hong. Vibrational energy flow analysis of corrected flexural waves in Timoshenko beam-Part Ⅰ:Theory of an energetic model [J]. Shock and Vibration.2006,13:137-165
[51]Yong-Ho Park, and Suk-Yoon Hong. Vibrational energy flow analysis of corrected flexural waves in Timoshenko beam-Part Ⅱ:Application to coupled Timoshenko beams [J]. Shock and Vibration.2006,13:167-196
[52]Young-Ho Park etal. Vibrational power flow models for transversely vibrating finite Mindlin plate [J]. Journal of Sound and Vibration.2008,317:800-840.
[53]曾勤谦.功率流有限元法及其在耦合结构中的应用[D].上海交通大学博士学位论文.1999
[54]孙丽萍.能量有限元法研究及其应用[D].哈尔滨工程大学博士学位论文.2004
[55]游进,孟光等.随机激励下框架结构能量有限元分析[J].上海交通大学学报.2009,Vo1.43，No.10：1632-1635
[56]游进,孟光等.耦合板结构随机能量有限元分析.振动与冲击.2009,Vo1.28,No.11：43-46
[57]游进,孟光等.L型耦合板相关激励下高频随机能量流分析[J].振动工程学报.2010,Vo1.23,No.1：60-63
[58]周振.基于能量有限元的柔性结构振动主动控制研究[D].山东大学硕士研究生学位论文.2011
[59]朱继清.基于能量流有限元的耦合结构振动传递特性分析[D].山东大学硕士研究生学位论文.2011
[60]郭浩男.耦合板结构的功率流有限元分析及在复杂加筋板结构中的应用[D].山东大学硕士研究生学位论文.2012
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[63]Michael Berling Skeen. An investigation of energy flow through coupled structures [D]. PhD. University of New South Wales, Australia,2008
[2]Mickol, J D, and Bernhard, R J.1986, An Investigation of Energy Transmission due to Flexural Wave Propagation in Lightweight, Built-Up Structures, Report No.0353-4 HL 86-40, Ray W. Herrick Laboratories, Purdue University, West Lafayette, Indiana
[3]Lyon R H, Mdidanik G. Power Flow between Linearly Coupled Oscillators [J]. J Acoust. Soc. Am.,1962,34(5):623-639
[4]Lyon R H, Scharton T D. Vibrational Energy Transmission in a Three Element Structure [J]. Journal of Sound and Vibration,1965,38:253-261
[5]Lyon R H. Statistical Energy Analysis of Dynamical System:Theory and Application [M]. Cambridge:MIT press,1975
[6]Craik R J M. The Prediction of Sound Transmission through Building Using Statistical Energy Analysis [J]. Journal of Sound and Vibration,1982,82(4):505-516
[7]Davies H G. Random Vibration of Distributed Systems Strongly Coupled at Discrete Points [J]. J Acoust. Soc. Am.,1973,54(2):507-515
[8]Lotz R, Crandall S H. Prediction and Measurement of the Proportionality Constant in Statistical Energy Analysis of Structures [J]. J Acoust. Soc. Am.,1973,54(2):516-524
[9]Maidanik G. Response of Ribbed Panel to Reverberant Acoustic Field [J]. J Acoust. Soc. Am.,1962,34(6):809-826
[10]Newland D E. Calculation of Power Flow between Coupled Oscillators [J]. Journal of Sound and Vibration,1966(3):262-276
[11]Newland D E. Power Flow between a Class of Coupled Oscillators [J]. J Acoust. Soc. Am., 1968,43(3):553-559
[12]Scharton T D, Lyon R H. Power Flow and Energy Sharing in Random Vibration [J]. J Acoust. Soc. Am.,1968,43(6):1332-1343
[13]Smith P W Jr. Response and Radiation of Structural Modes Excited by Sound [J]. J Acoust. Soc. Am.,1962,34(5):640-647
[14]Smith P W Jr. Statistical Models of Coupled Dynamical Systems and Transition from Weak to Strong Coupling [J].J Acoust. Soc. Am.,1979,65 (3):695-698
[15]Fahy F J, Yao D Y. Power Flow between Non-conservatively Coupled Oscillators [J]. Journal of Sound and Vibration,1987,114(1):1-11
[16]Chen G, Soong T T. Power Flow and Energy Balance between Nonconservatively Oscillators [J]. Journal of Sound and Vibration,1991,113:448-454
[17]Sun J C, etal. Power Flow and Energy Balance of Non-conservatively Coupled Structures, I:theory [J]. Journal of Sound and Vibration,1987,112(2):321-330
[18]孙进才等.统计能量分析(SEA)研究的新进展[J].自然科学进展,1998,8(2)：129-136
[19]Lase Y, Jezequel L. Analysis of A Dynamic System Based on A New Energetic Formulation. International Congress on Intensity Techniques,1990. France:145-150P
[20]Wohlenver J C, Bernhand R J. Mechanical energy flow models of rods and beams [J]. Journal of Sound and Vibration.1992,153(1):1-19P
[21]Cho P E, Bernhand R J. Coupling of continuous beam models. Proceedings of Inter-noise 92,1992.Toronto:487-492P
[22]Bouthier O M, Bernhard R J. Models of space-averaged energetics of plates [J]. AIAA Journal.1992,30(3):616-622P
[23]Bouthier O M, Bernhard R J. Simple models of energy flow in vibrating membranes [J]. Journal of Sound and Vibration.1995,182(1):129-147P
[24]Bouthier O M, Bernhard R J. Simple models of energetics of transversely vibrating plates [J]. Journal of Sound and Vibration.1995,182(1):149-164P
[25]Cho, P. E.. Energy flow analysis of coupled structures [D]. PhD. Thesis, Purdue University, USA,1993
[26]Moens I., Vandepitte D., Sas.p.. Experimental validation of energy flow analysis on a two dimensional frame Proceedings ISMA 21 conference, Belgium,1996
[27]Beale L S, Accorsi M L. Power flow in two and three dimensional frame structures [J]. Journal of Sound and Vibration,1995,18(4):685-702
[28]Wang X U, Koss L L. Frequency response functions for structural intensity, Part Ⅰ:Theory [J]. J Sound and Vibration,1995,185(2):229-334
[29]Wang X U, Koss L L. Frequency response functions for power transfer ratio and coherence [J]. J Sound and Vibration,1992,155(1):55-73
[30]Stiehl A L. Power flow analysis of beam members with multiple wave types [J]. Finite Element in Analysis and Design.1996,21(3):253-264
[31]Moens I, ed. Study of the applicability of the energy finite element method for plates. Proceedings Novem,2000. Frace
[32]Park D H, Hong S Y. Power flow models and analysis of in-plane waves in finite coupled thin plates. Journal of Sound and Vibration,2001,244(4):651-668
[33]Seong-Hoon Seo, Suk-Yoon Hong, Hyun-Gwon Kil. Power flow analysis of reinforced beam-plate coupled structures [J]. Journal of Sound and Vibration,2003,259(5):1109-1129
[34]Weiguo Zhang etal. High-Frequency vibration analysis of thin elastic plates under heavy fluid loading by an energy finite element formulation[J].Journal of Sound and Vibration, 2003,263(1):21-46
[35]Weiguo Zhang etal. An energy finite element formulation for high-frequency vibration analysis of externally fluid-loaded cylindrical shells with periodic circumferential stiffeners subjected to ax-symmetric excitation[J].Journal of Sound and Vibration,2005, 282:679-700
[36]Ju-Bum Han etal. Energy flow model for thin plate considering fluid loading with mean flow[J]. Journal of Sound and Vibration,2012,331:5326-5346.
[37]Weiguo Zhang etal. An alternative energy finite element formulation based on incoherent orthogonal waves and its validation for marine structures [J].Finite Element in Analysis and Design,2002,38:1095-1113.
[38]A. Wang, N. Vlahopoulos etal. Energy Finite Element Analysis of the NASA Aluminum Test-Bed Cylinder. SAE Paper No.2005-01-2372,2005 SAE Noise and Vibration Conference, Traverse City, MI
[39]G. Zhang, and N. Vlahopoulos. Validation of an EFEA formulation for computing the vibrational response of complex structures. SAE Paper No.2007-01-2324, SAE 2007 Transactions Journal of Passenger Cars-Mechanical Systems
[40]N. Vlahopoulos etal. Structure-Acoustic Simulations of Naval Vehicles using an Energy Finite Element Method. MAST Americas,2010, June 2010, Crystal City, VA
[41]K.K. Choi, and J. Dong. Parametric Design Sensitivity Analysis of High Frequency Structural-Acousitc Problemes Using Energy Finite Element Method. ASME 2003 Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Chicago, Illinois, USA,2003, DETC2003/DAC-48753
[42]N.H. Kim eatl. Energy Flow Analysis and design Sensitivity of Structural Problems at High-Frequencies [J]. Journal of Sound and Vibration.2004,269:213-250
[43]Weiguo Zhang etal. High Frequency Vibro-Acoustic Analysis Using Energy Finite Element Method. SAE 2009 world congress, Detroit, Michigan, USA, SAE International, 2009-01-0771
[44]Jun Dong and Kyung K. Choi etal. Sensitivity Analysis and Optimization Using Energy Finite Element and Boundary Element Methods [J]. AIAA Journal,2007,45:1187-1198
[45]Miaoxia Xie etal. Prediction of High-frequency Vibro-acoustic Coupling in Anechoic Chamber Using Energy Finite Element Method and Energy Boundary Element Method [J]. Computer Modeling in Engineering & Sciences,2012,85:65-78
[46]W. Zhang etal. Sound Package Analysis using Energy Finite Element Method, Noise and Vibration. Conference and Exhibition, St. Charles, Illinois, USA, SAE International, 2007-01-2309
[47]Sung-Min Lee. Energy Finite Element Method for High Frequency Vibration Analysis of Composite Rotorcraft Structures [D]. Ph.D. Thesis, University of Michigan, USA,2010.
[48]Seyed-Javid Moravaeji. Modeling of Trim Panels in the Energy Finite Element Analysis[D].Ph.D. Thesis, University of Michigan, USA,2008
[49]J.-H. Song etal. Vibrational energy flow analysis of penetration beam-plate coupled structures [J]. Journal of Mechanical Science and Technology 2011,25 (3):567-576.
[50]Yong-Ho Park, and Suk-Yoon Hong. Vibrational energy flow analysis of corrected flexural waves in Timoshenko beam-Part Ⅰ:Theory of an energetic model [J]. Shock and Vibration.2006,13:137-165
[51]Yong-Ho Park, and Suk-Yoon Hong. Vibrational energy flow analysis of corrected flexural waves in Timoshenko beam-Part Ⅱ:Application to coupled Timoshenko beams [J]. Shock and Vibration.2006,13:167-196
[52]Young-Ho Park etal. Vibrational power flow models for transversely vibrating finite Mindlin plate [J]. Journal of Sound and Vibration.2008,317:800-840.
[53]曾勤谦.功率流有限元法及其在耦合结构中的应用[D].上海交通大学博士学位论文.1999
[54]孙丽萍.能量有限元法研究及其应用[D].哈尔滨工程大学博士学位论文.2004
[55]游进,孟光等.随机激励下框架结构能量有限元分析[J].上海交通大学学报.2009,Vo1.43，No.10：1632-1635
[56]游进,孟光等.耦合板结构随机能量有限元分析.振动与冲击.2009,Vo1.28,No.11：43-46
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