基于关键时间点的能量等效静态载荷法及结构动态响应优化研究
|
摘要
传统的结构优化设计是静态载荷下的优化设计，而工程实际中的机械结构普遍承受动态载荷作用，因此静态优化设计的结果已不能满足结构动态性能的要求。而直接进行动态响应优化设计时，由于与时间相关的动态约束处理非常困难，往往导致计算规模大且难以收敛，所以该方法在工程实际应用中尚不可行。因此，人们考虑将动态载荷等效转化为静态载荷。目前基于位移的等效静态载荷法，是通过位移场等效将动态载荷转化为静态载荷，进而将结构动态响应优化问题转化为静态响应优化问题，但该方法是以最大位移点为等效转化时间点，没有考虑最大应力点时刻的动态响应，不能全面反映结构的动态性能，而等效转化关键时间点难以识别。同时基于位移等效静态载荷法存在计算复杂、优化效率低的问题，导致工程实用性差。另外，目前的结构动态响应优化是以位移响应为基础的，忽略了应力响应对结构动态性能的影响。
论文针对目前结构动态载荷等效静态转化时关键时间点难以识别的问题，提出了一种动态响应解空间谱元离散的关键时间点识别方法。应用谱元法离散以及在高斯-勒让德-罗巴托(GLL)点插值的高精度优势，将模态叠加法计算得到的结构动态响应（包括应力响应和位移响应）解空间进行谱元离散，并在GLL点进行Lagrange插值，得到了动态响应时间历程的高精度插值函数，构建了动态载荷作用下结构关键时间点识别的数学模型，并利用全局优化算法（DIRECT法）得到结构动态响应的绝对极大值点，即关键时间点，并以124杆平面桁架和悬臂梁为算例，验证了本文方法的有效性。
论文针对基于位移等效静态载荷法的优化计算复杂、工程实用性差的问题，提出了一种基于能量等效的动态载荷等效静态转化方法。基于载荷等效前后结构能量守恒的原理，通过在关键时间点构建动态载荷等效静态转化的数学模型，并采用全局优化算法（DIRECT法）搜索得到了关键时间点的等效载荷集。同样以124杆平面桁架为算例进行分析，验证了本文方法的有效性。
论文针对基于位移等效静态载荷法在进行结构动态优化时，单纯考虑位移响应的不足，提出了一种基于系统响应的结构动态优化设计方法。在构建结构优化模型时，同时考虑动态载荷作用的结构位移响应和应力响应，将分别从最大应力、最大位移角度识别的关键时间点等效转化得到的静态载荷集，作为结构动态优化的外载荷条件，进行结构优化设计。并以圆形中空截面梁、3杆桁架结构、10杆桁架结构等算例为研究对象进行验证，通过与文献分析对比，表明本文得到的优化结果更加精确。
论文以某柴油机活塞为应用对象，结合试验结果修正了活塞的热边界条件，利用有限元法仿真分析得到了活塞各个节点的温度；开展了活塞热机耦合静力学分析和瞬态动力学分析，得到了活塞动态响应解空间，分别应用本文提出的关键时间点识别方法和基于能量原理的等效载荷法进行活塞在关键时间点的动态载荷等效静态转化，并应用基于系统响应的结构优化设计方法，实现了活塞的动态响应优化设计，取得了良好的效果。
本文研究了基于能量原理的等效静态载荷法及其在结构动态响应优化中的应用，在载荷等效转化关键时间点识别、基于能量原理的动态载荷等效静态转化、基于系统响应的结构动态响应优化设计三方面取得了突破，获得了一系列具有理论意义和工程实用价值的结论与成果，拓展了等效静态载荷法的理论研究范畴，为动态响应优化相关理论研究开辟了新的途径。
Traditional structural optimization is carried out under static load, while mechanicalstructure of engineering practice was generally under locally dynamic load, therefore theresults based on static structure optimization design method cannot meet the requirements ofthe structural dynamic performance. Due to the time-related dynamic constraint of the directdynamic response optimization technology was difficult to deal with, the direct optimizationcalculation of dynamic response is large scale and difficult to converge, so it is not feasiblefor structural dynamic response optimization design directly in engineering applications.Therefore, it is considered to transform dynamic load into equivalent static load. At present,the Equivalent Static Load method based on displacement, transforms dynamic load intostatic load via equivalent displacement, and further transforms the structure dynamic responseproblem into structure static response problem. This method takes the maximumdisplacement point as the equivalent transformation time point, without considering thedynamic response of the maximum stress time point, and cannot fully reflect the dynamicperformance of the structure, and the equivalent transformation critical time point is difficultto identify. Whatever, the equivalent static load method based on displacement is short incomplex computation and insufficient optimization inefficient, which leads to poorpracticability? In addition the current dynamic response optimization is based ondisplacement, ignoring the effects of stress response on the structural dynamic performance.
Aiming at the problem that the critical time point at which the structure dynamic loadequivalent transformation is taken was difficult to identify, a method based on discretizingthe solution space of the dynamic response by the spectral element method to identify the critical time point is presented. The paper utilizes the discrete by spectral element method andthe high-precision of the Gauss–Legendre–Lobatto (GLL) point interpolation, calculatesthe spectral element discrete solution space of the structure dynamic response (stressresponse and displacement response included) resulting from the modal superposition methodcalculation, and obtain a high-precision interpolation function of dynamic response timehistory, by Lagrange interpolation at GLL point, constructed a mathematical model toidentify the structure of the critical time point under dynamic load, and get the absolutemaximum structure dynamic response value point by global optimization algorithm (DIRECTmethod), called critical time point. Finally, a124-plane truss and cantilever beam were takenas examples to demonstrate the effectiveness of this method.
Aiming at the complex optimization calculation and poor practicability of the EquivalentStatic Load method based on displacement, a method in which the dynamic load istransformed into static based on energy principle is presented. This method is based on theconservation of the energy before and after the load equivalent, constructs a mathematicalmodel of the transformation of the dynamic load into static load, and obtains the equivalentstatic load set by global optimization algorithm (DIRECT method) eventually. The124planetruss was analyzed as an example, to confirm the validity of this method.
Aiming at the insufficiency of simply considering the displacement response when thestructure is dynamic optimized by using the Equivalent Static Load method based ondisplacement, a structure dynamic optimized design method based on system response ispresented. The structure is optimized with the displacement response and the stress responsewhich are generated when the structure is under dynamic load, as constraint functions of theoptimization model, and the static load set, which is generated by equivalently transformationat critical time point identified from the point of stress and displacement, as the external loadcondition of structure dynamic optimization. A circular hollow section beam, a3-trussstructure, and a10-truss structure were taken as examples and compared with the literatures.The results show that, the optimization result obtained by this method is more accurate.
A diesel engine piston was used as the application study in this paper. The thermalboundary condition of the piston was corrected by the test results, and the temperature ofeach node of the piston was got by the finite element method. The dynamic response solutionspace of the piston was got by the thermo-mechanical coupling static analysis and thetransient kinetic analysis of the piston. The method to identify the critical time point and theEquivalent Load method based on energy principle presented in this paper were used totransform the dynamic load into static load equivalently at the critical time point, and then thestructure optimization design method based on system response was applied. Theoptimization of the dynamic response of the piston was achieved eventually, and achievedgood results.
By the investigation of the energy principle based Equivalent Static Load method and itsapplication for the dynamic response optimization in the paper, breakthroughs in the areas ofthe identification of critical time point of the transformation of equivalent load, thetransformation of dynamic load into static load based on energy principle and the structuredynamic response optimization design based on system response, are made, and a series ofconclusion and results with theoretical significance and practical value are received. Thepaper expands the scope of the theoretical study of Equivalent Load method, and opens a newway of theoretical study on dynamic response optimization theory.
论文针对目前结构动态载荷等效静态转化时关键时间点难以识别的问题，提出了一种动态响应解空间谱元离散的关键时间点识别方法。应用谱元法离散以及在高斯-勒让德-罗巴托(GLL)点插值的高精度优势，将模态叠加法计算得到的结构动态响应（包括应力响应和位移响应）解空间进行谱元离散，并在GLL点进行Lagrange插值，得到了动态响应时间历程的高精度插值函数，构建了动态载荷作用下结构关键时间点识别的数学模型，并利用全局优化算法（DIRECT法）得到结构动态响应的绝对极大值点，即关键时间点，并以124杆平面桁架和悬臂梁为算例，验证了本文方法的有效性。
论文针对基于位移等效静态载荷法的优化计算复杂、工程实用性差的问题，提出了一种基于能量等效的动态载荷等效静态转化方法。基于载荷等效前后结构能量守恒的原理，通过在关键时间点构建动态载荷等效静态转化的数学模型，并采用全局优化算法（DIRECT法）搜索得到了关键时间点的等效载荷集。同样以124杆平面桁架为算例进行分析，验证了本文方法的有效性。
论文针对基于位移等效静态载荷法在进行结构动态优化时，单纯考虑位移响应的不足，提出了一种基于系统响应的结构动态优化设计方法。在构建结构优化模型时，同时考虑动态载荷作用的结构位移响应和应力响应，将分别从最大应力、最大位移角度识别的关键时间点等效转化得到的静态载荷集，作为结构动态优化的外载荷条件，进行结构优化设计。并以圆形中空截面梁、3杆桁架结构、10杆桁架结构等算例为研究对象进行验证，通过与文献分析对比，表明本文得到的优化结果更加精确。
论文以某柴油机活塞为应用对象，结合试验结果修正了活塞的热边界条件，利用有限元法仿真分析得到了活塞各个节点的温度；开展了活塞热机耦合静力学分析和瞬态动力学分析，得到了活塞动态响应解空间，分别应用本文提出的关键时间点识别方法和基于能量原理的等效载荷法进行活塞在关键时间点的动态载荷等效静态转化，并应用基于系统响应的结构优化设计方法，实现了活塞的动态响应优化设计，取得了良好的效果。
本文研究了基于能量原理的等效静态载荷法及其在结构动态响应优化中的应用，在载荷等效转化关键时间点识别、基于能量原理的动态载荷等效静态转化、基于系统响应的结构动态响应优化设计三方面取得了突破，获得了一系列具有理论意义和工程实用价值的结论与成果，拓展了等效静态载荷法的理论研究范畴，为动态响应优化相关理论研究开辟了新的途径。
Traditional structural optimization is carried out under static load, while mechanicalstructure of engineering practice was generally under locally dynamic load, therefore theresults based on static structure optimization design method cannot meet the requirements ofthe structural dynamic performance. Due to the time-related dynamic constraint of the directdynamic response optimization technology was difficult to deal with, the direct optimizationcalculation of dynamic response is large scale and difficult to converge, so it is not feasiblefor structural dynamic response optimization design directly in engineering applications.Therefore, it is considered to transform dynamic load into equivalent static load. At present,the Equivalent Static Load method based on displacement, transforms dynamic load intostatic load via equivalent displacement, and further transforms the structure dynamic responseproblem into structure static response problem. This method takes the maximumdisplacement point as the equivalent transformation time point, without considering thedynamic response of the maximum stress time point, and cannot fully reflect the dynamicperformance of the structure, and the equivalent transformation critical time point is difficultto identify. Whatever, the equivalent static load method based on displacement is short incomplex computation and insufficient optimization inefficient, which leads to poorpracticability? In addition the current dynamic response optimization is based ondisplacement, ignoring the effects of stress response on the structural dynamic performance.
Aiming at the problem that the critical time point at which the structure dynamic loadequivalent transformation is taken was difficult to identify, a method based on discretizingthe solution space of the dynamic response by the spectral element method to identify the critical time point is presented. The paper utilizes the discrete by spectral element method andthe high-precision of the Gauss–Legendre–Lobatto (GLL) point interpolation, calculatesthe spectral element discrete solution space of the structure dynamic response (stressresponse and displacement response included) resulting from the modal superposition methodcalculation, and obtain a high-precision interpolation function of dynamic response timehistory, by Lagrange interpolation at GLL point, constructed a mathematical model toidentify the structure of the critical time point under dynamic load, and get the absolutemaximum structure dynamic response value point by global optimization algorithm (DIRECTmethod), called critical time point. Finally, a124-plane truss and cantilever beam were takenas examples to demonstrate the effectiveness of this method.
Aiming at the complex optimization calculation and poor practicability of the EquivalentStatic Load method based on displacement, a method in which the dynamic load istransformed into static based on energy principle is presented. This method is based on theconservation of the energy before and after the load equivalent, constructs a mathematicalmodel of the transformation of the dynamic load into static load, and obtains the equivalentstatic load set by global optimization algorithm (DIRECT method) eventually. The124planetruss was analyzed as an example, to confirm the validity of this method.
Aiming at the insufficiency of simply considering the displacement response when thestructure is dynamic optimized by using the Equivalent Static Load method based ondisplacement, a structure dynamic optimized design method based on system response ispresented. The structure is optimized with the displacement response and the stress responsewhich are generated when the structure is under dynamic load, as constraint functions of theoptimization model, and the static load set, which is generated by equivalently transformationat critical time point identified from the point of stress and displacement, as the external loadcondition of structure dynamic optimization. A circular hollow section beam, a3-trussstructure, and a10-truss structure were taken as examples and compared with the literatures.The results show that, the optimization result obtained by this method is more accurate.
A diesel engine piston was used as the application study in this paper. The thermalboundary condition of the piston was corrected by the test results, and the temperature ofeach node of the piston was got by the finite element method. The dynamic response solutionspace of the piston was got by the thermo-mechanical coupling static analysis and thetransient kinetic analysis of the piston. The method to identify the critical time point and theEquivalent Load method based on energy principle presented in this paper were used totransform the dynamic load into static load equivalently at the critical time point, and then thestructure optimization design method based on system response was applied. Theoptimization of the dynamic response of the piston was achieved eventually, and achievedgood results.
By the investigation of the energy principle based Equivalent Static Load method and itsapplication for the dynamic response optimization in the paper, breakthroughs in the areas ofthe identification of critical time point of the transformation of equivalent load, thetransformation of dynamic load into static load based on energy principle and the structuredynamic response optimization design based on system response, are made, and a series ofconclusion and results with theoretical significance and practical value are received. Thepaper expands the scope of the theoretical study of Equivalent Load method, and opens a newway of theoretical study on dynamic response optimization theory.
引文
[1]杨姝.复杂机械结构拓扑优化若干问题研究[D]，大连：大连理工大学，2009
[2]杜平安.MCAE计算机辅助机械工程[M]．北京：机械工业出版社，1996
[3]李佳.计算机辅助设计与制造(CAD/CAM)[M]，天津：天津大学出版社，2002
[4]源清,肖文，简析九十年代主流CAD造型基础技术[J]，计算机辅助设计与制造，1998(2):1-5
[5]罗浩,张新访.基于约束的参数化设计技术发展现状及前景[J]，中国机械工程，1995(5)21-24
[6]王贤坤.机械CAD／CAM技术应用与开发[M]．北京：机械工业出版社，2001
[7] Raymond H.Kurland, Understanding Variable-Driven Modeling,http://www.technicom.com.1994
[8] D.H.Brown Associates， Inc.SDRC’s Variational Analysis： A Design AnalysisBreakthrough. http://www.Dhbrown.com
[9]谭建国.使用ANSYS6.0进行有限元分析[M]，北京：北京大学出版社，2002
[10]张学玲,唐毅等.用实验模态与有限元方法识别结合面接触刚度的方法[J]，组合机床与自动化加工技术,2005(11):56-59
[11]冯培恩，邱清盈等．机械产品的广义优化设计进程研究[J]．中国科学，1999(E)，29(4)：338-346
[12]Maxwell C S．Scientific papers lI[M]．New York：Dover Publications，1952
[13]Michell A GM．The limits of economy of material in flame structures．PhilosophicalMagazine，1904
[14]Shanley F R．Weight strength Analysis of Air craft Structures．New York：McGraw—Hill，1952
[15]秦东晨，祁建中，张明成等，液压机横梁结构的优化设计[J]．锻压技术，2004(2)：49-52
[16]李泉永．机械结构优化设计的回顾与展望[J]．桂林电子工业学院学报,2000(4)：114-119
[17]Schmit L.A.，Farshi B.，Some approximation concepts for structural synthesis [J]，Journal of AIAA，1974，12：692-699．
[18]Schmit, L.A., Miura, H. Approximation concepts for efficient structural synthesis. NASACR-2552,1976
[19]L. A. Schmit, Jr., H. Miura, Approximation concepts for efficient structural synthesis,Tech. Report CR–2552, NASA,1976
[20]余俊,周济著.优化方法程序库OPB-2—原理及应用(第一版)[M].武汉:华中理工大学出版社,1997
[21]程耿东,顾元宪．序列二次规划在结构动力优化中的应用[J]．振动与冲击，1986，5(1)：12-20．
[22]K. A. High, R. D. LaRoche. Parallel nonlinear optimization techniques for chemicalprocess design problems. Computers&Chemical Engineering,1995,19(6-7):807-825
[23]陈忠.关于非线性规划问题的并行算法[J].长江大学学报(自科),2006,4(12):1-8
[24]童卫华,姜节胜,顾松年．一种以均方响应作为约束的动力学设计方法[J].应用力学学报,1996,13(3)：94-98．
[25]李强，周济，钟毅芳．机械系统动态优化设计的复合遗传算法[J]．1999，35(5)：27-30．
[26]唐文艳.结构优化中的遗传算法研究和应用[D],大连：大连理工大学,200201
[27]Pantelides C.P., Tzan S．R． Optimal design of dynamically constrained structures [J].Computers&Structures,1997,62(1):141-150
[28]Y. M. Xie, G. R. Steven.A Simple Evolutionary Procedure for StructuralOptimization．Computers＆Structures,l993,49:885-896
[29]Y M．Xie．G P．Steven．Optimal Design of Multiple Load Case Structures Using anEvolutionary Procedure．Engineering Computations,1994(11):295-302
[30]O.M．Querin，G．P. Steven，Y M．Xie．Evolutionary Structural Optimization (ESO)Using Bi-directional Algorithm．Engineering Computations,1998(15)：103l-1048
[31]杨灿军，陈鹰，路用祥．基于人机一体化思想的企业MIS[J]．系统工程理论与实践，1997,(10):73-77
[32]Li Quanyong, Zhou Wei．A Nonlinear Programming for Multi-objective Fuzzy OptimalDesign. Journal of Guilin Institute of Electronic Technology, Mar.1999，V01.19,No.1(80-84)
[33]刘小平等，机械结构动力学模型的模糊优化修正方法[J],振动工程学报,1995(9)
[34]H Tanaka, K Asai．Fuzzy Linear Programming Problems with Number[J]．Fuzzy Sets andSystems,1984,13(1):1-10
[35]王光远，陈树勋.工程结构系统软设计理论及其应用[M].北京:国防工业出版社，1996
[36]Richardson M．H．，Parameter estimate from frequency response measurements using therational fraction polynomial method, Proc．of the1st IMAC，1982
[37]JONES D R. Direct global optimization algorithm direct global optimization algorithm
[M]//Encyclopedia of Optimization，US：Springer，2001：431-440
[38]JONES D R，PERTTUMEN C D，STUCKMAN B E. Lipschitzian optimization withoutthe Lipschitz constant [J]. Journal of Optimization Theory and Applications,1993,79(1)：157-181
[39]卢熹.机械结构优化方法及应用研究[D],南京:东南大学,2004(8)
[40]张学玲,唐毅,徐燕申.用实验模态与有限元方法识别结合面接触刚度的方法[J],组合机床与自动化加工技术,2005,(11):56-59
[41]邓君.YC-6K型柴油机活塞故障分析及结构优化[D],武汉:武汉理工大学,2011(4)
[42]黄震,张继春.基于有限元的活塞优化设计分析[J],昆明理工大学学报(理工版),2006(6):76-80
[43]单峰,朱俊虎等.基于灵敏度分析的载货汽车车架结构优化[J],汽车技术,2013(1)
[44]Q.Hao，Y Sun．Car Door Optimization Based on Kriging Model[C]．Taiyuan：2010International Conference on Computer Application and System Modeling,2010
[45]Kane, Couro, Jouve, Franeois, Sehoenauer, Mare.Topology optimization in1inear andnonlinear elasticity, using genetic algorithms. American Society of MechanicalEngineers,Design Engineering Division (Publication) DE,1995(9):385-392．
[46]刘俊栋.基于HyperWorks柴油机结构优化的研究[D],天津:天津大学,2007(1)
[47]霍心达,郑明军.基于拓扑优化的某活塞轻量化分析[J],石家庄铁道大学学报(自然科学版),2013(3):87-91
[48]李鹏.内燃机关键件拓扑优化技术研究[D],太原:中北大学,2012(5)
[49]Niordson F.I. The optimum design of a Vibrating beams [J]. Quarterly of AppliedMathematics,1965,23(1):47-53
[50]M．S．Zarghamee．Optimum Frequency of Structures．AIAA Journal，1968，6(6)749-750．
[51]钱令希.工程结构优化设计[M].北京:水利电力出版社,1983
[52]Xie Y M.，Steven G P．Evolutionary Structural Optimization Springer-Verlag．Berlin：Germa.1997
[53]顾亮,胡琪琪.结构动力修改的拟力修改法[J],北京理工大学学报,1997,17(5):533-538
[54]J．R．Bakera，K．E．Rouch．Use of finite element structural models in analyzing machinetool chatter．Finite E1ements in Analysis and Design.2002,38:1029-1046
[55]陈新,孙庆鸿.基于动力学特征的磨床床身优化布局设计[J].制造技术与机床,2001(2)：21-22
[56]张宗兰.能量平衡法在机床结构动态参数优化中的应用[J].哈尔滨工业大学学报，1987,(4):37-43
[57]宋宝安.柴油机低噪声结构动态设计方法的研究[D]天津:天津大学学位论文,2007(7)
[58]Cheu T.C.，Wang B.P.，Chen T.Y．Design optimization of gas turbine blades withgeometry and natural frequency [J]. Computers&Structures，1989,32(1)：113-117
[59]姚英姿,樊俊星,莫云辉,邓召义.基于模糊理论和遗传算法的航空齿轮可靠性优化设计[J].上海大学学报(自然科学版),2004,10(4):363-366
[60]宁嘉.基于有限元法的直升机机体动力学建模与优化研究[D],南京:南京航空航天大学2013(3)
[61]G.J. Park. Analytic methods for design practice [M]. London.Springer,2007
[62]Byung-Soo Kang·Gyung-Jin Park·Jasbir S. Arora. A review of optimization ofstructures subjected to transient loads. Struct Multidisc Optim.2006(31):81-95
[63]Venkayya V.B., Knot N.S., Keddy V.S. Energy distribution in an optimum structuraldesign, AFFDL-TR-68-156,1969.
[64]Berke L., Khot N.S. Use of optimality criteria methods for large-scale system. AGRADLecture Series,1974, No.70.
[65]Kiusalaas J., Shaw R.C. An algorithm for optimal structural design with frequencyconstraints. Int J Num Eng,1978,13:283-295.
[66]林家浩.有频率禁区的结构优化设计[J].大连工学院学报,1981,20(1):27-36.
[67]宋海平,周传荣.模糊环境下多目标结构动力优化设计.应用力学学报,1998,15(3):139-142
[68]陈建军,车建文,崔明涛等.结构动力优化设计评述与展望.力学进展,2001,31(2):181-192
[69]Kang B-S, Park G-J, Arora JS. A review of optimization of structures subjected totransient loads. Struct Multidisc Optim,2006,31(2):81–95
[70]Qian Wang, Jasbir S. Arora. Alternative formulations for transient dynamic responseoptimization.46th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics&Materials Conference18-21, Austin, Texas,2005(8)
[71]Dong-Hoon Choi, Hong-Soo Park, Min-Soo Kim.A direct treatment of min-max dynamicresponse optimization problems. AIAA-93-1352-CP
[72]Chung-Ho Wang. Dynamic multi-response optimization using principal componentanalysis and multiple criteria evaluation of the grey relation model. Int J Adv Manuf Technol,2007(32):617-624
[73]J. Bausa, G. Tsatsaronis. Dynamic optimization of startup and load-Increasing processesin power plants D Part I: Method. Transactions of the ASME.2001,123(1):246-250
[74]Park S, Kapania RK, Kim SJ. Nonlinear transient response and second-order sensitivityusing time finite element method. AIAA J,1999,37(5):613-622
[75]Choi W S，Park G J． Structural Optimization Using Equivalent Static Loads at All TimeIntervals [J]. Computer Methods in Applied Mechanics and Engineering,2002,191(19-20):2105-2122
[76]Grandhi RV, Haftka RT, Watson LT. Design oriented identification of critical time’sintransient response. AIAA Pap,1984:1984-0899
[77]Tenk L.H., Hagiwara I. Static and vibration shapes&topology optimization usinghomogenization and mathematical programming. Comput Meth in Appl Meth&Eng,1993(109):143-154.
[78]Kapoor M. P. Kumarasamy K. Optimum configuration of transmission towers indynamic response regime. In: Proceedings International Symposium on Optimum StructuralDesign, Tucson, Ariz, Oct.1981.
[79]Rao S.S. Optimization of airplane wing structures under gust loads. Computers&Structures,1985,21(3):741-749.
[80]Lin J.H. Structural optimization on genetrical configuration and element sizing withstatic and dynamically constraints. Computers&Structures,1982,15(5):507-526.
[81]Arvind U. Raghunathan, Dharmashankar Subramanian, Lorenz T. Biegler, Tariq Samad.Dynamic optimization strategies for three-dimensional conflict resolution of Multiple Aircraft.Journal of guidance, control, and dynamics.,2004,27(4):586-594
[82]Y.A. Khulief，H.A1-Naser． Finite element dynamic analysis of drill strings [J]. FiniteElement in Analysis and Design，2005(41)：1270-1288．
[83]郭志全.基于有限元分析的机械产品结构动态设计及其工程应用[D],天津：天津大学学位论文，2006(12)
[84]高卫娟.船用柴油机机体动态应力分析方法研究[D]，武汉：武汉理工大学，2010(5)
[85]毛虎平.基于仿真模型的动态响应优化算法研究[D]，武汉：华中科技大学，2011()
[86]James M L. Vibration of Mechanical and Structural Systems (2nd ed)[M]. New York:Harper Collins,1994.
[87]Willmert KD, Fox RL (1972) Optimum Design of a Linear Multidegree-of-freedomShock Isolation System. Journal of Engineering for Industry, Transactions of the ASME94:465-471.
[88]Afimiwala KA, Mayne RW (1974) Optimal Design of an ImpactAbsorber. Journal of Engineering for Industry, Transactions of the ASME96:124-130.
[89]Fox RL, Kapoor MP (1970) Structural Optimization in the Dynamic Regime: AComputational Approach. AIAA Journal8:1798-1804.
[90]毛虎平，苏铁熊等.多元模型自适应与时间谱元法结合的动态优化[J]，计算机辅助设计与图形学学报，2013（11）：1725-1734
[91]毛虎平，吴义忠等.时间谱元法在动态响应优化中的应用[J]，振动工程学报，2013（6）：95-403
[92]Chen X., Kareem A. Equivalent Static Wind Loads for Buffeting Response of Bridge.Journal of Structural Engineering,2001,127(12):1467-1475.
[93]Wen H.M., Reddy T.Y. Reid S.R. Deformation and Failure of Clamped Beams underLow Speed Impact Loading. International Journal of Impact Engineering,1995,16(3):435-454.
[94]Letchford C.W., Killen G.P. Equivalent Static Wind Loads for Cantilevered GrandstandRoofs. Engineering Structures,2002,23:207-217.
[95]Zhang Q.L., Vrouwenvelder A., Wardenier, J. Dynamic Amplification Factors and EUDLof Bridges under Random Traffic Flows. Engineering Structures,2001(23):663-672.
[96]Kang B.S., Choi W.S. Park G.J. Structural Optimization under Equivalent Static LoadsTransformed from Dynamic Loads Based on Displacement. Computers&Structures,2001(79)2:145-154.
[97]Choi W.S., Park G.J. Structural Optimization Using Equivalent Static Loads at All theTime Intervals. Computer Methods in Applied Mechanics and Engineering,2002,191:2105-2122.
[98]Choi W.S., Park G.J., Shin M.J. Transformation of a Dynamic Load into an EquivalentStatic Load and shape optimization of the Road Arm. KSME Fall Conference，1995(1):609-614.
[99]Choi W.S., Kang S.C., Shin M.J. Transformation of a Dynamic Load into an EquivalentStatic Load and Shape Optimization of the Road Arm in Self-Propelled Howitzer.Transactions of the KSME A,1996,20(12):3767-3781.
[100] Shin M.J., Choi W.S., Park G.J. Transformation of a dynamic load into an equivalentstatic load and shape optimization of the road arm. PACAM V Conference, Puerto Rico,1997.
[101] Kim M.S., Choi D.H. Multibody Dynamic Response Optimization with ALM andApproximate Line Search. Multibody System Dynamics, Kluwer Academic Publishers,1997(1):47-64.
[102] Choi W.S. Transformation of dynamic loads into equivalent static loads andstructural optimization. Ph.D. Dissertation. Hanyang: Hanyang University,2002.
[103] Choi W.S., Park G.J. Structural Optimization Using Equivalent Static Loads at AlltheTime Intervals. Computer Methods in Applied Mechanics and Engineering,2002,191(19/20):2105-2122.
[104] Park G.J., Kang B.S. Validation of a Structural Optimization AlgorithmTransforming Dynamic Loads into Equivalent Static Loads. Journal of Optimization Theoryand Applications,2003,119(1):191-200.
[105] CHOI W S，PARK G J. Transformation of dynamic loads into equivalent static loadsbased on modal analysis [J]. Int. J. Numer. Meth. Engng.，1999，46：29-43.
[106] Kang B.S. On Optimality of Solution Obtained by Equivalent Static Load Methodand Its Application to Flexible Multibody Dynamic Systems. Ph.D. Dissertation. Hanyang:Hanyang University,2002.
[107]顾慧芝,聂君剑.载荷识别的等效点动标定法[J]，振动工程学报，1997(9)
[108]苏义脑,唐雪平.初弯曲纵横弯曲梁的等效载荷法及其应用[J]，力学与实践200401
[109]黄武龙.基于等效静态载荷方法的大型复杂结构的轻量化设计[D]，广州：广东工业大学，2013(5)
[110]杨志军.基于等效静态载荷原理的高速机构结构拓扑优化方法[J]．机械工程学报.2011,9(47)：119-126．
[111]赵礼辉.ESL法在汽车结构优化设计中的应用[D]，上海：上海交通大学，2009(2)
[112]芮强，王红岩.基于等效静态载荷法的结构动态优化[J],汽车工程2014(1)
[113]肖登红，王俊元.基于等效静态载荷法的平面四杆机构的结构动态优化设计[J]，机械设计与制造2011(4)
[114]刚宪约，张帆.基于载荷等效法的空气悬架客车有限元建模方法[J],山东理工大学学报(自然科学版)201109
[115]贺新峰，基于等效静态载荷和响应面法的搅拌车副车架疲劳设计[D]，湖南大学201204
[116]张培信.能量理论结构力学[M],上海科学技术出版社，2010（1）:1-3
[117] T.R.陶丘得著，李世昌译.结构力学能量原理[M],人民交通出版社，1984（12）：33-39
[118]刘晶波杜修力.结构动力学[M],机械工业出版社，2012（6）：15-20
[119] S.P.Timoshenko（美），D.H.Young，叶红玲，杨庆生等译.结构理论[M]，机械工业出版社2005（5）：103-105
[120]张洪信，管殿柱.有限元基础理论与ANSYS11.0应用[M],机械工业出版社，2012（7）：46-52
[121]王海波，陈伯望等.结构动力方程Newmark-β方法递推简化分析[J];四川大学学报(工程科学版),2008(5)
[122] CLOUGH R W. Dynamics of structures [M]. New York：McGraw-Hill，1993.
[123]毛虎平,吴义忠,陈立平.基于时间谱元法的动态响应优化[J].机械工程学报，2010，46(16)：79-87.
[124] A.T. Patera. A spectral element method for fluid dynamics; laminar flow in a channelexpansion. Journal of Computational Physics,54:468-488,1984.
[125] A. Bueno-Orovio and V．M．Pérez-García, Spectral smoothed boundary methods：the role of external boundary conditions，Numer. Methods for Partial Differential Eq.,22(2006), pp435–448．
[126] M.H. Kurdi and P.S. Beran. Spectral element method in time for rapidly actuatedsystems. Journal of Computational Physics,2007(3):1809-1835.
[127] R.D. Hendereon and G．E. Karniadakis Structured spectral element Methods forsimulation of turbulent flows. J. Comput. Phys.,122(1995) pp.191-217.
[128] D. Pathrig and G．E．Karuiadakist, Spectral element methods for elliptic problems inmooth domains， J. Comput. Phys.,122(1995), pp.83-95．
[129] J.S. Hestlmven，A stable penalty method for the compressible Navier-Stokesequations：II．one-dimensional domain decomposion schemes.，SIAM J. Sci. Comput.，18(1997), pp.658-685．
[130] Thomas J IL Hughes．The Finite Element Method--Linear Static and Dynamic FiniteElement Analysis [M]，Prentice-Hall，lnc，Englewood Cliffs，NewJersey,1987.
[131] POZRIKIDIS C. Introduction to finite and spectral element methods using matlab
[M]. New York：Chapman and Hall/CRC，2005.
[132] Kang B S, Shyy Y K, Hong Q Q. Implementation of Equivalent Static Load Methodin Flexible Multi-body Dynamic Systems[C].7th World Congresses of Structural andMultidisciplinary Optimization，2007．
[133] Kang B S，Choi W S，Park G J． Structural Optimization Under Equivalent StaticLoads Transformed from Dynamic Loads Based on Displacement［J］． Computers＆Structures，2001，79(2):145-154．
[134]黄国权.有限元法基础及ANSYS应用[M].北京:机械工业出版社,2004
[135](美)Saeed Moaveni.有限元分析:ANSYS理论与应用[M].北京:电子工业出版社,2008
[136]齐从谦，崔琼瑶，基于参数化技术的CAD创新设计方法研究[J].中国机械工程，2003,14（8）：681-683.
[137]崔志琴，杨瑞峰.复杂机械结构的参数化建模及模态分析[J].机械工程学报，2008，44(2)：234-237.
[138] Kocer FY, Arora JS (2002) optimal design of latticed towers subjected to earthquakeloading. J Struct Eng128:197-204.
[139] Oral S, Ider SK (1997) Optimum design of high-speed flexible robotic arms withdynamic behavior constraints. Comput Struct65:255-259.
[140] Haug EJ, Arora JS (1979) Applied optimal design: mechanical and structural systems.Wiley-Interscience, New York.
[141] Grandhi RV, Haftka RT, Watson LT (1986b). Efficient identification of criticalstresses in structures subjected to dynamic loads. Comput Struct22:373-386.
[142] Hsieh CC, Arora JS (1985a). An efficient method for dynamic response optimization.AIAA J23:1484–1486.
[143] Hsieh CC, Arora JS (1985b) A hybrid formulation for treatment of point-wise statevariable constraints in dynamic response optimization. Comput Methods Appl Mech Eng48:171-189.
[144] Hsieh CC, Arora JS (1985c) Structural design sensitivity analysis with generalboundary conditions—dynamic problem. Int J Numer Methods Eng21:267-283.
[145] Park KJ, Lee JN, Park GJ (2005) Structural shape optimization using equivalentstatic loads transformed from dynamic loads. Int J Numer Methods Eng63. DOI:10.1002/nme.1295.
[2]杜平安.MCAE计算机辅助机械工程[M]．北京：机械工业出版社，1996
[3]李佳.计算机辅助设计与制造(CAD/CAM)[M]，天津：天津大学出版社，2002
[4]源清,肖文，简析九十年代主流CAD造型基础技术[J]，计算机辅助设计与制造，1998(2):1-5
[5]罗浩,张新访.基于约束的参数化设计技术发展现状及前景[J]，中国机械工程，1995(5)21-24
[6]王贤坤.机械CAD／CAM技术应用与开发[M]．北京：机械工业出版社，2001
[7] Raymond H.Kurland, Understanding Variable-Driven Modeling,http://www.technicom.com.1994
[8] D.H.Brown Associates， Inc.SDRC’s Variational Analysis： A Design AnalysisBreakthrough. http://www.Dhbrown.com
[9]谭建国.使用ANSYS6.0进行有限元分析[M]，北京：北京大学出版社，2002
[10]张学玲,唐毅等.用实验模态与有限元方法识别结合面接触刚度的方法[J]，组合机床与自动化加工技术,2005(11):56-59
[11]冯培恩，邱清盈等．机械产品的广义优化设计进程研究[J]．中国科学，1999(E)，29(4)：338-346
[12]Maxwell C S．Scientific papers lI[M]．New York：Dover Publications，1952
[13]Michell A GM．The limits of economy of material in flame structures．PhilosophicalMagazine，1904
[14]Shanley F R．Weight strength Analysis of Air craft Structures．New York：McGraw—Hill，1952
[15]秦东晨，祁建中，张明成等，液压机横梁结构的优化设计[J]．锻压技术，2004(2)：49-52
[16]李泉永．机械结构优化设计的回顾与展望[J]．桂林电子工业学院学报,2000(4)：114-119
[17]Schmit L.A.，Farshi B.，Some approximation concepts for structural synthesis [J]，Journal of AIAA，1974，12：692-699．
[18]Schmit, L.A., Miura, H. Approximation concepts for efficient structural synthesis. NASACR-2552,1976
[19]L. A. Schmit, Jr., H. Miura, Approximation concepts for efficient structural synthesis,Tech. Report CR–2552, NASA,1976
[20]余俊,周济著.优化方法程序库OPB-2—原理及应用(第一版)[M].武汉:华中理工大学出版社,1997
[21]程耿东,顾元宪．序列二次规划在结构动力优化中的应用[J]．振动与冲击，1986，5(1)：12-20．
[22]K. A. High, R. D. LaRoche. Parallel nonlinear optimization techniques for chemicalprocess design problems. Computers&Chemical Engineering,1995,19(6-7):807-825
[23]陈忠.关于非线性规划问题的并行算法[J].长江大学学报(自科),2006,4(12):1-8
[24]童卫华,姜节胜,顾松年．一种以均方响应作为约束的动力学设计方法[J].应用力学学报,1996,13(3)：94-98．
[25]李强，周济，钟毅芳．机械系统动态优化设计的复合遗传算法[J]．1999，35(5)：27-30．
[26]唐文艳.结构优化中的遗传算法研究和应用[D],大连：大连理工大学,200201
[27]Pantelides C.P., Tzan S．R． Optimal design of dynamically constrained structures [J].Computers&Structures,1997,62(1):141-150
[28]Y. M. Xie, G. R. Steven.A Simple Evolutionary Procedure for StructuralOptimization．Computers＆Structures,l993,49:885-896
[29]Y M．Xie．G P．Steven．Optimal Design of Multiple Load Case Structures Using anEvolutionary Procedure．Engineering Computations,1994(11):295-302
[30]O.M．Querin，G．P. Steven，Y M．Xie．Evolutionary Structural Optimization (ESO)Using Bi-directional Algorithm．Engineering Computations,1998(15)：103l-1048
[31]杨灿军，陈鹰，路用祥．基于人机一体化思想的企业MIS[J]．系统工程理论与实践，1997,(10):73-77
[32]Li Quanyong, Zhou Wei．A Nonlinear Programming for Multi-objective Fuzzy OptimalDesign. Journal of Guilin Institute of Electronic Technology, Mar.1999，V01.19,No.1(80-84)
[33]刘小平等，机械结构动力学模型的模糊优化修正方法[J],振动工程学报,1995(9)
[34]H Tanaka, K Asai．Fuzzy Linear Programming Problems with Number[J]．Fuzzy Sets andSystems,1984,13(1):1-10
[35]王光远，陈树勋.工程结构系统软设计理论及其应用[M].北京:国防工业出版社，1996
[36]Richardson M．H．，Parameter estimate from frequency response measurements using therational fraction polynomial method, Proc．of the1st IMAC，1982
[37]JONES D R. Direct global optimization algorithm direct global optimization algorithm
[M]//Encyclopedia of Optimization，US：Springer，2001：431-440
[38]JONES D R，PERTTUMEN C D，STUCKMAN B E. Lipschitzian optimization withoutthe Lipschitz constant [J]. Journal of Optimization Theory and Applications,1993,79(1)：157-181
[39]卢熹.机械结构优化方法及应用研究[D],南京:东南大学,2004(8)
[40]张学玲,唐毅,徐燕申.用实验模态与有限元方法识别结合面接触刚度的方法[J],组合机床与自动化加工技术,2005,(11):56-59
[41]邓君.YC-6K型柴油机活塞故障分析及结构优化[D],武汉:武汉理工大学,2011(4)
[42]黄震,张继春.基于有限元的活塞优化设计分析[J],昆明理工大学学报(理工版),2006(6):76-80
[43]单峰,朱俊虎等.基于灵敏度分析的载货汽车车架结构优化[J],汽车技术,2013(1)
[44]Q.Hao，Y Sun．Car Door Optimization Based on Kriging Model[C]．Taiyuan：2010International Conference on Computer Application and System Modeling,2010
[45]Kane, Couro, Jouve, Franeois, Sehoenauer, Mare.Topology optimization in1inear andnonlinear elasticity, using genetic algorithms. American Society of MechanicalEngineers,Design Engineering Division (Publication) DE,1995(9):385-392．
[46]刘俊栋.基于HyperWorks柴油机结构优化的研究[D],天津:天津大学,2007(1)
[47]霍心达,郑明军.基于拓扑优化的某活塞轻量化分析[J],石家庄铁道大学学报(自然科学版),2013(3):87-91
[48]李鹏.内燃机关键件拓扑优化技术研究[D],太原:中北大学,2012(5)
[49]Niordson F.I. The optimum design of a Vibrating beams [J]. Quarterly of AppliedMathematics,1965,23(1):47-53
[50]M．S．Zarghamee．Optimum Frequency of Structures．AIAA Journal，1968，6(6)749-750．
[51]钱令希.工程结构优化设计[M].北京:水利电力出版社,1983
[52]Xie Y M.，Steven G P．Evolutionary Structural Optimization Springer-Verlag．Berlin：Germa.1997
[53]顾亮,胡琪琪.结构动力修改的拟力修改法[J],北京理工大学学报,1997,17(5):533-538
[54]J．R．Bakera，K．E．Rouch．Use of finite element structural models in analyzing machinetool chatter．Finite E1ements in Analysis and Design.2002,38:1029-1046
[55]陈新,孙庆鸿.基于动力学特征的磨床床身优化布局设计[J].制造技术与机床,2001(2)：21-22
[56]张宗兰.能量平衡法在机床结构动态参数优化中的应用[J].哈尔滨工业大学学报，1987,(4):37-43
[57]宋宝安.柴油机低噪声结构动态设计方法的研究[D]天津:天津大学学位论文,2007(7)
[58]Cheu T.C.，Wang B.P.，Chen T.Y．Design optimization of gas turbine blades withgeometry and natural frequency [J]. Computers&Structures，1989,32(1)：113-117
[59]姚英姿,樊俊星,莫云辉,邓召义.基于模糊理论和遗传算法的航空齿轮可靠性优化设计[J].上海大学学报(自然科学版),2004,10(4):363-366
[60]宁嘉.基于有限元法的直升机机体动力学建模与优化研究[D],南京:南京航空航天大学2013(3)
[61]G.J. Park. Analytic methods for design practice [M]. London.Springer,2007
[62]Byung-Soo Kang·Gyung-Jin Park·Jasbir S. Arora. A review of optimization ofstructures subjected to transient loads. Struct Multidisc Optim.2006(31):81-95
[63]Venkayya V.B., Knot N.S., Keddy V.S. Energy distribution in an optimum structuraldesign, AFFDL-TR-68-156,1969.
[64]Berke L., Khot N.S. Use of optimality criteria methods for large-scale system. AGRADLecture Series,1974, No.70.
[65]Kiusalaas J., Shaw R.C. An algorithm for optimal structural design with frequencyconstraints. Int J Num Eng,1978,13:283-295.
[66]林家浩.有频率禁区的结构优化设计[J].大连工学院学报,1981,20(1):27-36.
[67]宋海平,周传荣.模糊环境下多目标结构动力优化设计.应用力学学报,1998,15(3):139-142
[68]陈建军,车建文,崔明涛等.结构动力优化设计评述与展望.力学进展,2001,31(2):181-192
[69]Kang B-S, Park G-J, Arora JS. A review of optimization of structures subjected totransient loads. Struct Multidisc Optim,2006,31(2):81–95
[70]Qian Wang, Jasbir S. Arora. Alternative formulations for transient dynamic responseoptimization.46th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics&Materials Conference18-21, Austin, Texas,2005(8)
[71]Dong-Hoon Choi, Hong-Soo Park, Min-Soo Kim.A direct treatment of min-max dynamicresponse optimization problems. AIAA-93-1352-CP
[72]Chung-Ho Wang. Dynamic multi-response optimization using principal componentanalysis and multiple criteria evaluation of the grey relation model. Int J Adv Manuf Technol,2007(32):617-624
[73]J. Bausa, G. Tsatsaronis. Dynamic optimization of startup and load-Increasing processesin power plants D Part I: Method. Transactions of the ASME.2001,123(1):246-250
[74]Park S, Kapania RK, Kim SJ. Nonlinear transient response and second-order sensitivityusing time finite element method. AIAA J,1999,37(5):613-622
[75]Choi W S，Park G J． Structural Optimization Using Equivalent Static Loads at All TimeIntervals [J]. Computer Methods in Applied Mechanics and Engineering,2002,191(19-20):2105-2122
[76]Grandhi RV, Haftka RT, Watson LT. Design oriented identification of critical time’sintransient response. AIAA Pap,1984:1984-0899
[77]Tenk L.H., Hagiwara I. Static and vibration shapes&topology optimization usinghomogenization and mathematical programming. Comput Meth in Appl Meth&Eng,1993(109):143-154.
[78]Kapoor M. P. Kumarasamy K. Optimum configuration of transmission towers indynamic response regime. In: Proceedings International Symposium on Optimum StructuralDesign, Tucson, Ariz, Oct.1981.
[79]Rao S.S. Optimization of airplane wing structures under gust loads. Computers&Structures,1985,21(3):741-749.
[80]Lin J.H. Structural optimization on genetrical configuration and element sizing withstatic and dynamically constraints. Computers&Structures,1982,15(5):507-526.
[81]Arvind U. Raghunathan, Dharmashankar Subramanian, Lorenz T. Biegler, Tariq Samad.Dynamic optimization strategies for three-dimensional conflict resolution of Multiple Aircraft.Journal of guidance, control, and dynamics.,2004,27(4):586-594
[82]Y.A. Khulief，H.A1-Naser． Finite element dynamic analysis of drill strings [J]. FiniteElement in Analysis and Design，2005(41)：1270-1288．
[83]郭志全.基于有限元分析的机械产品结构动态设计及其工程应用[D],天津：天津大学学位论文，2006(12)
[84]高卫娟.船用柴油机机体动态应力分析方法研究[D]，武汉：武汉理工大学，2010(5)
[85]毛虎平.基于仿真模型的动态响应优化算法研究[D]，武汉：华中科技大学，2011()
[86]James M L. Vibration of Mechanical and Structural Systems (2nd ed)[M]. New York:Harper Collins,1994.
[87]Willmert KD, Fox RL (1972) Optimum Design of a Linear Multidegree-of-freedomShock Isolation System. Journal of Engineering for Industry, Transactions of the ASME94:465-471.
[88]Afimiwala KA, Mayne RW (1974) Optimal Design of an ImpactAbsorber. Journal of Engineering for Industry, Transactions of the ASME96:124-130.
[89]Fox RL, Kapoor MP (1970) Structural Optimization in the Dynamic Regime: AComputational Approach. AIAA Journal8:1798-1804.
[90]毛虎平，苏铁熊等.多元模型自适应与时间谱元法结合的动态优化[J]，计算机辅助设计与图形学学报，2013（11）：1725-1734
[91]毛虎平，吴义忠等.时间谱元法在动态响应优化中的应用[J]，振动工程学报，2013（6）：95-403
[92]Chen X., Kareem A. Equivalent Static Wind Loads for Buffeting Response of Bridge.Journal of Structural Engineering,2001,127(12):1467-1475.
[93]Wen H.M., Reddy T.Y. Reid S.R. Deformation and Failure of Clamped Beams underLow Speed Impact Loading. International Journal of Impact Engineering,1995,16(3):435-454.
[94]Letchford C.W., Killen G.P. Equivalent Static Wind Loads for Cantilevered GrandstandRoofs. Engineering Structures,2002,23:207-217.
[95]Zhang Q.L., Vrouwenvelder A., Wardenier, J. Dynamic Amplification Factors and EUDLof Bridges under Random Traffic Flows. Engineering Structures,2001(23):663-672.
[96]Kang B.S., Choi W.S. Park G.J. Structural Optimization under Equivalent Static LoadsTransformed from Dynamic Loads Based on Displacement. Computers&Structures,2001(79)2:145-154.
[97]Choi W.S., Park G.J. Structural Optimization Using Equivalent Static Loads at All theTime Intervals. Computer Methods in Applied Mechanics and Engineering,2002,191:2105-2122.
[98]Choi W.S., Park G.J., Shin M.J. Transformation of a Dynamic Load into an EquivalentStatic Load and shape optimization of the Road Arm. KSME Fall Conference，1995(1):609-614.
[99]Choi W.S., Kang S.C., Shin M.J. Transformation of a Dynamic Load into an EquivalentStatic Load and Shape Optimization of the Road Arm in Self-Propelled Howitzer.Transactions of the KSME A,1996,20(12):3767-3781.
[100] Shin M.J., Choi W.S., Park G.J. Transformation of a dynamic load into an equivalentstatic load and shape optimization of the road arm. PACAM V Conference, Puerto Rico,1997.
[101] Kim M.S., Choi D.H. Multibody Dynamic Response Optimization with ALM andApproximate Line Search. Multibody System Dynamics, Kluwer Academic Publishers,1997(1):47-64.
[102] Choi W.S. Transformation of dynamic loads into equivalent static loads andstructural optimization. Ph.D. Dissertation. Hanyang: Hanyang University,2002.
[103] Choi W.S., Park G.J. Structural Optimization Using Equivalent Static Loads at AlltheTime Intervals. Computer Methods in Applied Mechanics and Engineering,2002,191(19/20):2105-2122.
[104] Park G.J., Kang B.S. Validation of a Structural Optimization AlgorithmTransforming Dynamic Loads into Equivalent Static Loads. Journal of Optimization Theoryand Applications,2003,119(1):191-200.
[105] CHOI W S，PARK G J. Transformation of dynamic loads into equivalent static loadsbased on modal analysis [J]. Int. J. Numer. Meth. Engng.，1999，46：29-43.
[106] Kang B.S. On Optimality of Solution Obtained by Equivalent Static Load Methodand Its Application to Flexible Multibody Dynamic Systems. Ph.D. Dissertation. Hanyang:Hanyang University,2002.
[107]顾慧芝,聂君剑.载荷识别的等效点动标定法[J]，振动工程学报，1997(9)
[108]苏义脑,唐雪平.初弯曲纵横弯曲梁的等效载荷法及其应用[J]，力学与实践200401
[109]黄武龙.基于等效静态载荷方法的大型复杂结构的轻量化设计[D]，广州：广东工业大学，2013(5)
[110]杨志军.基于等效静态载荷原理的高速机构结构拓扑优化方法[J]．机械工程学报.2011,9(47)：119-126．
[111]赵礼辉.ESL法在汽车结构优化设计中的应用[D]，上海：上海交通大学，2009(2)
[112]芮强，王红岩.基于等效静态载荷法的结构动态优化[J],汽车工程2014(1)
[113]肖登红，王俊元.基于等效静态载荷法的平面四杆机构的结构动态优化设计[J]，机械设计与制造2011(4)
[114]刚宪约，张帆.基于载荷等效法的空气悬架客车有限元建模方法[J],山东理工大学学报(自然科学版)201109
[115]贺新峰，基于等效静态载荷和响应面法的搅拌车副车架疲劳设计[D]，湖南大学201204
[116]张培信.能量理论结构力学[M],上海科学技术出版社，2010（1）:1-3
[117] T.R.陶丘得著，李世昌译.结构力学能量原理[M],人民交通出版社，1984（12）：33-39
[118]刘晶波杜修力.结构动力学[M],机械工业出版社，2012（6）：15-20
[119] S.P.Timoshenko（美），D.H.Young，叶红玲，杨庆生等译.结构理论[M]，机械工业出版社2005（5）：103-105
[120]张洪信，管殿柱.有限元基础理论与ANSYS11.0应用[M],机械工业出版社，2012（7）：46-52
[121]王海波，陈伯望等.结构动力方程Newmark-β方法递推简化分析[J];四川大学学报(工程科学版),2008(5)
[122] CLOUGH R W. Dynamics of structures [M]. New York：McGraw-Hill，1993.
[123]毛虎平,吴义忠,陈立平.基于时间谱元法的动态响应优化[J].机械工程学报，2010，46(16)：79-87.
[124] A.T. Patera. A spectral element method for fluid dynamics; laminar flow in a channelexpansion. Journal of Computational Physics,54:468-488,1984.
[125] A. Bueno-Orovio and V．M．Pérez-García, Spectral smoothed boundary methods：the role of external boundary conditions，Numer. Methods for Partial Differential Eq.,22(2006), pp435–448．
[126] M.H. Kurdi and P.S. Beran. Spectral element method in time for rapidly actuatedsystems. Journal of Computational Physics,2007(3):1809-1835.
[127] R.D. Hendereon and G．E. Karniadakis Structured spectral element Methods forsimulation of turbulent flows. J. Comput. Phys.,122(1995) pp.191-217.
[128] D. Pathrig and G．E．Karuiadakist, Spectral element methods for elliptic problems inmooth domains， J. Comput. Phys.,122(1995), pp.83-95．
[129] J.S. Hestlmven，A stable penalty method for the compressible Navier-Stokesequations：II．one-dimensional domain decomposion schemes.，SIAM J. Sci. Comput.，18(1997), pp.658-685．
[130] Thomas J IL Hughes．The Finite Element Method--Linear Static and Dynamic FiniteElement Analysis [M]，Prentice-Hall，lnc，Englewood Cliffs，NewJersey,1987.
[131] POZRIKIDIS C. Introduction to finite and spectral element methods using matlab
[M]. New York：Chapman and Hall/CRC，2005.
[132] Kang B S, Shyy Y K, Hong Q Q. Implementation of Equivalent Static Load Methodin Flexible Multi-body Dynamic Systems[C].7th World Congresses of Structural andMultidisciplinary Optimization，2007．
[133] Kang B S，Choi W S，Park G J． Structural Optimization Under Equivalent StaticLoads Transformed from Dynamic Loads Based on Displacement［J］． Computers＆Structures，2001，79(2):145-154．
[134]黄国权.有限元法基础及ANSYS应用[M].北京:机械工业出版社,2004
[135](美)Saeed Moaveni.有限元分析:ANSYS理论与应用[M].北京:电子工业出版社,2008
[136]齐从谦，崔琼瑶，基于参数化技术的CAD创新设计方法研究[J].中国机械工程，2003,14（8）：681-683.
[137]崔志琴，杨瑞峰.复杂机械结构的参数化建模及模态分析[J].机械工程学报，2008，44(2)：234-237.
[138] Kocer FY, Arora JS (2002) optimal design of latticed towers subjected to earthquakeloading. J Struct Eng128:197-204.
[139] Oral S, Ider SK (1997) Optimum design of high-speed flexible robotic arms withdynamic behavior constraints. Comput Struct65:255-259.
[140] Haug EJ, Arora JS (1979) Applied optimal design: mechanical and structural systems.Wiley-Interscience, New York.
[141] Grandhi RV, Haftka RT, Watson LT (1986b). Efficient identification of criticalstresses in structures subjected to dynamic loads. Comput Struct22:373-386.
[142] Hsieh CC, Arora JS (1985a). An efficient method for dynamic response optimization.AIAA J23:1484–1486.
[143] Hsieh CC, Arora JS (1985b) A hybrid formulation for treatment of point-wise statevariable constraints in dynamic response optimization. Comput Methods Appl Mech Eng48:171-189.
[144] Hsieh CC, Arora JS (1985c) Structural design sensitivity analysis with generalboundary conditions—dynamic problem. Int J Numer Methods Eng21:267-283.
[145] Park KJ, Lee JN, Park GJ (2005) Structural shape optimization using equivalentstatic loads transformed from dynamic loads. Int J Numer Methods Eng63. DOI:10.1002/nme.1295.