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大连理工大学本科毕业设计（论文）多自由度结构动力学直接积分法的稳定性分析StabilityAnalysisofDirectIntegrationAlgorithmsAppliedtoMDOFStructuralDynamics学院（系）：运载工程与力学学部专业：工程力学系学生姓名：施志伟学号：200731034指导教师：杨迪雄评阅教师：完成日期：大连理工大学DalianUniversityofTechnology多自由度非线性结构动力学直接积分法的稳定性分析-I-摘要直接积分法是解决结构动力学问题的重要方法，精度和稳定性是它的两个重要指标，本文利用自动控制理论与结构动力学知识有机结合，研究探讨了直接积分法的稳定性问题以及稳定性条件。本文由浅入深，首先介绍了结构动力学的基础知识、然后介绍了几种常见的直接积分法，包括Newmark法、Wilson-θ法、Runge-Kutta法和HHT-α法，并且用时域分析中矩阵特征值分析的方法分析了Newmark法的稳定性，给出了Newmark法的稳定性条件；接着详细讲解了自动控制理论中的重要知识，包括反馈控制、控制系统的数学模型（主要介绍了微分方程和传递函数）、开环系统、闭环系统以及根轨迹等；最后将结构动力学知识与自动控制理论结合，以MATLAB为工具，分析了多自由度系统在线弹性结构和非线性结构下的稳定性。通过改变振动微分方程的形式，运用自动控制理论的闭环系统及根轨迹概念，可以分析直接积分法处理非线性动力学问题时的稳定性，本文中主要分析了Newmark法的稳定性。在线弹性系统中，根据叠加原理我们可以研究单自由度系统的稳定性来得到多自由度系统的稳定性，本文以单自由度系统为例分析了多自由度系统的稳定性；在非线性系统中，本文以简单的二自由度结构为例，分析了多自由度系统的稳定性。关键词：多自由度系统；直接积分法；非线性；稳定性；控制理论多自由度非线性结构动力学直接积分法的稳定性分析-II-StabilityAnalysisofDirectIntegrationAlgorithmsAppliedtoMDOFStructuralDynamicsAbstractDirectintegrationalgorithmsaretheimportantmethodtosolvetheproblemofstructuredynamics.Precisionandstabilityareitstwoimportantindexes,Thispapercombineautomaticcontroltheorywithstructuraldynamicsknowledgetostudythedirectintegrationalgorithms’stabilityproblemandstabilityconditions.Thispaperfirstintroducethebasicknowledgeofstructuredynamics,thenintroduceseveraldirectintegrationalgorithms,includingNewmarkmethod,Wilson–θmethod,Runge-KuttamethodandHHT-αmethod.TheconditionofstabilityofNewmarkmethodisgivenbytheeigenvalueanalysisofmatrixoftime-domainanalysis,Explaintheimportantknowledgeofautomaticcontroltheory,includingfeedbackcontrol,controlsystem’smathematicalmodel(mainlyintroducedifferentialequationandtransferfunction),theopenloopsystem,closed-loopsystemandrootlocus,etc.;Atlastcombinestructuraldynamicsknowledgewithautomaticcontroltheory,useMATLABfortools,analyzesthestabilityoftheMDOFsystemunderthelinearelasticstructureandthenonlinearstructure.Bychangingtheformofthevibrationdifferentialequation,usingautomaticcontroltheoryofclosed-loopsystemandrootlocusconcept,youcananalyzethestabilityofthedirectintegrationalgorithmswithnonlineardynamicproblems.Inthispaper,weanalyzethestabilityoftheNewmarkintegrationalgorithm.Inthelinearelasticstructuresystem,basedontheprinciple,wecangetthestabilityofMDOFsystembyanalyzingtheSDOFsystem.TakingtheSDOFsystemasanexample,wegetthestabilityofMDOFsysteminthispaper.Takingthesimpletwo-degree-of-freedomsystemasanexample,weanalyzethestabilityoftheMDOFsystem.KeyWords：MDOF；DirectIntegrationAlgorithm；Nonlinear；Stability;ControlTheory多自由度非线性结构动力学直接积分法的稳定性分析-III-目录摘要.....................................................................................................................................IAbstract.....................................................................................................................................II引言....................................................................................................................................11结构动力学基础。..............................................................................................................41.1结构动力学简介.......................................................................................................41.2结构动力学运动微分方程.....................................................................41.2.1单自由度系统................................................................................................41.2.2多自由度系统................................................................................................41.3动力响应的求解方法...............................................................................................52直接积分法..........................................................................................................................62.1直接积分法的原理...................................................................................................62.2直接积分法的分类...................................................................................................62.3常用的直接积分法...................................................................................................72.3.1Newmark法...................................................................................................72.3.2Wilson-θ法...................................................................................................82.3.3Runge-Kutta法...............................................................................................102.3.4HHT-α法.................................................................................................102.4时域积分法的稳定性分析.....................................................................................113自动控制理论基础..............................................................................................................133.1自动控制理论发展综述.........................................................................................133.2反馈控制.................................................................................................................133.3控制系统的数学模型....................................................................