叠层橡胶支座与悬臂柱串联体系动力稳定性分析
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摘要
针对国内外不少工程的隔震支座都放在地下室悬臂柱顶部的问题，为保证悬臂柱-隔震支座串联系统的安全性，对该系统的动力稳定性进行了分析。围绕这一主题，本文主要在以下几个方面进行了研究：
1．对结构振动控制进行了分类说明，并主要阐述了基础隔震控制体系在国内外研究现状及趋势；给出了结构动力稳定问题的定义与分类，对其研究方法作了简要但全面的介绍，并回顾了它的研究历史；最后论述了本课题的提出、课题研究现状及本课题的意义。
2．对动力稳定理论作了全面的介绍，引入了参数共振的概念，并推导了动力稳定问题的微分方程——Mathieu-Hill方程，对Floquet理论作了简单介绍。着重论述了有限元法在动力稳定问题中的应用，并采用Lagrangian方程推导出单元矩阵的表达式(单元刚度矩阵、单元质量矩阵和单元几何刚度矩阵)。并得出结构的静力学问题、动力学问题和静力屈曲问题仅为动力稳定控制方程的一个特殊情形。最后采用Floquet理论求得考虑阻尼及不考虑阻尼时确定动力失稳区间的控制方程。
3．回顾了Kelly等人和周锡元等人对橡胶支座的稳定性研究，然后介绍了周锡元等人对叠层橡胶支座和R/C柱的串联体系以及用两个不同截面的叠层橡胶支座组成组合橡胶支座的隔震体系提出的实用静力稳定分析公式，由后者的计算公式，可进一步求得考虑R/C柱P-Δ效应的串联体系静力稳定公式。
4．介绍了本人基于动力稳定理论编制的有限元程序DSA；然后阐述了频谱分析的基础理论和快速傅利叶变换(FFT)算法，并以FFT为基础，基于Matlab5.3编制了对某地震波进行频谱分析的计算程序，并对三个地震波进行了频谱分析。
5．基于动力稳定有限元分析程序DSA计算了叠层橡胶支座与柱串联体系的动力稳定性。算例一表明虽然采用了较方便的离散单元法，结果仍可满足工程要求。算例二讨论了随静力载荷因子及柱截面的不同，串联体系动力失稳区间的变化规律。通过地震波的频谱分析，结合体系的动力失稳区间分析，可判断在某竖向地震作用下，串联体系并无发生失稳的可能。最后在算例三中给出了满足动力稳定条件的最小柱截面范围。
At present, the laminated rubber bearings are often mounted on the top of the basement cantilever columns in most of projects in china and abroad. In order to guarantee the engineering security, the dynamic stability of serial system is analyzed. Highlighting this topic, the thesis tries to investigate the following essential aspects:
1. Structure vibration control is classified and explained and the current researching situation and the tendency of base-isolated control system in china and abroad are mainly detailed. The definition and classification of the theory of dynamic stability are provide and the brief but overall introduction to its research approach is given and its research history is reviewed; Lastly, questions such as proposing of the problem, current research-situation and the meanings of the problem are discussed.
2. A comprehensive introduction to the theory of dynamic stability is given. The concept of parametric resonance is introduced. The system differential equations(Mathieu-Hill equations)for dynamic stability analysis is derived. Floquet's theory is introduced. The application of finite element method(FEM) to dynamic stability analysis is described emphatically. The Lagrangian equations is used to derive the necessary element matrices(element stiffness matrix, element mass matrix and element geometric stiffness matrix). It is also shown that system equations for static structural mechanics, structural dynamics and static buckling are only a special case of the system equations of dynamic instability. Finally, Floquet's theory is used to generate- dynamic stability system equations for the purpose of determining both undamped and damped instability regions.
3. A investigation review of static stability of the laminated rubber bearings are presented according to the documents of J.M.Kelly et al. and Zhou Xiyuan et al.. The serial isolation system consisting of laminated rubber bearing and R/C column and the combined bearing consisting of an up small one and down larger one are introduced and their calculation formulas of static stability is derived. With the latter calculation formula, the static stability formula of the serial system considering P-A effect of R/C column can be acquired.
4. The FEM program DSA(Dynamic Stability Analysis) developed on the basis of the dynamic stability theory, the basic theory of spectrum analysis and the Fast Fourier Transform (FFT) algorithm is introduced. The spectral analyses of three earthquake waves are made using FFT algorithm. All the process is realized by programming in Matlab5.3.
5. The dynamic stability of serial system of laminated rubber bearing with column is calculated on the basis of FEM Analysis program DSA. The first example indicates that
though adopting the simple dispersed unit, the calculation result can still meet the project request. The second one discusses the rules of dynamic instability regions of serial system with different static load factor and different column cross-section size. Through the frequency spectrum analysis of earthquake waves and the dynamic instability regions analysis, the dynamic stability of serial system can be judged under a certain earthquake wave. Finally, the range of least column cross-section size meeting the dynamic stability condition is provided at the third example.
1．对结构振动控制进行了分类说明，并主要阐述了基础隔震控制体系在国内外研究现状及趋势；给出了结构动力稳定问题的定义与分类，对其研究方法作了简要但全面的介绍，并回顾了它的研究历史；最后论述了本课题的提出、课题研究现状及本课题的意义。
2．对动力稳定理论作了全面的介绍，引入了参数共振的概念，并推导了动力稳定问题的微分方程——Mathieu-Hill方程，对Floquet理论作了简单介绍。着重论述了有限元法在动力稳定问题中的应用，并采用Lagrangian方程推导出单元矩阵的表达式(单元刚度矩阵、单元质量矩阵和单元几何刚度矩阵)。并得出结构的静力学问题、动力学问题和静力屈曲问题仅为动力稳定控制方程的一个特殊情形。最后采用Floquet理论求得考虑阻尼及不考虑阻尼时确定动力失稳区间的控制方程。
3．回顾了Kelly等人和周锡元等人对橡胶支座的稳定性研究，然后介绍了周锡元等人对叠层橡胶支座和R/C柱的串联体系以及用两个不同截面的叠层橡胶支座组成组合橡胶支座的隔震体系提出的实用静力稳定分析公式，由后者的计算公式，可进一步求得考虑R/C柱P-Δ效应的串联体系静力稳定公式。
4．介绍了本人基于动力稳定理论编制的有限元程序DSA；然后阐述了频谱分析的基础理论和快速傅利叶变换(FFT)算法，并以FFT为基础，基于Matlab5.3编制了对某地震波进行频谱分析的计算程序，并对三个地震波进行了频谱分析。
5．基于动力稳定有限元分析程序DSA计算了叠层橡胶支座与柱串联体系的动力稳定性。算例一表明虽然采用了较方便的离散单元法，结果仍可满足工程要求。算例二讨论了随静力载荷因子及柱截面的不同，串联体系动力失稳区间的变化规律。通过地震波的频谱分析，结合体系的动力失稳区间分析，可判断在某竖向地震作用下，串联体系并无发生失稳的可能。最后在算例三中给出了满足动力稳定条件的最小柱截面范围。
At present, the laminated rubber bearings are often mounted on the top of the basement cantilever columns in most of projects in china and abroad. In order to guarantee the engineering security, the dynamic stability of serial system is analyzed. Highlighting this topic, the thesis tries to investigate the following essential aspects:
1. Structure vibration control is classified and explained and the current researching situation and the tendency of base-isolated control system in china and abroad are mainly detailed. The definition and classification of the theory of dynamic stability are provide and the brief but overall introduction to its research approach is given and its research history is reviewed; Lastly, questions such as proposing of the problem, current research-situation and the meanings of the problem are discussed.
2. A comprehensive introduction to the theory of dynamic stability is given. The concept of parametric resonance is introduced. The system differential equations(Mathieu-Hill equations)for dynamic stability analysis is derived. Floquet's theory is introduced. The application of finite element method(FEM) to dynamic stability analysis is described emphatically. The Lagrangian equations is used to derive the necessary element matrices(element stiffness matrix, element mass matrix and element geometric stiffness matrix). It is also shown that system equations for static structural mechanics, structural dynamics and static buckling are only a special case of the system equations of dynamic instability. Finally, Floquet's theory is used to generate- dynamic stability system equations for the purpose of determining both undamped and damped instability regions.
3. A investigation review of static stability of the laminated rubber bearings are presented according to the documents of J.M.Kelly et al. and Zhou Xiyuan et al.. The serial isolation system consisting of laminated rubber bearing and R/C column and the combined bearing consisting of an up small one and down larger one are introduced and their calculation formulas of static stability is derived. With the latter calculation formula, the static stability formula of the serial system considering P-A effect of R/C column can be acquired.
4. The FEM program DSA(Dynamic Stability Analysis) developed on the basis of the dynamic stability theory, the basic theory of spectrum analysis and the Fast Fourier Transform (FFT) algorithm is introduced. The spectral analyses of three earthquake waves are made using FFT algorithm. All the process is realized by programming in Matlab5.3.
5. The dynamic stability of serial system of laminated rubber bearing with column is calculated on the basis of FEM Analysis program DSA. The first example indicates that
though adopting the simple dispersed unit, the calculation result can still meet the project request. The second one discusses the rules of dynamic instability regions of serial system with different static load factor and different column cross-section size. Through the frequency spectrum analysis of earthquake waves and the dynamic instability regions analysis, the dynamic stability of serial system can be judged under a certain earthquake wave. Finally, the range of least column cross-section size meeting the dynamic stability condition is provided at the third example.
引文
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[3]．周福霖．工程结构减震控制[M]．地震出版社，1997，4
[4]．毛剑琴，卜庆忠等．结构振动控制的新进展．控制理论与应用，2001，18(5)：647～651
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[10].Ken P. Chong and Alison B. Flatau. Dynamic smart material and structural systems. Advances in structural dynamics, 2000, 1:19～30
[11].Shih Chi Liu and Weiling Chiang. Earthquake hazard mitigation research: new technologies and international engineering. Advances in structural dynamics, 2000, 1: 79～93
[12]．高德川，王依民，邹黎明．智能材料及其应用．2000(4)：17～19
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[2]．阎维明，周福霖，谭平．土木工程结构振动控制的研究进展．世界地震工程，1997，13(2)：8～20
[3]．周福霖．工程结构减震控制[M]．地震出版社，1997，4
[4]．毛剑琴，卜庆忠等．结构振动控制的新进展．控制理论与应用，2001，18(5)：647～651
[5]．M．帕兹，结构动力学—理论与计算，同济大学出版社，1993，9
[6]．欧进萍、王光远，结构随机振动，高等教育出版社，1998，5
[7]. Ray W. Clouyh, Joseph. Penzien. Dynamics of structures Mc Graw-Hill book Company, 1993
[8]．刘文锋．结构控制技术及最新进展(下)．世界地震工程，1997，13(4)：34～40
[9]. Michael D. Symans, Michael C. Constantinou. Semi-active control systems for seismic protetion of structures: a state-of-the-art review. Engineering Structure 1999, (21):469～487
[10].Ken P. Chong and Alison B. Flatau. Dynamic smart material and structural systems. Advances in structural dynamics, 2000, 1:19～30
[11].Shih Chi Liu and Weiling Chiang. Earthquake hazard mitigation research: new technologies and international engineering. Advances in structural dynamics, 2000, 1: 79～93
[12]．高德川，王依民，邹黎明．智能材料及其应用．2000(4)：17～19
[13]. Martel R.R. The effects of earthquakes on buildings with a flexible first story. Bull of the seismological Soc of America, 1929, 19(3)
[14]．苏经宇，曾德民等。叠层钢板橡胶隔震体系设计与分析。工程抗震。1996年12月，第四期。
[15]．R．I．Skinner，W．H．Robinsow，G．H．Mcverry。谢礼立，周雍年，赵兴全译。工程隔震概论。北京：地震出版社。1996年12月。
[16]．杜永峰，李慧，周茗如，屠锦敏。基础隔震结构响应衰减的动力凝聚法。第一届全国结构控制会议论文集。1998年10月。
[17]．杜永峰，严克明，孙荣镐。多层框架隔震参数的优化确定。工程力学增刊1995年Vol．2：1273～1281
[18]．黄永林，孔建国，章熙海等。基础隔震技术的发展及其对未来建筑设计崽想的影响。工程抗震。2000年3月，第1期，24～29。
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