基于等效线性化法的弱非线性体系下的地震响应分析
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摘要
对弱非线性单自由度非自由振动系统结构在白噪声地震激励下的随机响应问题进行了研究。首先,建立了结构的弱非线性响应方程;然后根据能量平衡原理对运动方程进行了等效线性化;最后获得了金井清谱(Kanai-Tajimi谱)下等效线性化方程的位移响应及速度响应的方差,将弱非线性随机响应问题简化为求解线性随机响应问题,并给出了算例验证。结果表明,该方法可以准确得到弱非线性随机响应问题的解。同时,建立了单自由度结构弱非线性随机响应频域分析的统一解析解法。
In this paper, the stochastic response of the structure of the weakly nonlinear single degree of freedom system under the excitation of white noise excitation is studied. Firstly, the weakly nonlinear motion equation of the structure is established. Then according to the principle of energy balance, the equation of motion of the equivalent is linearized. Finally, the displacement response and the variance of the velocity response of the Tajimi-Kanai(Tajimi-Kanai spectrum) equivalent linearization equation are obtained. The complex weak nonlinear stochastic response is simplified to the problem of linear stochastic response, and a numerical example is given. Thus a unified analytical solution to the frequency domain equation of the single degree of freedom structure with weak nonlinear response is established.
In this paper, the stochastic response of the structure of the weakly nonlinear single degree of freedom system under the excitation of white noise excitation is studied. Firstly, the weakly nonlinear motion equation of the structure is established. Then according to the principle of energy balance, the equation of motion of the equivalent is linearized. Finally, the displacement response and the variance of the velocity response of the Tajimi-Kanai(Tajimi-Kanai spectrum) equivalent linearization equation are obtained. The complex weak nonlinear stochastic response is simplified to the problem of linear stochastic response, and a numerical example is given. Thus a unified analytical solution to the frequency domain equation of the single degree of freedom structure with weak nonlinear response is established.
引文
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[17]方同.工程随机振动[M].北京:国防工业出版社,1995.(Fang Tong.Vibration of engineering[M].Beijing:National Defence Industry Press,1995(in Chinese)).
[18]Lin Y K.Some observations on stochastic averaging method[J].Probabilistic Engineering Mechanics,1986,1(1):23-27.
[19]朱位秋.随机振动[M].北京:科学出版社,1992.(Zhu Weiqiu.Random vibration[M].Beijing:Science Press,1992(in Chinese)).
[20]刘晶波,杜修力,欧进萍.结构动力学[M].北京:机械工业出版社,2005.(Liu Jingbo,Du Xiuli,Ou Jinping.The structural dynamics[M].Beijing:Mechanical Industry Press,2005(in Chinese)).
[21]庄表中,梁以德,张佑启.结构随机振动[M].北京:国防工业出版社,1995.(Zhuang Biaozhong,Liang Yide,Zhang Youqi.Structure random vibration[M].Beijing:National Defence Industry Press,1995(in Chinese)).
[22]Yang C Y.结构随机振动[M].邱法维,潘景龙,李暄,译.哈尔滨:哈尔滨工业大学出版社,1990.(Yang C Y.Structure random vibration[M].Qiu Fawei,Pan Jinglong,Li Xuan,translated.Harbin:Harbin Industrial University Press,1990(in Chinese)).
[23]邹湘经.结构动力学[M].哈尔滨:哈尔滨工业大学出版社,1996(Zou Xiangjing.Structural dynamics[M].Harbin:Harbin Industrial University Press,1996(in Chinese)).
[24]尼格姆N C.随机振动概论[M].候成慧,周鉴如,陈文良,等译.上海:上海交通大学出版社,1985.(Nigel N C.Bishop overview of random vibration[M].Hou Chenghui,Zhou Jianru,Chen Wenliang,et al translated.Shanghai:Shanghai Jiao Tong University Press,1985(in Chinese)).
[25]欧进萍,王光远.结构随机振动[M].北京:高等教育出版社,1998.(Ou Jinping,Wang Guangyuan.The structure random vibration[M].Beijing:Higher Education Press,1998(in Chinese)).
[26]奈弗A H,穆克D T.非线性振动[M].宋家骕,罗惟德,陈守吉,译.北京:高等教育出版社,1990.(Nayfeh A H,Mook D T.Nonlinear vibration[M].Song Jiasu,Luo Weide,Chen Shouji,translated.Beijing:Higher Education Press,1990(in Chinese)).
[27]卢成江,赵欢,吴知丰.非线性结构地震随机响应方差的简化计算方法[J].武汉理工大学学报,2010(9):222-225.(Lu Chengjiang,Zhao Huan,Wu Zhifeng.Nonlinear structure seismic random response variance of thesimplified calculation method[J].Journal of Wuhan University of Technology,2010(9):222-225(in Chinese)).
[28]钟万勰.一个高效结构随机响应算法系列—虚拟激励法[J].自然科学进展,1996(4):394-401.(Zhong Wanxie.An efficient structure random response algorithm series the pseudo-excitation method[J].Progress in Natural Science,1996(4):394-401(in Chinese)).
[29]李杰,廖松涛.线性随机结构在随机激励下动力响应分析[J].力学学报,Acta Mechanica Sinica,2002,34(3):416-424.(Li Jie,Liao Songtao.Linear stochastic structure dynamic response analysis under random excitation[J].Acta Mechanica Sinica,2002,34(3):416-424(in Chinese)).
[30]张振浩,杨伟军.非线性体系动力可靠性分析的等效Duffing体系法[J].土木建筑与环境工程,2012,34(3):70-75.(Zhang Zhenhao,Yang Weijun.Dynamic reliability analysis of the equivalent nonlinear system method of Duffing system[J].Journal of Civil Architectural&Environmental Engineering,2012,34(3):70-75(in Chinese)).
[31]褚亦清,李翠英.非线性振动分析[M].北京:北京理工大学出版社,1996.(Chu Yiqing,Li Cuiying.Nonlinear vibration analysis[M].Beijing:Peking University Press,1996(in Chinese)).
[2]Nayfch A H,Mook D T.Nonlinear oscillations[M].New York:John Wiley,1979.
[3]Meirovitch L.振动分析基础[M].上海:上海交通大学出版社,1975.(Meirovitch L.Vibration analysis foundation[M].Shanghai:Shanghai Jiao Tong University Press,1975(in Chinese)).
[4]马尔金.非线性振动理论中的李雅普诺夫与邦加来方法[M].北京:科学出版社,1959.(Malkin.Nonlinear vibration theory of lyapunov to bangka to methods[M].Beijing:Science Press,1959(in Chinese)).
[5]Nayfeh A H.Introduction to perturbation techniques[M].New York:John Wiley&Sons,1981.
[6]Roberts J B,Spanos P D.Random vibration and statistical linearization[M].New York:Dover Publication,1988.
[7]Bogoliuboff N,Kryloff N,Lefschetz S.Introduction to non linear mechanics[M].New York:Princeton University Press,1947.
[8]Booton R C.Nonlinear control systems with random inputs[J].Ire Transactions on Circuit Theory,1954,1(1):9-18.
[9]Caughey T K.Equivalent linearization techniques[J].Journal of the Acoustical Society of America,1963,35(12):1706-1711.
[10]Brückner A,Lin Y K.Generalization of the equivalent linearization method for non-linear random vibration problems[J].International Journal of Non-Linear Mechanics,1987,22(3):227-235.
[11]裴多利,俞载道.地基土-非线性单自由度结构相互作用体系在随机地震作用下的加权等效线性化分析法[J].防灾减灾工程学报,1988(1):3-13.(Pei Duoli,Yu Zaidao.Weighted equivalent linearization analysis method for interaction system of ground soil nonlinear single degree of freedom under random earthquake[J].Journal of Disaster Prevention and Mitigation Engineering,1988(1):3-13(in Chinese)).
[12]吴再光,韩国城,林皋,等.非线性土层平稳随机地震反应分析的等价线性化法[J].水利学报,1988(8):68-73.(Wu Zaiguang,Han Guocheng,Lin Gao,et al.The nonlinear soil stationary random seismic response analysis of equivalent linearization method[J].Journal of Water Conservancy,1988(8):68-73(in Chinese)).
[13]吴再光,韩国城,林皋,等.非线性土层随机地震反应的概率平均等价线性化法[J].岩土工程学报,1989,11(4):9-16.(Wu Zaiguang,Han Guocheng,Lin Gao,et al.Random seismic response of nonlinear soil probabilistic average equivalent linearization method[J].Journal of Geotechnical Engineering,1989,11(4):9-16(in Chinese)).
[14]吴再光,林皋,韩国城,等.水平成层地基非平稳随机地震反应分析[J].土木工程学报,1992,25(3):60-67.(Wu Zaiguang,Lin Gao,Han Guocheng,et al.Horizontal stratified foundation non-stationary random seismic response analysis[J].Journal of Civil Engineering,1992,25(3):60-67(in Chinese)).
[15]闻邦椿,李以农,韩清凯.非线性振动理论中的解方法及工程应用[M].沈阳:东北大学出版社,2001.(Wen Bangchun,Li Yinong,Han Qingkai.Analytical methods and engineering application of the theory of nonlinear vibration[M].Shenyang:Northeastern University Press,2001(in Chinese)).
[16]包戈留包夫H H,米特罗波尔斯基ЮА.非线性振动理论中的的渐进方法[M].金福临,译.上海:上海科学技术出版社,1963:117-125.(БОГОИОБОВH H,.МитропольскийЮА.Incremental method in nonlinear vibration theory[M].Jin Fulin,translated.Shanghai:Shanghai Science and Technology Publishing Press,1963:117-125(in Chinese)).
[17]方同.工程随机振动[M].北京:国防工业出版社,1995.(Fang Tong.Vibration of engineering[M].Beijing:National Defence Industry Press,1995(in Chinese)).
[18]Lin Y K.Some observations on stochastic averaging method[J].Probabilistic Engineering Mechanics,1986,1(1):23-27.
[19]朱位秋.随机振动[M].北京:科学出版社,1992.(Zhu Weiqiu.Random vibration[M].Beijing:Science Press,1992(in Chinese)).
[20]刘晶波,杜修力,欧进萍.结构动力学[M].北京:机械工业出版社,2005.(Liu Jingbo,Du Xiuli,Ou Jinping.The structural dynamics[M].Beijing:Mechanical Industry Press,2005(in Chinese)).
[21]庄表中,梁以德,张佑启.结构随机振动[M].北京:国防工业出版社,1995.(Zhuang Biaozhong,Liang Yide,Zhang Youqi.Structure random vibration[M].Beijing:National Defence Industry Press,1995(in Chinese)).
[22]Yang C Y.结构随机振动[M].邱法维,潘景龙,李暄,译.哈尔滨:哈尔滨工业大学出版社,1990.(Yang C Y.Structure random vibration[M].Qiu Fawei,Pan Jinglong,Li Xuan,translated.Harbin:Harbin Industrial University Press,1990(in Chinese)).
[23]邹湘经.结构动力学[M].哈尔滨:哈尔滨工业大学出版社,1996(Zou Xiangjing.Structural dynamics[M].Harbin:Harbin Industrial University Press,1996(in Chinese)).
[24]尼格姆N C.随机振动概论[M].候成慧,周鉴如,陈文良,等译.上海:上海交通大学出版社,1985.(Nigel N C.Bishop overview of random vibration[M].Hou Chenghui,Zhou Jianru,Chen Wenliang,et al translated.Shanghai:Shanghai Jiao Tong University Press,1985(in Chinese)).
[25]欧进萍,王光远.结构随机振动[M].北京:高等教育出版社,1998.(Ou Jinping,Wang Guangyuan.The structure random vibration[M].Beijing:Higher Education Press,1998(in Chinese)).
[26]奈弗A H,穆克D T.非线性振动[M].宋家骕,罗惟德,陈守吉,译.北京:高等教育出版社,1990.(Nayfeh A H,Mook D T.Nonlinear vibration[M].Song Jiasu,Luo Weide,Chen Shouji,translated.Beijing:Higher Education Press,1990(in Chinese)).
[27]卢成江,赵欢,吴知丰.非线性结构地震随机响应方差的简化计算方法[J].武汉理工大学学报,2010(9):222-225.(Lu Chengjiang,Zhao Huan,Wu Zhifeng.Nonlinear structure seismic random response variance of thesimplified calculation method[J].Journal of Wuhan University of Technology,2010(9):222-225(in Chinese)).
[28]钟万勰.一个高效结构随机响应算法系列—虚拟激励法[J].自然科学进展,1996(4):394-401.(Zhong Wanxie.An efficient structure random response algorithm series the pseudo-excitation method[J].Progress in Natural Science,1996(4):394-401(in Chinese)).
[29]李杰,廖松涛.线性随机结构在随机激励下动力响应分析[J].力学学报,Acta Mechanica Sinica,2002,34(3):416-424.(Li Jie,Liao Songtao.Linear stochastic structure dynamic response analysis under random excitation[J].Acta Mechanica Sinica,2002,34(3):416-424(in Chinese)).
[30]张振浩,杨伟军.非线性体系动力可靠性分析的等效Duffing体系法[J].土木建筑与环境工程,2012,34(3):70-75.(Zhang Zhenhao,Yang Weijun.Dynamic reliability analysis of the equivalent nonlinear system method of Duffing system[J].Journal of Civil Architectural&Environmental Engineering,2012,34(3):70-75(in Chinese)).
[31]褚亦清,李翠英.非线性振动分析[M].北京:北京理工大学出版社,1996.(Chu Yiqing,Li Cuiying.Nonlinear vibration analysis[M].Beijing:Peking University Press,1996(in Chinese)).