非线性流滞阻尼器耗能结构随机地震响应和首超时间分析
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摘要
对非线性流滞阻尼器耗能结构在Kanai-Tajimi谱地震激励下的随机响应及其随机失效时间和动力可靠性进行了系统研究。首先建立了结构的非线性运动方程;然后,基于随机平均法,将结构响应幅值近似为一维markov扩散过程,获得了扩散过程漂移系数和扩散系数的解析表达式;其次,利用扩散过程与FPK方程的对应关系,获得了幅值平稳概率密度函数和幅值任意阶矩的解析表达式;再次,利用幅值与结构位移和速度的相互转化关系,获得了结构位移与速度的平稳联合概率密度函数和位移、速度方差以及位移期望穿越率的解析表达式;最后,利用扩散过程的后向Kolmogrov方程,基于首超失效模型,建立了结构动力可靠性函数方程和结构随机失效时间统计矩方程,并利用一维扩散过程的边界分类性质,将统计矩方程的奇异定性边界条件转化为等价的定量边界条件,进而获得了失效时间任意阶统计矩的解析解,并利用此矩,对结构动力可靠性和失效时间概率分布函数进行了近似分析,给出了算例,从而建立了结构非线性随机地震响应及其随机失效时间和动力可靠性的分析方法。
The random response,first-passage time and dynamic reliability of structures with nonlinear fluid viscous dampers subjected to Kanai-Tajimi random earthquake model excitation are studied systematically. The nonlinear dynamic equations of structures are established; the structural diffusion process by using the stochastic averaging method and the analytic formulas of the drift and diffusion coefficients are gotten; The exact solutions of stationary probability density and arbitrary-order moments of amplitude are derived by using the mutual relationship between diffusion process and FPK equation; the exact solutions of joint stationary probability density and the mean-square values of displacement and velocity, and the exact solution of the expected rate of displacement threshold crossings are achieved by using the mutual relationships between structural amplitude,phase,displacement and velocity. According to the Kolmogorov backward equation of amplitude diffusion and first-passage model, the equations of structural dynamic reliability function and the statistical moments for structural first-passage time are established.By using the classification characteristics of boundaries of diffusion process, the singular qualitative boundary condition of the moment equation is transformed into the equal quantitative boundary condition and then the analytic solutions of arbitrary-order moments of first-passage time are obtained. The structural dynamic reliability and the probability distribution function of first-passage time are analyzed approximately by utilizing the moments of first-passage time, An example is given out.
The random response,first-passage time and dynamic reliability of structures with nonlinear fluid viscous dampers subjected to Kanai-Tajimi random earthquake model excitation are studied systematically. The nonlinear dynamic equations of structures are established; the structural diffusion process by using the stochastic averaging method and the analytic formulas of the drift and diffusion coefficients are gotten; The exact solutions of stationary probability density and arbitrary-order moments of amplitude are derived by using the mutual relationship between diffusion process and FPK equation; the exact solutions of joint stationary probability density and the mean-square values of displacement and velocity, and the exact solution of the expected rate of displacement threshold crossings are achieved by using the mutual relationships between structural amplitude,phase,displacement and velocity. According to the Kolmogorov backward equation of amplitude diffusion and first-passage model, the equations of structural dynamic reliability function and the statistical moments for structural first-passage time are established.By using the classification characteristics of boundaries of diffusion process, the singular qualitative boundary condition of the moment equation is transformed into the equal quantitative boundary condition and then the analytic solutions of arbitrary-order moments of first-passage time are obtained. The structural dynamic reliability and the probability distribution function of first-passage time are analyzed approximately by utilizing the moments of first-passage time, An example is given out.
引文
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[2]Singh M P,et al,Seismic analysis and design with maxwell dampers[J].Journal of Engineering Mechanics,2003,129(3):273—282.
[3]Terenzi G.Dynamics of SDOF systems with non-linear viscous damping[J].Journal of Engineering Mechanics,1999,125(8):956—963.
[4]Lin W H,et al.Earthquake response of elastic SDOF systems with nonlinear fluid viscous dampers[J].Earthquake Engi-neering and Structure Dynamics,2002,31:1623—1642.
[5]Lin W H,et al.Earthquake response of elastic single-degree-of-freedom systems with nonlinear viscoelastic dampers[J].Journal of Engineering Mechanics,2003,129(6):599—606.
[6]朱位秋.非线性随机振动理论的近期发展[J].力学进展,1994,24(2):163—172.
[7]Zhu W Q.Recent developments and application of the sto-chastic averaging method in random vibration[J].Applied Mechanics Reviens,1996,49(10):72—80.
[8]Zhu Weiqiu,Cai Guoqiang.Nonlinear stochastic dynamics:a survey of recent development[J].Acta Mechauica Siuia,2002,18(6):551—566.
[9]欧进萍等.结构被动耗能减振效果的参数影响[J].地震工程与工程振动,1998,18(1):60—70.
[10]朱位秋著.随机振动[M].北京:科学出版社,1998:291;236;520;525.
[11]Cai G Q,Lin Y K.On Statistics of first-passage failure[J].Journal of Applied Mechanics,1994,61(1):93—99.
[12]刘维信.机械可靠性设计[M].北京:清华大学出版社,1996:74—78.
[13]李创第,黄天立,李暾,邹万杰.TMD控制优化设计与振动台试验研究[J].土木工程学报,2006,39(7):19—25.
[14]李创第,黄东梅,李暾,邹万杰.剪切型基础平动与转动结构随机地震响应分析的复模态法[J].振动与冲击2005,24(5):67—70.
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