频率相关黏性阻尼理论的时程积分计算
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摘要
黏性阻尼理论存在每循环消耗能量与外激励频率相关的缺陷,而复阻尼理论的运动方程通解中包含有发散项。依据频域转化原则,将复阻尼运动方程等价转化为频率相关黏性阻尼运动方程,克服复阻尼理论时域不稳定性的缺点,同时保证了每循环消耗能量与外激励频率无关。在此基础上,基于速度与位移关系假定和能量守恒准则,构建了适用于时程计算的频率相关黏性阻尼运动方程。结合常平均加速度法,给出了频率相关黏性阻尼运动方程时程积分计算的递推表达式。算例分析表明,频率相关黏性阻尼理论时程计算结果稳定收敛,可有效避免复阻尼理论时程计算中的发散问题。
Viscous damping theory has some disadvantages,one of which is that energy consumption per cycle is related to external excitation frequencies. The general solution to the complex damping theory's motion equation contains divergent items. Here,with the principle of frequency domain conversion,the complex damping motion equation was equivalently converted into a frequency dependent viscous damping motion equation to overcome the disadvantage of the complex damping theory having instability in time domain,and guarantee energy consumption per cycle being independent upon external excitation frequencies. Using the relationship assumption of velocity and displacement and the principle of energy conservation,the frequency dependent viscous damping motion equation being suitable for time history calculation was built. Combining with the constant average acceleration method,the recursion expression for time history integration calculation of the frequency dependent viscous damping motion equation was derived. The calculation examples showed that the time history integration calculation results of the frequency dependent viscous damping theory are stably convergent to effectively avoid divergent problems in time history integration calculation of complex damping theory.
Viscous damping theory has some disadvantages,one of which is that energy consumption per cycle is related to external excitation frequencies. The general solution to the complex damping theory's motion equation contains divergent items. Here,with the principle of frequency domain conversion,the complex damping motion equation was equivalently converted into a frequency dependent viscous damping motion equation to overcome the disadvantage of the complex damping theory having instability in time domain,and guarantee energy consumption per cycle being independent upon external excitation frequencies. Using the relationship assumption of velocity and displacement and the principle of energy conservation,the frequency dependent viscous damping motion equation being suitable for time history calculation was built. Combining with the constant average acceleration method,the recursion expression for time history integration calculation of the frequency dependent viscous damping motion equation was derived. The calculation examples showed that the time history integration calculation results of the frequency dependent viscous damping theory are stably convergent to effectively avoid divergent problems in time history integration calculation of complex damping theory.
引文
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[3]朱镜清,朱敏.复阻尼地震反应谱的计算方法及其它[J].地震工程与工程振动,2000,20(2):20-23.ZHU Jingqing,ZHU Min.Calculation of complex damping response spectra from earthquake records[J].Earthquake Engineering and Engineering Vibration,2000,20(2):20-23.
[4]朱镜清,朱敏.复阻尼地震反应谱计算的再研究[J].地震工程与工程振动,2001,21(2):36-40.ZHU Jingqing,ZHU Min.Further studies on calculation of complex damping seismic response spectra[J].Earthquake Engineering and Engineering Vibration,2001,21(2):36-40.
[5]王建平,刘宏昭,原大宁,等.含复阻尼振动系统的对偶原则及其数值方法研究[J].振动工程学报,2004,17(1):62-65.WANG Jianping,LIU Hongzhao,YUAN Daning,et al.Dual principle of oscillation systems with complex damping and its numerical method[J].Journal of Vibration Engineering,2004,17(1):62-65.
[6]何钟怡.复本构理论中的对偶原则[J].固体力学学报,1994,15(2):177-180.HE Zhongyi.The dual principle in theory of complex constitutive equations[J].Chinese Journal of Solid Mechanics,1994,15(2):177-180.
[7]胡海岩.结构阻尼模型及系统时域动响应[J].振动工程学报,1992,5(1):8-15.HU Haiyan.Structural damping modal and system dynamic response at time domain[J].Journal of Vibration Engineering,1992,5(1):8-15.
[8]INAUDI J A,KELLY J M.Linear hysteretic damping and the Hilbert transform[J].Journal of Engineering Mechanics,1995,121(5):626-632.
[9]朱镜清.频率相关黏性阻尼理论及有关问题的解[J].振动与冲击,1992,11(4):1-7.ZHU Jingqing.Theory and some related problems[J].Journal of Vibration and Shock,1992,11(4):1-7.
[10]周锡元,董娣,苏幼坡.非正交阻尼线性振动系统的复振型地震响应叠加分析方法[J].土木工程学报,2003,36(5):30-36.ZHOU Xiyuan,DONG Di,SU Youpo.New method for linear systems with non-classical damping under ground motion[J].China Civil Engineering Journal,2003,36(5):30-36.
[11]CHEN L Y,CHEN J T,CHEN C H,et al.Free vibration of a SDOF system with hysteretic damping[J].Mechanics Research Communication,1994,21(6):599-604.
[12]曹树谦,张文德,萧龙翔.振动结构模态分析[M].天津:天津大学出版社,2001.
[13]NEWMARK N M.A method of computation for structural dynamics[J].Journal of Engineering Mechanics Division,1959,85(3):67-94.
[14]刘庆林,傅学怡.基于复阻尼假定的不同材料阻尼特性混合结构抗震分析反应谱CCQC法[J].土木工程学报,2011,44(3):61-71.LIU Qinglin,FU Xueyi.A response spectrum CCQC method for seismic analysis of structures of multiple material damping characteristics based on complex damping assumption[J].China Civil Engineering Journal,2011,44(3):61-71.