功率谱二次正交化法在随机地震动响应的应用

Application of quadratic orthogonalization method of response power spectrum to random ground motion response

针对平稳激励下线性结构随机地震动响应方差和谱矩时频域法无解析解或时域法解析解复杂的问题,提出了结构响应功率谱的二次正交化法,并成功获得线性结构基于李鸿晶随机地震谱的系列响应(结构层位移和层间位移)的0~2阶谱矩和方差的简明封闭解。综合虚拟激励法和复模态方法,提出了线性结构频率响应特征值函数的二次正交法,即将频率响应特征值函数表示为振动复特征值和频域变量二次方和的线性组合;以李鸿晶随机地震动谱为例,基于留数定律获得该谱的二次正交式,进而获得结构地震动系列响应功率谱的二次正交式;获得了建筑结构随机地震动系列响应方差及0~2阶谱矩的统一简明封闭解。利用本文方法对一单自由度结构和一多自由度TMD耗能结构地震动响应进行分析,并与虚拟激励法进行了对比研究,结果表明本文所提方法为封闭解且可用于验证虚拟激励法谱矩分析时的精度。此外,本文方法可适用于各种线性结构基于各类平稳随机地震谱的随机响应封闭解的分析。

In order to solve the problem that the frequency domain method has no analytical solution or the time domain method has complex analytical solution for the variance and spectral moment response of linear structures under stationary excitation,the quadratic orthogonalization method of power spectral density function(RPSDF)of response is proposed innovatively,and the concise closed solution of 0~2 order spectral moment and square difference of linear structures are obtained successfully.A quadratic orthogonal method for the eigenvalue function of frequency response of linear structures is proposed by combining the pseudo excitation method and the complex mode method.The eigenvalue function of frequency response of the structure is expressed as a linear combination of the complex eigenvalues of the structure vibration and the quadratic sum of the frequency domain variables.The quadratic orthogonal form of the spectrum is obtained based on the residue law for Li Hongjing’s random ground motion spectrum.The unified and concise closed form solutions of the variance and 0~2 spectral moment of the series response of building structures are obtained.The seismic responses of a single degree of freedom(SDOF)structure and a multi degrees of freedom(MDOF)energy dissipation structure with TMD are analyzed by using the proposed method.The results show that the proposed method is correct and efficient.In addition,this method can be applied to the analysis of closed solutions of random responses of linear structures based on all kinds of stationary random seismic spectra.