结构在水平与竖向随机地震同时作用下的相关函数和谱密度

对单自由度结构在水平与竖向地震同时作用下的随机稳定性、响应及其相关函数和谱密度函数进行系统研究。首先利用Stratonovich和It随机微分方程与响应矩微分方程的互相关转化关系,建立了结构响应矩方程;然后根据Hurwitz随机稳定准则,获得了结构一阶和二阶响应矩渐近稳定的解析判别式;继而,利用复模态法获得了结构响应二阶矩的解析瞬态解和平稳解;最后利用It随机微分方程解具有的非可料函数性质,获得了结构位移、速度响应的自相关函数、互相关函数以及谱密度函数、互谱密度函数的解析解,给出了算例,并综合分析了各种参数对结构响应、稳定性以及相关函数和谱密度函数的影响。

CORRELATION FUNCTIONS AND SPECTRAL DENSITY FUNCTIONS OF STRUCTURE SUBJECTED TO HORIZONTAL-VERTICAL RANDOM EARTHQUAKE EXCITATIONS

This paper studies the random response, stability, correlation functions, and spectral density functions of structures with single degree of freedom subjected to horizontal-vertical random earthquake excitations. According to the relationships between stochastic differential equations and moment differential equations, the moment differential equations of structural response are established. Then, using the Hurwitz’s random stability criterion, this paper obtains the exact formula of stability for the first and second-order moments of structural response. Furtherly, using the complex mode theory, the exact transient and steady-state solutions of the second-order moments are obtained. Lastly, this paper derives the exact solutions to the auto-correlation functions, cross-correlation functions, auto-spectral density functions, cross-correlation functions of structural displacement and velocity responses according to the non-anticipating function characteristics of the solutions of It stochastic differential equations. An example is given to analyze the influence of various parameters on structural response, stability, correlation functions and spectral density functions.