随机激励下滞迟系统的稳态响应闭合解

THE CLOSED-FORM SOLUTION OF STEADY STATE RESPONSE OF HYSTERETIC SYSTEM UNDER STOCHASTIC EXCITATION

滞迟系统属于一类典型的强非线性系统,滞迟力不仅取决于系统的瞬时变形,还与变形历程有关.虽然滞迟系统的随机振动问题已被广泛研究,但至今尚未得到滞迟系统随机响应概率密度函数的精确闭合解.本文运用迭代加权残值法获得了高斯白噪声激励下Bouc-Wen滞迟系统稳态响应概率密度函数的近似闭合解.首先,运用等效线性化法求出系统的稳态高斯概率密度函数;然后以此构造权函数,应用加权残值法求得了系统指数多项式形式的非高斯概率密度函数;最后引入迭代的过程,逐步优化权函数,提高计算所得结果的精度.以随机地震激励下钢纤维陶粒混凝土结构的稳态响应作为算例,其中Bouc-Wen模型的参数是基于拟静力学试验数据,并应用最小二乘法辨识获得.与Monte Carlo模拟结果相比,等效线性化法得到的结果精度较差;由加权残值法得到的结果能够表现出非线性特征,但其精度依然无法令人满意;采用迭代加权残值法得到的近似闭合解与Monte Carlo模拟的结果吻合非常好;对于较强随机激励情形,采用渐进迭代加权残值法具有较高的求解效率,所获得的理论解析解具有较高的精度.结果表明,所获得的近似闭合解不仅对于土木工程领域具有重要的实际应用价值,而且还可作为检验其他非线性系统随机响应预测方法的精度的标准.

The hysteretic system is one of the typical strongly nonlinear systems.Hysteretic force depends not only on the instantaneous deformation but also on the past history of deformation.In the last few decades,random vibration of hysteretic system has been studied extensively,but no closed-form solution of random response of hysteretic systems is available so far.In this paper,the newly developed nonlinear random vibration scheme called iterative method of weighted residuals is explored to obtain the closed-form solution of steady-state probability density function(PDF)of the Bouc-Wen hysteretic system under Gaussian white noise excitation.First,a Gaussian PDF is obtained with equivalent linearization technique,which is used as a weighting function.Then,the method of weighted residuals is utilized to determine the non-Gaussian PDF of exponential polynomial type.Finally,an iterative procedure is introduced to improve the accuracy of the solutions obtained from the method of weighted residuals.As an illustrative example,the steadystate stochastic response of the steel fiber reinforced ceramsite concrete column under random excitation is studied,in which the hysteretic parameters associated with Bouc-Wen hysteretic model is identified from the pseudo-static test by using the method of least square.Compared to the Monte Carlo results,the accuracy of results obtained from equivalent linearization method is poor.The results obtained from weight residue method can show the nonlinearity of Bouc-Wen systems,but its accuracy is still unsatisfactory.The iterative method of weight residuals can lead to results with higher accuracy.In the case of stronger random excitation,the progressive iterative method of weighted residuals has high e_ciency.The obtained solutions agree well with the Monte Carlo simulation data.The proposed closed-form solution of PDF of Bouc-Wen hysteretic system not only is significant to the civil engineering,but also can be a benchmark to examine the accuracy of solutions obtained by other methods.