非线性复合随机振动方法研究及其工程应用
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摘要
本文主要研究非线性随机结构在非平稳随机地震作用下的随机反应,即非线性复合随机振动系统的随机反应。
在总结国内外相关研究概况后,本文将求解非线性随机结构的改进摄动法与虚拟激励法结合,形成改进摄动虚拟激励法,求解单自由度非线性复合随机振动系统的反应;在采用虚拟激励法时,本文直观地给出了虚拟反应。为计算单自由度非线性复合随机系统在随机作用下的随机反应,本文依次做了如下工作:对线性结构,采用摄动虚拟激励法进行计算,考虑到空间随机场与随机过程的独立性,对摄动虚拟激励法进行修正;对非线性随机结构在非平稳激励下的随机反应,采用等效随机线性化法和改进摄动虚拟激励法求解,非线性随机复合系统的随机反应计算方法以线性复合随机系统的方法为基础。最后给出Monte Carlo法模拟非线性复合随机系统的地震反应的实现过程。
对多自由度线性复合随机结构的随机反应,本文提出基于数论选点的直接模态摄动法、基于Monte Carlo选点的直接模态摄动法、基于数论选点的时程分析法求解复合随机系统的随机反应。为采用模态叠加法,需要求解随机模态,本文采用直接模态摄动法求解随机模态,本文在这方面工作如下:
采用直接模态摄动法求解有阻尼确定振动系统的复模态。计算结果表明,这种方法概念清晰、省时,计算结果也很精确。当直接模态摄动法求出的复模态用于复模态的模态叠加法时,计算结果也是相当精确、省时的。由于土木工程结构的阻尼一般很小,求解有阻尼系统复模态的直接模态摄动法应用于土木工程中是很实用的一种方法。
提出基于数论选点的直接模态摄动法和基于Monte Carlo选点的直接模态摄动法求解随机结构的随机模态。首先求解无阻尼随机结构的随机模态,随后求解有阻尼随机结构的随机模态。这两种方法利用数论选点法和Monte Carlo选点法选取结构样本,然后采用直接模态摄动法计算样本结构相对于均值结构的摄动量,从而得到样本结构的模态,最后采用统计方法求得随机模态。对于较少随机变量确定的随机结构,基于数论选点的直接模态摄动法计算效率较基于Monte Carlo选点的直接模态摄动法计算效率高,可以很快很准确地得到计算结果。
提出基于数论选点的直接模态摄动法、基于Monte Carlo选点的直接模态摄动法和基于数论选点的时程分析法求解复合随机系统的随机反应,并与基于Monte Carlo选点的FOSS法、基于Monte Carlo选点的时程分析法的计算结果进行比较。当随机变量较少,变异性很小时,基于数论选点的直接模态摄动法、基于数论选点的时程分析法能迅速、准确地给出结构的随机反应。
在前述研究基础上,将基于数论选点的直接模态摄动法、基于Monte Carlo选点的直接模态摄动法和基于数论选点的时程分析法用于求解非线性土层的复合随机地震反应。对土层的随机场,以工程实例,采用相关函数法求解了实际场地的波动因子,以便采用随机有限元法将随机场数值离散。对随机地震,采用基于Hartley正交基的K—L分解法将功率谱表示的随机地震波表示为独立随机变量和确定过程的线性组合。在此基础上,将等效线性化法和基于数论选点的直接模态摄动法、基于Monte Carlo选点的直接模态摄动法、基于数论选点的时程分析法结合求解随机土层的地震反应。
在论文的最后,利用Monte Carlo法、摄动虚拟激励法、频域传递函数摄动法研究结构随机阻尼对结构位移响应的影响。
The main effort of this paper is taking some researches on the response of non-linear stochastic structures under non-stationary stochastic earthquake, that is to say, the stochastic response of non-linear compound stochastic vibration system (CSVS)。Based on the publications about CSVS from home and abroad, combinated improved perturbation method to solve non-linear stochastic structures with pseudo excitation method, the improved perturbation pseudo excitation method to solve nonlinear CSVS system of SDOF(single degree of freedom) has been formed. When applying pseudo excitation method, the pseudo response has been expressed in figures. In order to analyze the response of CSVS of SDOF, the researches as follow have been taken: the linear CSVS has been solved by perturbation pseudo excitation method(PPEM) which has been corrected according to independence between space random field and time random progress; as to the nonlinear CSVS, it has been performed by equivalent stochastic linear method and improved PPEM,which is based on linear CSVS. At last, the process of Monte Carlo (MC) method for simulating nonlinear CSVS has been presented.
For CSVS of linear MDOF(multi degree of freedom), it has been proposed that the method of direct modal perturbation (DMP) based on selecting point by number-theoretic method(NTM), the DMP method based on selecting point by MC and time history processing analysis(THPA) based on selecting point by NTM to solve its stochastic response. As known that the method of modal superstition (MMS) can be to find srochastic response, here DMP can be uses to find stochastic mode. The researches about this part are as follow:
The DMP method has been used to find complex modes of determined dynamical system with damping. As shown in the computational results, the concept of DMP method is much clearer,and it's time-saving and more accurate. When applying the method to MMS, the computational results are also very accurate. Because the damping of civil engineering structures is usually very small, DMP method is appropriate to find the complex modes of the structures. DMP based on selecting point by NTM or based on selecting point by MC have been proposed to find the stochastic modes of stochastic structures. These two methods can be used to solve stochastic modes of un-damping stochastic structures or damping stochastic structures. Firstly, the structures sampling can be formed by selecting point by NTM or MC method. And then, DMP method can be used to calculate stochastic perturbation value of stochastic structure to mean structure. Finally, the stochastic modes can be found by statistic method. When stochastic structures relies on few stochastic variables or stochastic variables with small coefficients of variation, DMP method based on selecting point by NTM would be more efficient than DMP method based on selecting point by MC and more quickly and accurately to find the computational results.
In order to find stochastic response of CSVS, the DMP method based on selecting point by NTM, the DMP method based on selecting point by MC and the method of THPA based on selecting point by NTM have been proposed. The computational results of the FOSS method based on selecting point by MC, and the method of THPA based on selecting point by MC have been compared. When stochastic structures relies on few stochastic variables or stochastic variables with small coefficients of variation,stochastic response of CSVS can be quickly and accurately found by the DMP method based on selecting point by NTM, and the method of THPA based on selecting point by NTM.
According to the research about linear CSV, it is naturally to apply the DMP method based on selecting point by NTM, the DMP method based on selecting point by MC and the method of THPA based on selecting point by NTM to solve stochastic seismic response of nonlinear soil with CSVS. The relative function method has been applied to solve fluctuating factor of real soil layer random field and the stochastic finite method applied for discreted random field. Stationary stochastic earthquake has been expressed in linear combination between independent stochastic variable and determined process of Karhunen-Loeve expansion based on Hartley orthogonal basic function. Based on work of these preparations, the equivalent linear method and the methods above have been used to find seismic response of nonlinear stochastic soil layer under non-stationary stochastic earthquake.
At last, the MC method,PPEM, and the frequency transfer function perturbation method have been used to analyze the effect of structural random damping onstructural displacement.
在总结国内外相关研究概况后,本文将求解非线性随机结构的改进摄动法与虚拟激励法结合,形成改进摄动虚拟激励法,求解单自由度非线性复合随机振动系统的反应;在采用虚拟激励法时,本文直观地给出了虚拟反应。为计算单自由度非线性复合随机系统在随机作用下的随机反应,本文依次做了如下工作:对线性结构,采用摄动虚拟激励法进行计算,考虑到空间随机场与随机过程的独立性,对摄动虚拟激励法进行修正;对非线性随机结构在非平稳激励下的随机反应,采用等效随机线性化法和改进摄动虚拟激励法求解,非线性随机复合系统的随机反应计算方法以线性复合随机系统的方法为基础。最后给出Monte Carlo法模拟非线性复合随机系统的地震反应的实现过程。
对多自由度线性复合随机结构的随机反应,本文提出基于数论选点的直接模态摄动法、基于Monte Carlo选点的直接模态摄动法、基于数论选点的时程分析法求解复合随机系统的随机反应。为采用模态叠加法,需要求解随机模态,本文采用直接模态摄动法求解随机模态,本文在这方面工作如下:
采用直接模态摄动法求解有阻尼确定振动系统的复模态。计算结果表明,这种方法概念清晰、省时,计算结果也很精确。当直接模态摄动法求出的复模态用于复模态的模态叠加法时,计算结果也是相当精确、省时的。由于土木工程结构的阻尼一般很小,求解有阻尼系统复模态的直接模态摄动法应用于土木工程中是很实用的一种方法。
提出基于数论选点的直接模态摄动法和基于Monte Carlo选点的直接模态摄动法求解随机结构的随机模态。首先求解无阻尼随机结构的随机模态,随后求解有阻尼随机结构的随机模态。这两种方法利用数论选点法和Monte Carlo选点法选取结构样本,然后采用直接模态摄动法计算样本结构相对于均值结构的摄动量,从而得到样本结构的模态,最后采用统计方法求得随机模态。对于较少随机变量确定的随机结构,基于数论选点的直接模态摄动法计算效率较基于Monte Carlo选点的直接模态摄动法计算效率高,可以很快很准确地得到计算结果。
提出基于数论选点的直接模态摄动法、基于Monte Carlo选点的直接模态摄动法和基于数论选点的时程分析法求解复合随机系统的随机反应,并与基于Monte Carlo选点的FOSS法、基于Monte Carlo选点的时程分析法的计算结果进行比较。当随机变量较少,变异性很小时,基于数论选点的直接模态摄动法、基于数论选点的时程分析法能迅速、准确地给出结构的随机反应。
在前述研究基础上,将基于数论选点的直接模态摄动法、基于Monte Carlo选点的直接模态摄动法和基于数论选点的时程分析法用于求解非线性土层的复合随机地震反应。对土层的随机场,以工程实例,采用相关函数法求解了实际场地的波动因子,以便采用随机有限元法将随机场数值离散。对随机地震,采用基于Hartley正交基的K—L分解法将功率谱表示的随机地震波表示为独立随机变量和确定过程的线性组合。在此基础上,将等效线性化法和基于数论选点的直接模态摄动法、基于Monte Carlo选点的直接模态摄动法、基于数论选点的时程分析法结合求解随机土层的地震反应。
在论文的最后,利用Monte Carlo法、摄动虚拟激励法、频域传递函数摄动法研究结构随机阻尼对结构位移响应的影响。
The main effort of this paper is taking some researches on the response of non-linear stochastic structures under non-stationary stochastic earthquake, that is to say, the stochastic response of non-linear compound stochastic vibration system (CSVS)。Based on the publications about CSVS from home and abroad, combinated improved perturbation method to solve non-linear stochastic structures with pseudo excitation method, the improved perturbation pseudo excitation method to solve nonlinear CSVS system of SDOF(single degree of freedom) has been formed. When applying pseudo excitation method, the pseudo response has been expressed in figures. In order to analyze the response of CSVS of SDOF, the researches as follow have been taken: the linear CSVS has been solved by perturbation pseudo excitation method(PPEM) which has been corrected according to independence between space random field and time random progress; as to the nonlinear CSVS, it has been performed by equivalent stochastic linear method and improved PPEM,which is based on linear CSVS. At last, the process of Monte Carlo (MC) method for simulating nonlinear CSVS has been presented.
For CSVS of linear MDOF(multi degree of freedom), it has been proposed that the method of direct modal perturbation (DMP) based on selecting point by number-theoretic method(NTM), the DMP method based on selecting point by MC and time history processing analysis(THPA) based on selecting point by NTM to solve its stochastic response. As known that the method of modal superstition (MMS) can be to find srochastic response, here DMP can be uses to find stochastic mode. The researches about this part are as follow:
The DMP method has been used to find complex modes of determined dynamical system with damping. As shown in the computational results, the concept of DMP method is much clearer,and it's time-saving and more accurate. When applying the method to MMS, the computational results are also very accurate. Because the damping of civil engineering structures is usually very small, DMP method is appropriate to find the complex modes of the structures. DMP based on selecting point by NTM or based on selecting point by MC have been proposed to find the stochastic modes of stochastic structures. These two methods can be used to solve stochastic modes of un-damping stochastic structures or damping stochastic structures. Firstly, the structures sampling can be formed by selecting point by NTM or MC method. And then, DMP method can be used to calculate stochastic perturbation value of stochastic structure to mean structure. Finally, the stochastic modes can be found by statistic method. When stochastic structures relies on few stochastic variables or stochastic variables with small coefficients of variation, DMP method based on selecting point by NTM would be more efficient than DMP method based on selecting point by MC and more quickly and accurately to find the computational results.
In order to find stochastic response of CSVS, the DMP method based on selecting point by NTM, the DMP method based on selecting point by MC and the method of THPA based on selecting point by NTM have been proposed. The computational results of the FOSS method based on selecting point by MC, and the method of THPA based on selecting point by MC have been compared. When stochastic structures relies on few stochastic variables or stochastic variables with small coefficients of variation,stochastic response of CSVS can be quickly and accurately found by the DMP method based on selecting point by NTM, and the method of THPA based on selecting point by NTM.
According to the research about linear CSV, it is naturally to apply the DMP method based on selecting point by NTM, the DMP method based on selecting point by MC and the method of THPA based on selecting point by NTM to solve stochastic seismic response of nonlinear soil with CSVS. The relative function method has been applied to solve fluctuating factor of real soil layer random field and the stochastic finite method applied for discreted random field. Stationary stochastic earthquake has been expressed in linear combination between independent stochastic variable and determined process of Karhunen-Loeve expansion based on Hartley orthogonal basic function. Based on work of these preparations, the equivalent linear method and the methods above have been used to find seismic response of nonlinear stochastic soil layer under non-stationary stochastic earthquake.
At last, the MC method,PPEM, and the frequency transfer function perturbation method have been used to analyze the effect of structural random damping onstructural displacement.
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