摘要
众多的研究工作表明，结构参数的随机变异性可以引起结构随机动力响应的大幅度涨
落，结构力学参数的随机性还可能成为主导因素，在结构系统模型中引入随机性的概念，采
用随机结构系统模型是较确定性结构系统模型更为合理的一种选择。
本文的研究工作主要由两个部分组成；第一部分是分别用分段线性化方法和等效随机系
统方法对含随机参数非线性结构动态响应统计量的求解；第二部分是建立了用扩阶随机有限
元方法求解含随机参数粘弹性问题的计算模型。主要工作如下：
1．用Monte-Carlo数值模拟技术对含随机参数结构频率响应函数进行分析。在单自由情况
中，质量和刚度的随机能削弱频响函数的峰值；对多自由度情况，参数的变化将引起频
响函数的大幅值增加，从两个自由度的数值模拟可以看出，在随机抽样调查中，质量和
刚度的小变异，会给频响函数带来大的突变，这是按照灵敏度的方法所不能预示的，也
是试验测试中应该引起注意的。
2．用分段线性化方法计算了含随机参数非线性结构的动态响应。计算结果可以看出本文解
和Monte-Carlo模拟解十分吻合，曲线几乎重合，说明方法可靠，而且计算时间相比
Monte-Carlo法少很多。对随机和确定结构计算的结果表明，当考虑参数随机时，位移、
速度响应将发生大的变化。在相同条件下，结构非线性时的响应时程和线性时还是有大
的差异的。正负幅值的大小，以及出现的时刻都发生了变化。因此，在研究随机结构时，
有必要对非线性情况进行分析。
3．必须注意到，计算非线性随机结构所采用的分段线性化方法仅适用于单自由度质量随机
时的情况，刚度和阻尼随机时，由于涉及到求解等效刚度、等效阻尼以及等效刚度均方
差、等效阻尼均方差，而不可解。其本质上是参数的随机和非线性不耦合时，才适用，
当参数的随机和非线性耦合时，质量、刚度、阻尼随机，以及多自由度的情况，将采用
等效随机系统分析方法加以解决。本文推导了相应的多自由度的计算公式，给出了算法，
编制了非线性迭代计算程序。从计算结果可以看出本文解和Monte-Carlo模拟解十分吻
合，而且计算时间相比Monte-Carlo法少很多。本文提出的等效随机系统分析法，原理
清楚，方法简单，算例的精度较高，有望在处理非线性随机动力问题中得到较大发展。
4．介绍了随机有限元方法的研究现状，对于线弹性问题，方法趋于成熟，但是用随机的思
想来进行粘弹性结构的分析的研究工作迄今少有见诸报导；本文在空间上采用有限元
法，在时域上采用差分的方法，首次建立了用扩阶随机有限元方法求解含随机参数粘弹
性问题的计算模型，推导了相应的有限元公式，给出了算法，编制了有限元程序。计算
结果可以看出，本文解和Monte-Carlo模拟的结果吻合，计算时间相比Monte-Carlo法少。
随机和确定结构计算的结果表明，当考虑参数随机时，位移响应将发生变化，有必要考
虑随机的情况。
Many investigations show that randomicity of structures?parameter will bring large value of stochastic dynamic response of structures. Randomicity of structures?mechanics parameter may be dominant factors. Therefore, introduction of randomicity into system model of structure and using random system model are more reasonable than that of determinate system model.
The work in this dissertation mainly consists of two parts. In the first part, the dynamic response of nonlinear structures with uncertain physical parameters is studied by means of subsection linearization method and equivalent random systems method, separately. In the second part, a method for analyzing the response of viscoelastic structures with uncertain physical parameters is proposed, with FEM in space domain and discrete method in time domain. The main results of this study are lists as follows:
1. The statistical characteristic of frequency response functions of single and two degrees of freedom structures with uncertain physical parameters are obtained by using Monte-Carlo method, and some results are presented. For the single degree of freedom systems, the randomicity of mass and stiffness can considerably weaken the peak value of frequency response functions. While for the two degrees of freedom structures, the randomicity of parameters will increase the peak value of frequency response functions largely. The numerical value simulation of two degrees of freedom structures shows that the small randomicity of mass and stiffness will bring the frequency response functions large perturbation. This fact can not be indicated by the method of sensitivity of the determine model, and must also be considered in measurement.
2. Dynamic response of nonlinear structures with uncertain physical parameters is studied by subsection linearization method. The numerical results show that the responses of the proposed approach are much close to the Monte Carlo solutions and that the time of calculation of subsection linearizaton method is much less than that of the Monte Carlo. The results of random structures and determinate structures show that when the ranmomicity of parameters is considered, response of displacement and velocity will occur large changes. In the same condition, there is great difference between response of nonlinear structure and that of linear structure. Thereby, during studying random structures, it is necessary to analyze the nonlinear chrarcteristics.
3. It should be pointed out that the subsection linearization method is only suitable for the single degree of freedom system with random mass. When the stiffness and damp are both stochastic, on account of coming down to seek equivalent stiffness, equivalent damp, mean square deviation of equivalent stiffness and mean square deviation of equivalent damp, that method is not applicable. In nature, it is appropriate for the case that the randomicity and nonlinearization of parameters are not coupling. When the randomicity and nonlinearization of parameters are coupled, mass, stiffhess, and damp are stochastic, equivalent random systems method is adopted. In this paper, the formulates are deduced, the arithmetic is presented, and numerical iterative programs of nonlinear structures are compiled. The numerical results show that the responses of the proposed approach are much close to the Monte Carlo solutions and that the time of calculation of equivalent random systems method is much less than that of the Monte
Carlo. The equivalent random systems method that is clear in principle, simple in plan, and high in precision of example, will be developed in the
2
disposal of dynamic response of structures with uncertain physical parameters.
4. Developments of study on stochastic finite element method (SFEM) are introduced. For the linear elastic problem, the SEEM is to mature. But the work that analyses viscoelastic random structures is almost not covered heretofore. In this paper, by order expansion stochastic finite element method, a calculation model used in analyzing th
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