The dynamic analysis of systems has several applications in the project or monitoring and controlling machines and equipments. In particular, the analysis of nonlinear systems has academic and industrial appeal for owning many particularities such as structural integrity, models updating, stability and real-time predictability. Those particularities complicate the study of these models. However, the development of more representative mathematical models requires normally costly computations. By considering the uncertainties associated with the fabrication process (geometrical dimensions, physical and material properties), as well as those related to the operational conditions (boundary conditions, external forces, etc.), it complicates further the analysis of those nonlinear systems. Thus, this paper proposes a methodology to analyze nonlinear systems subjected to uncertainties using the stochastic finite element method. In view of the high computational cost needed to construct the confidence region predicted with the stochastic nonlinear computational model, it is proposed herein a new model reduction method based on the construction of an adaptive iterative enriched basis to deal with the stochastic nonlinear system addressed herein composed by a thin rectangular plate used as an application. The results demonstrate clearly the efficiency and accuracy of the proposed method as an efficient tool to approximate the nonlinear responses of more complex nonlinear systems subjected to uncertainties. Also, it demonstrates the relevance of taking into account the uncertainties in the modeling of nonlinear systems to consider more realistic situations.

系统的动态分析在项目或监视和控制机器设备中有多个应用。特别地，非线性系统的分析具有许多特殊性，例如结构完整性，模型更新，稳定性和实时可预测性，在学术和工业上都具有吸引力。这些特性使这些模型的研究复杂化。但是，开发更具代表性的数学模型通常需要昂贵的计算。通过考虑与制造过程相关的不确定性（几何尺寸，物理和材料属性）以及与操作条件相关的不确定性（边界条件，外力等），这使得那些非线性系统的分析变得更加复杂。从而，本文提出了一种使用随机有限元方法分析具有不确定性的非线性系统的方法。考虑到构造用随机非线性计算模型预测的置信区域所需的高计算成本，在此提出一种基于自适应迭代丰富基础的构造以处理本文所解决的随机非线性系统的新模型简化方法。由用作应用的矩形薄板制成。结果清楚地证明了该方法的有效性和准确性，该方法可作为一种有效工具来逼近不确定性更复杂的非线性系统的非线性响应。也，