An efficient iterative model reduction method for stochastic systems having geometric nonlinearities
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(2020) 42:602
TECHNICAL PAPER
An efficient iterative model reduction method for stochastic systems having geometric nonlinearities M. H. Belonsi1 · A. M. G. de Lima2 · T. Trevilato2 · R. A. Borges3 Received: 17 March 2020 / Accepted: 13 October 2020 © The Brazilian Society of Mechanical Sciences and Engineering 2020
Abstract 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. Keywords Nonlinear systems · Reduction methods · Uncertainties · Stochastic finite element
1 Introduction It is well-known that, undesirable vibrations can cause fatigue, wear, noise and, in some situations, the collapse of structures. Thus, for years, due to the relevance of vibrational study’s applicability, the analysis of vibrations has been focused, for motives of safety, convenience and simplicity, only on linear problems in mechanics, and it has been Technical editor: José Roberto de França Arruda. * M. H. Belonsi [email protected] 1
State University of Goiás, Campus Morrinhos, Morrinhos, GO, Brazil
2
School of Mechanical Engineering, Federal University of Uberlândia, Campus Santa Mônica, Uberlândia, MG, Brazil
3
School of Industrial Mathematics, Federal University of Goiás, Catalão, GO, Brazil
considered enough to understand the vibrational characteristics of these systems [1]. Silva et al. [2] have pointed out that, the structures’ modernization, as well as the considerable increase of velocity in machine equipment’s
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