利用实值二次谐波平衡公式计算多项式动力系统周期轨道的库普曼-希尔稳定性

IF 2.8 3区 工程技术 Q2 MECHANICS International Journal of Non-Linear Mechanics Pub Date : 2024-09-10 DOI:10.1016/j.ijnonlinmec.2024.104894
Fabia Bayer , Remco I. Leine , Olivier Thomas , Aurélien Grolet
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引用次数: 0

摘要

在本文中,我们将最近为周期解的数值稳定性分析而引入的 Koopman-Hill 投影法推广到经典的实值谐波平衡 (HBM) 公式中。我们将其纳入渐近数值方法(ANM)延续框架,为通过 HBM 获得的频率响应曲线提供了一种高效的数值稳定性分析工具。希尔矩阵承载着稳定性信息,是 HBM 求解过程的副产品,用传统方法分析希尔矩阵往往具有计算上的挑战性。为了解决这个问题,我们将 Koopman-Hill 投影稳定性方法从复值公式推广到实值公式,该方法使用矩阵指数从希尔矩阵中提取单调矩阵。此外,我们还提出了一种差分重铸程序,使实值希尔矩阵在 ANM 延续框架内立即可用。以非线性 von Kármán 梁为例,我们证明了这些修改提高了频率响应曲线稳定性分析的计算效率。
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Koopman–Hill stability computation of periodic orbits in polynomial dynamical systems using a real-valued quadratic harmonic balance formulation

In this paper, we generalize the Koopman–Hill projection method, which was recently introduced for the numerical stability analysis of periodic solutions, to be included immediately in classical real-valued harmonic balance (HBM) formulations. We incorporate it into the Asymptotic Numerical Method (ANM) continuation framework, providing a numerically efficient stability analysis tool for frequency response curves obtained through HBM. The Hill matrix, which carries stability information and follows as a by-product of the HBM solution procedure, is often computationally challenging to analyze with traditional methods. To address this issue, we generalize the Koopman–Hill projection stability method, which extracts the monodromy matrix from the Hill matrix using a matrix exponential, from complex-valued to real-valued formulations. In addition, we propose a differential recast procedure, which makes this real-valued Hill matrix immediately available within the ANM continuation framework. Using as an example a nonlinear von Kármán beam, we demonstrate that these modifications improve computational efficiency in the stability analysis of frequency response curves.

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来源期刊
CiteScore
5.50
自引率
9.40%
发文量
192
审稿时长
67 days
期刊介绍: The International Journal of Non-Linear Mechanics provides a specific medium for dissemination of high-quality research results in the various areas of theoretical, applied, and experimental mechanics of solids, fluids, structures, and systems where the phenomena are inherently non-linear. The journal brings together original results in non-linear problems in elasticity, plasticity, dynamics, vibrations, wave-propagation, rheology, fluid-structure interaction systems, stability, biomechanics, micro- and nano-structures, materials, metamaterials, and in other diverse areas. Papers may be analytical, computational or experimental in nature. Treatments of non-linear differential equations wherein solutions and properties of solutions are emphasized but physical aspects are not adequately relevant, will not be considered for possible publication. Both deterministic and stochastic approaches are fostered. Contributions pertaining to both established and emerging fields are encouraged.
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