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<p>The recovery of the probability density function (PDF) of the response of a stochastic dynamic system is a quite thorny problem, since the PDF of structural response, especially the power spectral density (PSD), usually has a complex shape. This extremely irregular PDF poses a major challenge for many well-known PDF modeling methods. To this end, this paper proposes to employ the B-spline representation of the PDF to solve uncertainty quantification (UQ) problems for stochastic structural dynamics, and develops a novel B-spline-based maximum entropy method (MEM). For a general problem, a parameterized nonlinear mapping is firstly introduced to transform the PSD to the bounded interval [0, 1]. Then, we can represent the PDF of the transformed PSD by using the B-spline function and employ the MEM to estimate the undetermined coefficients. Finally, the PDF of the original PSD can be derived. The parameters of the nonlinear mapping are the key to improve computational accuracy, which is determined in a data-driven way by using the metaheuristic algorithm (the Beluga whale optimization algorithm). Two toy examples and one engineering problem are used to test the performance of the proposed method. The results show that the proposed method can capture complex PDF characteristics of the PSD of the stochastic dynamic response, if they exist, and is markedly superior to the compared methods.</p>

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An effective approach for uncertainty analysis of power spectral density of stochastic vibration problems via B-spline theory and maximum entropy method

  • 期刊论文

  • 发表日期：

    2026年1月2日

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  • Yangfan Li,
  • Yang Duan,
  • Wanxin He,
  • Michael Beer

摘要

The recovery of the probability density function (PDF) of the response of a stochastic dynamic system is a quite thorny problem, since the PDF of structural response, especially the power spectral density (PSD), usually has a complex shape. This extremely irregular PDF poses a major challenge for many well-known PDF modeling methods. To this end, this paper proposes to employ the B-spline representation of the PDF to solve uncertainty quantification (UQ) problems for stochastic structural dynamics, and develops a novel B-spline-based maximum entropy method (MEM). For a general problem, a parameterized nonlinear mapping is firstly introduced to transform the PSD to the bounded interval [0, 1]. Then, we can represent the PDF of the transformed PSD by using the B-spline function and employ the MEM to estimate the undetermined coefficients. Finally, the PDF of the original PSD can be derived. The parameters of the nonlinear mapping are the key to improve computational accuracy, which is determined in a data-driven way by using the metaheuristic algorithm (the Beluga whale optimization algorithm). Two toy examples and one engineering problem are used to test the performance of the proposed method. The results show that the proposed method can capture complex PDF characteristics of the PSD of the stochastic dynamic response, if they exist, and is markedly superior to the compared methods.