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<p>Addressing the issues of long computation time and oscillating solutions in the process of solving stochastic reliability analysis, this study proposes an efficient and stable algorithm based on the combination of K–L decomposition and probability density evolution method using the QUICK difference scheme. Firstly, the stochastic evolution rate of random variables is calculated based on K-L decomposition. Secondly, the probability density evolution process is calculated by QUICK difference scheme and adaptive flux limiter. Thirdly, the equivalent stiffness iteration method is introduced to linearize the system. Finally, the reliability is calculated by using adaptive Simpson’s law. In order to verify the effectiveness of the method, a reliability model of nonlinear gear system was established considering the effects of dimensionless stiffness, damping and excitation frequency. Compared with Monte Carlo method, the proposed method can guarantee the calculation accuracy and reduce the calculation time by 99.3%. Results show that the difference of probability density evolution and reliability of dimensionless excitation frequency is the largest in the chaotic response state and the nonperiodic response state caused by three parameters, while the difference of dimensionless damping is mainly reflected in the initial operation of the system, and there is almost no difference in dimensionless stiffness. This method guarantees the calculation accuracy and restrains the numerical oscillation on the premise of reducing the calculation cost, and provides a new idea for the reliability design of complex mechanical systems.</p> Graphical abstract <p></p>

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Stochastic dynamic reliability optimization of nonlinear gear systems using Karhunen–Loève representation and probability density evolution theory

  • 期刊论文

  • 发表日期：

    2026年5月8日

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  • Hongchuan Cheng,
  • Zhaoyang Shi,
  • Guilong Fu,
  • Jianxun Du,
  • Zhiwu Shang,
  • Xiafei Shi,
  • Xingbao Huang

摘要

Addressing the issues of long computation time and oscillating solutions in the process of solving stochastic reliability analysis, this study proposes an efficient and stable algorithm based on the combination of K–L decomposition and probability density evolution method using the QUICK difference scheme. Firstly, the stochastic evolution rate of random variables is calculated based on K-L decomposition. Secondly, the probability density evolution process is calculated by QUICK difference scheme and adaptive flux limiter. Thirdly, the equivalent stiffness iteration method is introduced to linearize the system. Finally, the reliability is calculated by using adaptive Simpson’s law. In order to verify the effectiveness of the method, a reliability model of nonlinear gear system was established considering the effects of dimensionless stiffness, damping and excitation frequency. Compared with Monte Carlo method, the proposed method can guarantee the calculation accuracy and reduce the calculation time by 99.3%. Results show that the difference of probability density evolution and reliability of dimensionless excitation frequency is the largest in the chaotic response state and the nonperiodic response state caused by three parameters, while the difference of dimensionless damping is mainly reflected in the initial operation of the system, and there is almost no difference in dimensionless stiffness. This method guarantees the calculation accuracy and restrains the numerical oscillation on the premise of reducing the calculation cost, and provides a new idea for the reliability design of complex mechanical systems.

Graphical abstract