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<p>Quasi-zero stiffness (QZS) vibration isolators are widely used in low-frequency applications for their superior vibration attenuation. However, strong nonlinearity leads to complex, non-Gaussian steady-state responses, particularly near resonance, challenging conventional uncertainty quantification (UQ) methods. In this study, a Mixture Density Network (MDN) is introduced to the UQ of QZS systems, enabling direct learning of the conditional probability distribution of the steady-state peak displacement response given uncertain input parameters. This approach accurately predicts response distributions across the full frequency range while requiring approximately 15% of the computation time of traditional Monte Carlo simulations in the studied cases. The trained MDN captures the progressive transition of probability mass from high-response to low-response branches, revealing multimodal response evolution that has been rarely reported in existing UQ studies of QZS systems. Sobol’ sensitivity analysis is then performed using parameters derived from the MDN, and the results are further validated against gradient-based sensitivity estimates, showing excellent agreement. The study demonstrates an alternative methodological framework for efficiently quantifying uncertainty in strongly nonlinear QZS systems, offering both theoretical insight and practical guidance for vibration isolation design.</p>

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Application of mixture density network on solving uncertainty qualification problem in quasi-zero stiffness vibration isolator

  • 期刊论文

  • 发表日期：

    2026年4月2日

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  • Junhan Zhu,
  • Jin Wu,
  • Junhan An,
  • Huan He

摘要

Quasi-zero stiffness (QZS) vibration isolators are widely used in low-frequency applications for their superior vibration attenuation. However, strong nonlinearity leads to complex, non-Gaussian steady-state responses, particularly near resonance, challenging conventional uncertainty quantification (UQ) methods. In this study, a Mixture Density Network (MDN) is introduced to the UQ of QZS systems, enabling direct learning of the conditional probability distribution of the steady-state peak displacement response given uncertain input parameters. This approach accurately predicts response distributions across the full frequency range while requiring approximately 15% of the computation time of traditional Monte Carlo simulations in the studied cases. The trained MDN captures the progressive transition of probability mass from high-response to low-response branches, revealing multimodal response evolution that has been rarely reported in existing UQ studies of QZS systems. Sobol’ sensitivity analysis is then performed using parameters derived from the MDN, and the results are further validated against gradient-based sensitivity estimates, showing excellent agreement. The study demonstrates an alternative methodological framework for efficiently quantifying uncertainty in strongly nonlinear QZS systems, offering both theoretical insight and practical guidance for vibration isolation design.