中国物理B ›› 2026, Vol. 35 ›› Issue (1): 10201-010201.doi: 10.1088/1674-1056/ade669

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Probabilistic distribution and stochastic P-bifurcation of a nonlinear energy-regenerative suspension system with time-delayed feedback control

Zhao-Bin Zeng(曾昭彬)1, Ya-Hui Sun(孙亚辉)1,2,†, and Yang Liu(刘洋)2,‡   

  1. 1 School of Mathematics and Statistics, Guangdong University of Technology, Guangzhou 510520, China;
    2 Exeter Small-Scale Robotics Laboratory, Engineering Department, University of Exeter, North Park Road, Exeter, EX4 4QF, United Kingdom
  • 收稿日期:2025-05-12 修回日期:2025-06-15 接受日期:2025-06-20 发布日期:2026-01-09
  • 通讯作者: Ya-Hui Sun, Yang Liu E-mail:sunyh@gdut.edu.cn;Y.Liu2@exeter.ac.uk
  • 基金资助:
    Project supported by the National Natural Science Foundation of China (Grant No. 12002089), the Science and Technology Projects in Guangzhou (Grant No. 2023A04J1323), and UKRI Horizon Europe Guarantee (Marie Skłodowska- Curie Fellowship) (Grant No. EP/Y016130/1).

Probabilistic distribution and stochastic P-bifurcation of a nonlinear energy-regenerative suspension system with time-delayed feedback control

Zhao-Bin Zeng(曾昭彬)1, Ya-Hui Sun(孙亚辉)1,2,†, and Yang Liu(刘洋)2,‡   

  1. 1 School of Mathematics and Statistics, Guangdong University of Technology, Guangzhou 510520, China;
    2 Exeter Small-Scale Robotics Laboratory, Engineering Department, University of Exeter, North Park Road, Exeter, EX4 4QF, United Kingdom
  • Received:2025-05-12 Revised:2025-06-15 Accepted:2025-06-20 Published:2026-01-09
  • Contact: Ya-Hui Sun, Yang Liu E-mail:sunyh@gdut.edu.cn;Y.Liu2@exeter.ac.uk
  • Supported by:
    Project supported by the National Natural Science Foundation of China (Grant No. 12002089), the Science and Technology Projects in Guangzhou (Grant No. 2023A04J1323), and UKRI Horizon Europe Guarantee (Marie Skłodowska- Curie Fellowship) (Grant No. EP/Y016130/1).

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摘要： Energy-regenerative suspension combined with piezoelectric and electromagnetic transduction has evolved into a core technological pathway in advancing automotive design paradigms. With the aim of improving energy harvesting performance, time-delayed feedback control is widely used in an energy-regenerative suspension system under different external disturbances in this paper. Meanwhile, limited research has addressed the stochastic dynamics of time-delayed nonlinear energy-regenerative suspension systems. Different from previous studies, this work studies the stochastic response and $P$-bifurcation of the nonlinear energy-regenerative suspension system with time-delayed feedback control. Firstly, an approximately equivalent dimension reduction system is established by the variable transformation method, and then the stationary probability density function of amplitude is obtained by the stochastic averaging method. Secondly, the precision of the method used in this work is verified by comparing the numerical solutions with the analytical results. Finally, based on the stationary probability density function, the influence of system parameters on stochastic $P$-bifurcation and the mean output power is discussed.

关键词: Probabilistic distribution and stochastic P-bifurcation of a nonlinear energy-regenerative suspension system with time-delayed feedback control

Abstract: Energy-regenerative suspension combined with piezoelectric and electromagnetic transduction has evolved into a core technological pathway in advancing automotive design paradigms. With the aim of improving energy harvesting performance, time-delayed feedback control is widely used in an energy-regenerative suspension system under different external disturbances in this paper. Meanwhile, limited research has addressed the stochastic dynamics of time-delayed nonlinear energy-regenerative suspension systems. Different from previous studies, this work studies the stochastic response and $P$-bifurcation of the nonlinear energy-regenerative suspension system with time-delayed feedback control. Firstly, an approximately equivalent dimension reduction system is established by the variable transformation method, and then the stationary probability density function of amplitude is obtained by the stochastic averaging method. Secondly, the precision of the method used in this work is verified by comparing the numerical solutions with the analytical results. Finally, based on the stationary probability density function, the influence of system parameters on stochastic $P$-bifurcation and the mean output power is discussed.

Key words: energy-regenerative suspension, stochastic P-bifurcation, stochastic resonance, time-delayed feedback control

中图分类号:  (Delay and functional equations)

  • 02.30.Ks
02.30.Jr (Partial differential equations) 02.50.-r (Probability theory, stochastic processes, and statistics) 02.60.Cb (Numerical simulation; solution of equations)