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

doi:10.1088/1674-1056/ade669

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.

Keywords: energy-regenerative suspension stochastic P-bifurcation stochastic resonance time-delayed feedback control

Received: 12 May 2025
Revised: 15 June 2025
Accepted manuscript online: 20 June 2025

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

Fund: 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).

Corresponding Authors: Ya-Hui Sun, Yang Liu
E-mail: sunyh@gdut.edu.cn;Y.Liu2@exeter.ac.uk

Cite this article:

Zhao-Bin Zeng(曾昭彬), Ya-Hui Sun(孙亚辉), and Yang Liu(刘洋) Probabilistic distribution and stochastic P-bifurcation of a nonlinear energy-regenerative suspension system with time-delayed feedback control 2026 Chin. Phys. B 35 010201

[1] Kasseris E P and Heywood J B 2007 SAE Transactions 626
[2] Wang R, Chen Z, Xu H, Schmidt K, Gu F and Ball A D 2014 IEEE Proceedings of 20th International Conference on Automation and Computing p. 32
[3] Fan Y and Sun J 2017 Mechanics 23 92
[4] Hrovat D 1997 Automatica 33 1781
[5] Abdelkareem M A, Makrahy M M, Abd-El-Tawwab A M, El-Razaz A, Kamal Ahmed Ali M and Moheyeldein M 2018 Proceedings of the Institution of Mechanical Engineers, Part K: Journal of Multi-body Dynamics 232 84
[6] Rana R S and Adhyaru D M 2023 Journal of Circuits, Systems and Computers 32 2350312
[7] Kim H J, Kang S M, Kim SMand Kim J H 2024 International Journal of Automotive Technology 25 627
[8] Wang H, Jasim A and Chen X 2018 Applied Energy 212 1083
[9] AliMK A, Fuming P, Younus H A, AbdelkareemMA, Essa F, Elagouz A and He X J 2018 Applied Energy 211 461
[10] Wei C and Jing X 2017 A Renewable and Sustainable Energy Reviews 74 1
[11] Rehman A U, Ryu S W, Park H and Jung J W 2024 IEEE Transactions on Industrial Informatics 20 6060
[12] Sun Y H, Zeng Z B and Huang M L 2024 Engineering Structures 312 118211
[13] Abdelkareem M A, Xu L, Guo X, Ali M K A, Elagouz A, Hassan M A, Essa F and Zou J 2018 Mechanical Systems and Signal Processing 110 307
[14] Guan Y, Li W, Kozak D and Zhao J 2024 Reliability Engineering & System Safety 249 110206
[15] Huang D, Han J, Li W, Deng H and Zhou S 2023 Commun. Nonlinear Sci. Numer. Simul. 119 107086
[16] Liang Q, Yu W X and Xiao Q M 2025 Chin. Phys. B 34 040502
[17] Han J, Huang D, Li W, Yang G and Gubeljak N 2023 International Journal of NonLinear Mechanics 157 104518
[18] Khoshnoud F, Zhang Y, Shimura R, Shahba A, Jin G, Pissanidis G, Chen Y K and De Silva CW2015 IEEEASME Transactions on Mechatronics 20 2513
[19] Cui T H, Li J, Yuan Q, Wei Y Q, Dai S Q, Li P D, Zhou F, Zhang J Q, Chen L and Feng M 2023 Chin. Phys. Lett. 40 080501
[20] Huang B, Hsieh C Y, Golnaraghi F and Moallem M 2015 Journal of Sound and Vibration 357 16
[21] Wei C and Taghavifar H 2017 Energy 134 279
[22] Zhang H, Guo X, Fang Z, Xu L and Zhang J 2015 Journal of Vibration, Measurement & Diagnosis 35 225
[23] Wang R, Gu F, Cattley R and Ball A D 2016 Energies 9 386
[24] Zou J, Guo X, Abdelkareem M A, Xu L and Zhang J 2019 Mechanical Systems and Signal Processing 127 345
[25] Liang H, Hao G and Olszewski O Z 2021 Sensors and Actuators A: Physical 331 112743
[26] Ibrahim P, Arafa M and Anis Y 2021 Sensors 21 5611
[27] Wang M, Yin P, Li Z, Sun Y, Ding J, Luo J, Xie S, Peng Y and Pu H 2020 Energy Conversion and Management 220 113050
[28] Zhao Z, Wang T, Zhang B, Shi J, et al. 2015 Mathematical Problems in Engineering 2019 1
[29] Xie X and Wang Q 2015 Energy 86 385
[30] Xiao H,Wang X and John S 2015 Mechanical Systems and Signal Processing 58 355
[31] Kou F, Jing Q, Chen C and Wu J 2020 Shock and Vibration 2020 1
[32] Zuo L, Scully B, Shestani J and Zhou Y 2010 Smart Materials and Structures 19 045003
[33] Li C and Peter W T 2013 Smart Materials and Structures 22 065024
[34] Zhang J, Liu J, Liu B and Li M 2022 Actuators 11 189
[35] Suda Y and Shiiba T 1996 Vehicle System Dynamics 25 641
[36] Zhang Z, Zhang X, Chen W, Rasim Y, Salman W, Pan H, Yuan Y and Wang C 2016 Applied Energy 178 177
[37] Liu Y, Xu L and Zuo L 2017 IEEEASME Transactions on Mechatronics 22 1933
[38] Satpute N, Iwaniec M, Narina R and Satpute S 2020 Vibrations in Physical Systems 31
[39] Taghavifar H 2021 Mechanical Systems and Signal Processing 146 106988
[40] Gu K and Niculescu S I 2003 Journal of Dynamic Systems, Measurement, and Control 125 158
[41] Wang Q, Wu H and Yang Y 2022 Nonlinear Dynamics 1
[42] Cermák C and Nechvátal L 2020 Physica D 404 132339
[43] Belhaq M, Ghouli Z and Hamdi M 2018 Nonlinear Dynamics 94 2537
[44] Sun X, Xu J and Fu J 2017 Mechanical Systems and Signal Processing 87 206
[45] Yang T and Cao Q 2019 Nonlinear Dyn. 96 1511
[46] Jin Y and Meng J 2020 Commun. Nonlinear Sci. Numer. Simul. 90 105333
[47] Zhang Y, Jin Y and Li Y 2021 Physica D 422 132908
[48] Wu K, Ren C and Atay F M 2024 Energy 300 131578
[49] Hamdi M and Belhaq M 2018 Recent Trends in Applied Nonlinear Mechanics and Physics pp. 69–83
[50] Nguyen C H, Nguyen D S and Halvorsen E 2014 J. Phys.: Conf. Ser. 557 012114
[51] Fokou I M, Buckjohn C N D, Siewe M S and Tchawoua C 2018 Commun. Nonlinear Sci. Numer. Simul. 56 177
[52] Stratonovich R L 1967 Topics in the Theory of Random Noise, Vol. 2 (CRC Press)
[53] Zhu W 1996 Applied Mechanics Reviews 49 S72
[54] Guan Y, Li W, Huang D and Gubeljak N 2023 Chaos, Solitons and Fractals 173 113641
[55] Yang Y G, Huang M L, Guo S L and Sun Y H 2023 Mechanical Systems and Signal Processing 186 109837
[56] Yang J, Sanjuáan M A, Liu H, Litak G and Li X 2016 Commun. Nonlinear Sci. Numer. Simul. 41 104
[57] Wang R, Ding Y and Kang Y 2024 The European Physical Journal Special Topics 1
[58] Sun Y H, Sun Y, Yang Y G and Xu W 2023 Probabilistic Engineering Mechanics 73 103464
[59] Liao Z J, Sun Y H and Liu Y 2024 Chin. Phys. B 33 070205

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

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