Chaotic characteristics for a class of hydro-pneumatic near-zero frequency vibration isolators under dry friction and noise excitation

doi:10.1007/s10483-025-3243-6

Abstract

Abstract:

Hydro-pneumatic near-zero frequency (NZF) vibration isolators have better performance at isolating vibration with low frequencies and heavy loadings when the nonlinear fluidic damping is introduced and the pressurized gas pressure is properly adjusted. The nonlinear characteristics of such devices make their corresponding dynamic research involve chaotic dynamics. Chaos may bring negative influence and disorder to the structure and low-frequency working efficiency of isolators, which makes it necessary to clarify and control the threshold ranges for chaos generation in advance. In this work, the chaotic characteristics for a class of hydro-pneumatic NZF vibration isolators under dry friction, harmonic, and environmental noise excitations are analyzed by the analytical and numerical methods. The parameter ranges for the generation of chaos are obtained by the classical and random Melnikov methods. The chaotic characteristics and thresholds of the parameters in the systems with or without noise excitation are discussed and described. The analytical solutions and the influence of noise and harmonic excitation about chaos are tested and further analyzed through many numerical simulations. The results show that chaos in the system can be induced or inhibited with the adjustment of the magnitudes of harmonic excitation and noise intensity.

Key words: chaos, vibration isolator, Melnikov method, noise excitation

2010 MSC Number:

O175

Zhouchao WEI, Yuxi LI, T. KAPITANIAK, Wei ZHANG. Chaotic characteristics for a class of hydro-pneumatic near-zero frequency vibration isolators under dry friction and noise excitation. Applied Mathematics and Mechanics (English Edition), 2025, 46(4): 647-662.

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Figures/Tables 8

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References 64

[1]	GAO, X. and CHEN, Q. Static and dynamic analysis of a high static and low dynamic stiffness vibration isolator utilising the solid and liquid mixture. Engineering Structures, 99, 205–213 (2015)
[2]	HO, C., LANG, Z. Q., and BILLINGS, S. A. A frequency domain analysis of the effects of nonlinear damping on the Duffing equation. Mechanical Systems and Signal Processing, 45, 49–67 (2014)
[3]	DING, H. and JI, J. C. Vibration control of fluid-conveying pipes: a state-of-the-art review. Applied Mathematics and Mechanics (English Edition), 44(9), 1423–1456 (2023) https://doi.org/10.1007/s10483-023-3023-9
[4]	ZHANG, J. E., MAO, X. Y., DING, H., and CHEN, L. Q. Discussion on isolation of flexible beams with various support configurations. Acta Mechanica Sinica, 41, 523474 (2025)
[5]	MAO, X. Y., DING, H., and CHEN, L. Q. Vibration of flexible structures under nonlinear boundary conditions. Journal of Applied Mechanics, 84, 111006 (2017)
[6]	ZHANG, Z., ZHANG, Y. W., and DING, H. Vibration control combining nonlinear isolation and nonlinear absorption. Nonlinear Dynamics, 100, 2121–2139 (2020)
[7]	GATTI, G. Statics and dynamics of a nonlinear oscillator with quasi-zero stiffness behaviour for large deflections. Communications in Nonlinear Science and Numerical Simulation, 83, 105143 (2020)
[8]	LU, Z. Q., LIU, W. H., DING, H., and CHEN, L. Q. Energy transfer of an axially loaded beam with a parallel-coupled nonlinear vibration isolator. Journal of Vibration and Acoustics, 144, 051009 (2022)
[9]	CARRELLA, A. Passive Vibration Isolators with High-Static-Low-Dynamic Stiffness, Ph. D. dissertation, University of Southampton, Southampton (2008)
[10]	YAN, B., YU, N., and WU, C. Y. A state-of-the-art review on low-frequency nonlinear vibration isolation with electromagnetic mechanisms. Applied Mathematics and Mechanics (English Edition), 43(7), 1045–1062 (2022) https://doi.org/10.1007/s10483-022-2868-5
[11]	LIU, C. R., ZHAO, R., YU, K. P., and LIAO, B. P. In-plane quasi-zero-stiffness vibration isolator using magnetic interaction and cables: theoretical and experimental study. Applied Mathematical Modelling, 96, 497–522 (2021)
[12]	MAO, X. Y., YIN, M. M., DING, H., GENG, X. F., SHEN, Y. J., and CHEN, L. Q. Modeling, analysis, and simulation of X-shape quasi-zero-stiffness-roller vibration isolators. Applied Mathematics and Mechanics (English Edition), 43(7), 1027–1044 (2022) https://doi.org/10.1007/s10483-022-2871-6
[13]	ZHOU, J. H., ZHOU, J. X., PAN, H. B., WANG, K., CAI, C. Q., and WEN, G. L. Multi-layer quasi-zero-stiffness meta-structure for high-efficiency vibration isolation at low frequency. Applied Mathematics and Mechanics (English Edition), 45(7), 1189–1208 (2024) https://doi.org/10.1007/s10483-024-3157-6
[14]	XU, K. F., NIU, M. Q., ZHANG, Y. W., and CHEN, L. Q. An active high-static-low-dynamic-stiffness vibration isolator with adjustable buckling beams: theory and experiment. Applied Mathematics and Mechanics (English Edition), 45(3), 425–440 (2024) https://doi.org/10.1007/s10483-024-3087-6
[15]	YU, X., CHAI, K., LIU, Y. B., and LIU, S. Y. Bifurcation and singularity analysis of HSLDS vibration isolation system with elastic base. Journal of Vibroengineering, 20, 2197–2211 (2018)
[16]	LIU, C. R., ZHANG, W., YU, K. P., LIU, T., and ZHENG, Y. Quasi-zero-stiffness vibration isolation: designs, improvements and applications. Engineering Structures, 301, 117282 (2024)
[17]	LIU, C. R., TANG, J., YU, K. P., LIAO, B. P., HU, R. P., and ZANG, X. On the characteristics of a quasi-zero stiffness vibration isolator with viscoelastic damper. Applied Mathematical Modelling, 88, 367–381 (2020)
[18]	WANG, K., ZHOU, J. X., CHANG, Y. P., OUYANG, H. J., XU, D. L., and YANG, Y. A nonlinear ultra-low-frequency vibration isolator with dual quasi-zero-stiffness mechanism. Nonlinear Dynamics, 101, 755–773 (2020)
[19]	YANG, J., JIANG, J. Z., and NEILD, S. A. Dynamic analysis and performance evaluation of nonlinear inerter-based vibration isolators. Nonlinear Dynamics, 99, 1823–1839 (2020)
[20]	YAN, B., YU, N., WANG, Z. H., WU, C. Y., WANG, S., and ZHANG, W. M. Lever-type quasi-zero stiffness vibration isolator with magnetic spring. Journal of Sound and Vibration, 527, 116865 (2022)
[21]	YAN, B., WANG, X. J., MA, H. Y., LU, W. Q., and LI, Q. C. Hybrid time-delayed feedforward and feedback control of lever-type quasi-zero-stiffness vibration isolators. IEEE Transactions on Industrial Electronics, 71, 2810–2819 (2024)
[22]	LIU, C. R., WANG, Y. W., ZHANG, W., YU, K. P., MAO, J. J., and SHEN, H. Nonlinear dynamics of a magnetic vibration isolator with higher-order stable quasi-zero-stiffness. Mechanical Systems and Signal Processing, 218, 111584 (2024)
[23]	KOCAK, K. and YILMAZ, C. Design of a compliant lever-type passive vibration isolator with quasi-zero-stiffness mechanism. Journal of Sound and Vibration, 558, 117758 (2023)
[24]	LIU, C. R., ZHAO, R., YU, K. P., LEE, H., and LIAO, B. P. A quasi-zero-stiffness device capable of vibration isolation and energy harvesting using piezoelectric buckled beams. Energy, 233, 121146 (2021)
[25]	HAN, H. S., SOROKIN, V., TANG, L. H., and CAO, D. Q. Lightweight origami isolators with deployable mechanism and quasi-zero-stiffness property. Aerospace Science and Technology, 121, 107319 (2022)
[26]	LU, Z. Q., BRENNAN, M., DING, H., and CHEN, L. Q. High-static-low-dynamic-stiffness vibration isolation enhanced by damping nonlinearity. Science China-Technological Sciences, 62, 1103–1110 (2019)
[27]	WEN, G. L., LIN, Y., and HE, J. F. A quasi-zero-stiffness isolator with a shear-thinning viscous damper. Applied Mathematics and Mechanics (English Edition), 43(3), 311–326 (2022) https://doi.org/10.1007/s10483-022-2829-9
[28]	GAO, X. and TENG, H. D. Dynamics and nonlinear effects of a compact near-zero frequency vibration isolator with HSLD stiffness and fluid damping enhancement. International Journal of Non-linear Mechanics, 128, 103632 (2021)
[29]	YU, H. and GAO, X. Modeling and stiffness properties for the hydro-pneumatic near-zero frequency vibration isolator with piecewise smooth stiffness. Journal of Vibration Engineering and Technologies, 10, 527–539 (2022)
[30]	HAN, H. S., WANG, W. Q., YU, B. H., TANG, L. H., WANG, Y. L., and CAO, D. Q. Integrated design of quasi-zero-stiffness vibration isolators based on bifurcation theory. Aerospace Science and Technology, 146, 108940 (2024)
[31]	HAO, Z. F., CAO, Q. J., and WIERCIGROCH, M. Nonlinear dynamics of the quasi-zero-stiffness SD oscillator based upon the local and global bifurcation analyses. Nonlinear Dynamics, 87, 987–1014 (2017)
[32]	LI, Y. L. and XU, D. L. Spectrum reconstruction of quasi-zero stiffness floating raft systems. Chaos Solitons and Fractals, 93, 123–129 (2016)
[33]	DONG, Y. Y., HAN, Y. W., and ZHANG, Z. J. On the analysis of nonlinear dynamic behavior of an isolation system with irrational restoring force and fractional damping. Acta Mechanica, 230, 2563–2579 (2019)
[34]	HaN, N., ZHANG, H. F., LU, P. P., and LIU, Z. X. Resonance response and chaotic analysis for an irrational pendulum system. Chaos Solitons and Fractals, 182, 114812 (2024)
[35]	LI, S. B., MA, W. S., ZHANG, W., and HAO, Y. X. Melnikov method for a class of planar hybrid piecewise-smooth systems. International Journal of Bifurcation and Chaos, 26, 1650030 (2016)
[36]	LI, S. B., SHEN, C., ZHANG, W., and HAO, Y. X. The Melnikov method of heteroclinic orbits for a class of planar hybrid piecewise-smooth systems and application. Nonlinear Dynamics, 85, 1091–1104 (2016)
[37]	LI, S. B., WU, H. L., and CHEN, J. E. Global dynamics and performance of vibration reduction for a new vibro-impact bistable nonlinear energy sink. International Journal of Non-linear Mechanics, 139, 103891 (2022)
[38]	TIAN, R. L., ZHOU, Y. F., WANG, Y. Z., FENG, W. J., and YANG, X. Y. Chaotic threshold for non-smooth system with multiple impulse effect. Nonlinear Dynamics, 85, 1849–1863 (2016)
[39]	TIAN, R. L., WANG, T., ZHOU, Y. F., LI, J., and ZHU, S. T. Heteroclinic chaotic threshold in a nonsmooth system with jump discontinuities. International Journal of Non-linear Mechanics, 30, 2050141 (2020)
[40]	ZHOU, B. L., JIN, Y. F., and XU, H. D. Subharmonic resonance and chaos for a class of vibration isolation system with two pairs of oblique springs. Applied Mathematical Modelling, 108, 427–444 (2022)
[41]	ZHOU, B. L., JIN, Y. F., and XU, H. D. Global dynamics for a class of tristable system with negative stiffness. Chaos Solitons and Fractals, 162, 112509 (2022)
[42]	LI, S. B., XU, R., and KONG, L. Y. Suppressing homoclinic chaos for a class of vibro-impact oscillators by non-harmonic periodic excitations. Nonlinear Dynamics, 112, 10845–10870 (2024)
[43]	PENG, R. Y., LI, Q. H., and ZHANG, W. Chaos analysis of SD oscillator with two-frequency excitation. Nonlinear Dynamics, 112, 7649–7677 (2024)
[44]	HUANG, X. Y., CAO, Q. J., and LENCI, S. The complicated behaviours of a novel model of smooth and discontinuous dynamics with quasi-zero-stiffness property. Nonlinear Dynamics, 112, 20879–20902 (2024)
[45]	CHEN, L. C., YUAN, Z., QIAN, J. M., and SUN, J. Q. Random vibration of hysteretic systems under Poisson white noise excitations. Applied Mathematics and Mechanics (English Edition), 44(2), 207–220 (2023) https://doi.org/10.1007/s10483-023-2941-6
[46]	TIAN, R. L., GUAN, H. T., LU, X. H., ZHANG, X. L., HAO, H. N., FENG, W. J., and ZHANG, G. L. Dynamic crushing behavior and energy absorption of hybrid auxetic metamaterial inspired by Islamic motif art. Applied Mathematics and Mechanics (English Edition), 44(3), 345–362 (2023) https://doi.org/10.1007/s10483-023-2962-9
[47]	XIA, L., SUN, J. J., YING, Z. G., HUAN, R. H., and ZHU, W. Q. Dynamics and response reshaping of nonlinear predator-prey system undergoing random abrupt disturbances. Applied Mathematics and Mechanics (English Edition), 42(8), 1123–1134 (2021) https://doi.org/10.1007/s10483-021-2755-8
[48]	BULSARA, A. R., SCHEVE, W. C., and JACOBS, E. W. Homoclinic chaos in systems perturbed by weak Langevin noise. Physical Review A, 41, 668–681 (1990)
[49]	FREY, M. and SIMIU, E. Noise-induced chaos and phase space flux. Physica D: Nonlinear Phenomena, 63, 321–340 (1993)
[50]	LIN, H. and YIM, S. C. S. Analysis of a nonlinear system exhibiting chaotic, noisy chaotic, and random behaviors. Journal of Applied Mechanics, 63, 509–516 (1996)
[51]	LI, Y. M., ZHANG, F., and SHAO, X. Z. Chaos and chaos control of the Frenkel-Kontorova model with dichotomous noise. International Journal of Bifurcation and Chaos, 27, 1750052 (2017)
[52]	TIAN, R. L., ZHAO, Z. J., and XU, Y. Variable scale-convex-peak method for weak signal detection. Science China-Technological Sciences, 64, 331–340 (2021)
[53]	LI, Y. X., WEI, Z. C., KAPITANIAK, T., and ZHANG, W. Stochastic bifurcation and chaos analysis for a class of ships rolling motion under non-smooth perturbation and random excitation. Ocean Engineering, 266, 112859 (2022)
[54]	WEI, Z. C., LI, Y. X., KAPITANIAK, T., and ZHANG, W. Analysis of chaos and capsizing of a class of nonlinear ship rolling systems under excitation of random waves. Chaos, 34, 043106 (2024)
[55]	LIU, D. and XU, Y. Random disordered periodical input induced chaos in discontinuous systems. International Journal of Bifurcation and Chaos, 29, 1950002 (2019)
[56]	LI, Y. X., WEI, Z. C., ZHANG, W., and YI, M. Melnikov-type method for a class of hybrid piecewise-smooth systems with impulsive effect and noise excitation: homoclinic orbits. Chaos, 32, 073119 (2022)
[57]	WEI, Z. C., LI, Y. X., MOROZ, I., and ZHANG, W. Melnikov-type method for a class of hybrid piecewise-smooth systems with impulsive effect and noise excitation: heteroclinic orbits. Chaos, 32, 103127 (2022)
[58]	LI, Y. X., WEI, Z. C., ZHANG, W., and KAPITANIA, K. T. Melnikov-type method for chaos in a class of hybrid piecewise-smooth systems with impact and noise excitation under unilateral rigid constraint. Applied Mathematical Modelling, 122, 506–523 (2023)
[59]	SIMIU, E. A unified theory of deterministic and noise-induced transitions: Melnikov processes and their application in engineering, physics and neuroscience. AIP Conference Proceedings, 502, 266–271 (1999)
[60]	YANG, X. L., XU, Y., and SUN, Z. K. Effect of Gaussian white noise on the dynamical behaviors of an extended Duffing-van der Pol oscillator. International Journal of Bifurcation and Chaos, 16, 2587–2600 (2006)
[61]	RAVINDRA, B. and MALLIK, A. K. Performance of non-linear vibration isolators under harmonic excitation. Journal of Sound and Vibration, 170, 325–337 (1994)
[62]	YU, C. Y., FU, Q., ZHANG, J. R., and ZHANG, J. The vibration isolation characteristics of torsion bar spring with negative stiffness structure. Mechanical Systems and Signal Processing, 180, 109378 (2022)
[63]	WIGGINS, S. Global Bifurcations and Chaos, Springer, New York (1988)
[64]	WOLF, A., SWIFT, J. R., SWINNEY, H. L., and VASTANO, J. A. Determining Lyapunov exponents from a time series. Physica D: Nonlinear Phenomena, 16, 285–317 (1985)