On a broadband vibration isolator with tunable stiffness: from quasi-zero-stiffness to zero-stiffness behavior

doi:10.1007/s10483-026-3351-9

摘要/Abstract

Abstract:

A novel vibration isolation system designed for superior performance in low-frequency environments is proposed in this work. The isolator is based on a unique hexagonal arrangement of linear springs, allowing for an adjustable geometric configuration via the initial inclination angle. Based on the principle of Lagrangian mechanics, the equation of motion governing the structural dynamics is rigorously derived. The system is modeled as a strongly nonlinear single-degree-of-freedom dynamical system, loaded with a normalized payload and subject to harmonic base excitation. To analyze the steady-state response, the harmonic balance method is employed, providing accurate predictions of the payload’s vibration amplitude and displacement transmissibility as functions of both the base excitation amplitude and frequency. The analysis reveals a direct relationship between the isolator’s geometric and stiffness parameters and its load-bearing capacity, leading to the identification of three distinct operational regimes. Depending on the unloaded initial inclination angle, the equivalent stiffness ratio, and the payload design configuration, the system can exhibit one of three vibration isolation modes: (i) the quasi-zero stiffness (QZS) isolation mode, (ii) the zero linear stiffness with controllable nonlinear stiffness, and (iii) the full-band perfect zero stiffness. The vibration isolation performance of the proposed structure is thoroughly discussed for all three oscillation modes in terms of frequency response curves, displacement transmissibility, and time-domain responses. The key novel finding is that this structure can operate as a full-band, high-performance vibration isolator when the initial inclination angle is designed to be a right angle, enabling full isolation of the maximum possible payload. Moreover, the analytical results and numerical simulations demonstrate that the isolator’s displacement transmissibility $T$ with the unit dB tends to $− ∞$ as the air-damping coefficient approaches zero, enabling ideal vibration isolation across the entire excitation frequency range. These analytical insights are validated through comprehensive numerical simulations, which show excellent agreement with the theoretical predictions.

Key words: nonlinear vibration isolation, quasi-zero stiffness (QZS) structure, full-band vibration isolator, harmonic balance method, displacement transmissibility

中图分类号:

O326

. [J]. Applied Mathematics and Mechanics (English Edition), 2026, 47(2): 255-282.

N. A. SAEED, Lei HOU, Haiming YI, A. A. SHUKUR, S. M. ALAMRY, S. M. EL-SHOURBAGY. On a broadband vibration isolator with tunable stiffness: from quasi-zero-stiffness to zero-stiffness behavior[J]. Applied Mathematics and Mechanics (English Edition), 2026, 47(2): 255-282.

图/表 23

参考文献 65

[1]	MEAD, D. J. Passive Vibration Control, 1st ed., Wiley, Chichester (1998)
[2]	IBRAHIM, R. A. Recent advances in nonlinear passive vibration isolators. Journal of Sound and Vibration, 314, 371–452 (2008)
[3]	VAKAKIS, A. F., GENDELMAN, O. V., BERGMAN, L. A., MCFARLAND, D. M., KERSCHEN, G., and LEE, Y. S. Nonlinear Targeted Energy Transfer in Mechanical and Structural Systems, Springer, Dordrecht (2008)
[4]	GENDELMAN, O. V. Transition of energy to a nonlinear localized mode in a highly asymmetric system of two oscillators. Nonlinear Dynamics, 25, 237–253 (2001)
[5]	FENG, X., JING, X., and GUO, Y. Vibration isolation with passive linkage mechanisms. Nonlinear Dynamics, 106, 1891–1927 (2021)
[6]	ALUJEVIĆ, N., ČAKMAK, D., WOLF, H., and JOKIĆ, M. Passive and active vibration isolation systems using inerter. Journal of Sound and Vibration, 418(31), 163–183 (2018)
[7]	CARRELLA, A., BRENNAN, M. J., and WATERS, T. P. Static analysis of a passive vibration isolator with quasi-zero-stiffness characteristic. Journal of Sound and Vibration, 301(3-5), 678–689 (2007)
[8]	KOVACIC, I., BRENNAN, M. J., and WATERS, T. P. A study of a nonlinear vibration isolator with a quasi-zero stiffness characteristic. Journal of Sound and Vibration, 315(3), 700–711 (2008)
[9]	MA, Z., ZHOU, R., and YANG, Q. Recent advances in quasi-zero stiffness vibration isolation systems: an overview and future possibilities. Machines, 10(9), 813 (2022)
[10]	CARRELLA, A., BRENNAN, M., WATERS, T., and LOPES, V. Force and displacement transmissibility of a nonlinear isolator with high-static-low-dynamic-stiffness. International Journal of Mechanical Sciences, 55(1), 22–29 (2012)
[11]	LE, T. D. and AHN, K. K. Experimental investigation of a vibration isolation system using negative stiffness structure. International Journal of Mechanical Sciences, 70, 99–112 (2013)
[12]	HU, Y., ZHANG, H., WANG, K., FANG, Y., and MA, C. Analytical analysis of vibration isolation characteristics of quasi-zero stiffness suspension backpack. International Journal of Dynamics and Control, 12, 4387–4397 (2024)
[13]	HUANG, X. C., LIU, X. T., SUN, J. Y., ZHANG, Z. Y., and HUA, H. X. Vibration isolation characteristics of a nonlinear isolator using Euler buckled beam as negative stiffness corrector: a theoretical and experimental study. Journal of Sound and Vibration, 333(4), 1132–1148 (2014)
[14]	HUANG, X. C., LIU, X. T., and HUA, H. X. Effects of stiffness and load imperfection on the isolation performance of a high-static-low-dynamic-stiffness non-linear isolator under base displacement excitation. International Journal of Non-Linear Mechanics, 65, 32–43 (2014)
[15]	ZHOU, J., WANG, X., XU, D., and BISHOP, S. Nonlinear dynamic characteristics of a quasi-zero stiffness vibration isolator with cam-roller-spring mechanisms. Journal of Sound and Vibration, 346, 53–69 (2015)
[16]	GATTI, G., BRENNAN, M. J., and TANG, B. Some diverse examples of exploiting the beneficial effects of geometric stiffness nonlinearity. Mechanical Systems and Signal Processing, 125, 4–20 (2019)
[17]	SUN, X., JING, X., XU, J., and CHENG, L. Vibration isolation via a scissor-like structured platform. Journal of Sound and Vibration, 333, 2404–2420 (2014)
[18]	ZHANG, W. and ZHAO, J. Analysis on nonlinear stiffness and vibration isolation performance of scissor-like structure with full types. Nonlinear Dynamics, 86, 17–36 (2016)
[19]	ZOU, W., CHENG, C., MA, R., HU, Y., and WANG, W. Performance analysis of a quasi-zero stiffness vibration isolation system with scissor-like structures. Archive of Applied Mechanics, 91(1), 117–133 (2020)
[20]	LU, Z. Q., GU, D. H., DING, H., LACARBONARA, W., and CHEN, L. Q. Nonlinear vibration isolation via a circular ring. Mechanical Systems and Signal Processing, 136, 106490 (2020)
[21]	QIU, Y., ZHU, Y., LUO, Z., GAO, Y., and LI, Y. The analysis and design of nonlinear vibration isolators under both displacement and force excitations. Archive of Applied Mechanics, 91(5), 2159–2178 (2021)
[22]	YE, K., JI, J. C., and BROWN, T. A novel integrated quasi-zero stiffness vibration isolator for coupled translational and rotational vibrations. Mechanical Systems and Signal Processing, 149, 107340 (2021)
[23]	HAO, Z. and CAO, Q. The isolation characteristics of an archetypal dynamical model with stable-quasi-zero-stiffness. Journal of Sound and Vibration, 340, 61–79 (2015)
[24]	ZHOU, J., WANG, K., XU, D., OUYANG, H., and LI, Y. A six degrees-of-freedom vibration isolation platform supported by a hexapod of quasi-zero-stiffness struts. Journal of Vibration and Acoustics, 139(3), 034502 (2017)
[25]	ZHOU, J., WANG, K., XU, D., OUYANG, H., and FU, Y. Vibration isolation in neonatal transport by using a quasi-zero-stiffness isolator. Journal of Vibration and Control, 24(15), 3278–3291 (2018)
[26]	WANG, X., ZHOU, J., XU, D., OUYANG, H., and DUAN, Y. Force transmissibility of a two-stage vibration isolation system with quasi-zero stiffness. Nonlinear Dynamics, 87(1), 633–646 (2017)
[27]	GUO, L., WANG, X., FAN, R. L., and BI, F. R. Review on development of high-static-low-dynamic-stiffness seat cushion mattress for vibration control of seating suspension system. Applied Sciences, 10, 2887 (2020)
[28]	LIU, C., TANG, J., YU, K., LIAO, B., HU, R., and ZANG, X. On the characteristics of a quasi-zero-stiffness vibration isolator with viscoelastic damper. Applied Mathematical Modelling, 88, 367–381 (2020)
[29]	CHANG, Y., ZHOU, J., WANG, K., and XU, D. A quasi-zero-stiffness dynamic vibration absorber. Journal of Sound and Vibration, 494, 115859 (2021)
[30]	LIU, Y., WANG, X., XUE, Y., DENG, E., WANG, Y., SONG, C., and FENG, Q. Dynamic analysis of quasi-zero stiffness vibration isolation system coupled with frequency adjustable dynamic vibration absorber. Archive of Applied Mechanics, 92, 3631–3647 (2022)
[31]	MENG, Q., HOU, L., WANG, A, LIN, R., LI, Z., ZHONG, S., CHEN, Y., SAEED, N. A., MOHAMED, A. F., and AWWAD, E. M. Subharmonic response suppression of a quasi-zero stiffness system. Journal of Sound and Vibration, 594, 118674 (2025)
[32]	MENG, Q., HOU, L., LIN, R., CHEN, Y., SAEED, N. A., FOULY, A., and AWWAD, E. M. On a quasi-zero stiffness vibration isolator with multiple zero stiffness points for mass load deviation. Applied Mathematical Modelling, 145, 116112 (2025)
[33]	SAEED, N. A., ELLABBAN, Y. Y., HOU, L., ZHONG, S., and DURAIHEM, F. Z. Geometric nonlinear dynamics of a quasi-zero stiffness isolator integrated with an energy harvester: monostable, perfect zero-linear stiffness, and bistable oscillation modes. Chaos, Solitons & Fractals, 199(1), 116633 (2025)
[34]	BANERJEE, P., BALAJI, P. S., and MURUGAN, S. A dual-function compliant metastructure based on quasi-zero-stiffness for combined vibration isolation and vibration energy harvesting. International Journal of Dynamics and Control, 13, 135 (2025)
[35]	JING, X. The Bio-Inspired X-Structure/Mechanism Approach for Exploring Nonlinear Benefits in Engineering, Springer Nature, Singapore (2024)
[36]	LIU, C., JING, X., and LI, F. Vibration isolation using a hybrid lever-type isolation system with an X-shape supporting structure. International Journal of Mechanical Sciences, 98, 169–177 (2015)
[37]	CHAI, Y., JING, X., and CHAO, X. X-shaped mechanism based enhanced tunable QZS property for passive vibration isolation. International Journal of Mechanical Sciences, 218, 107077 (2022)
[38]	WU, Z., LIU, W., LI, F., and ZHANG, C. Band-gap property of a novel elastic metamaterial beam with X-shaped local resonators. Mechanical Systems and Signal Processing, 134, 106357 (2019)
[39]	BIAN, J. and JING, X. Analysis and design of a novel and compact X-structured vibration isolation mount (X-mount) with wider quasi-zero-stiffness range. Nonlinear Dynamics, 101, 2195–2222 (2020)
[40]	BIAN, J. and JING, X. A nonlinear X-shaped structure based tuned mass damper with multi-variable optimization (X-absorber). Communications in Nonlinear Science and Numerical Simulation, 99, 105829 (2021)
[41]	ZHOU, S., LIU, Y., JIANG, Z., and REN, Z. Nonlinear dynamic behavior of a bio-inspired embedded X-shaped vibration isolation system. Nonlinear Dynamics, 110, 153–175 (2022)
[42]	XIONG, X., WANG, Y., LI, J., and LI, F. Internal resonance analysis of bio-inspired X-shaped structure with nonlinear vibration absorber. Mechanical Systems and Signal Processing, 185, 109809 (2023)
[43]	SAEED, N. A., ELLABBAN, Y. Y., MOATIMID, G. M., HOU, L., and MOHAMED, A. F. Nonlinear interactions of an n-layer X-shape low-frequency vibration isolator equipped with a nonlinear vibration absorber at 1:1 internal resonance: analytical and numerical investigations. Physica Scripta, 99(10), 105207 (2024)
[44]	SAEED, N. A., ELLABBAN, Y. Y., HOU, L., MOATIMID, G. M., ZHONG, S., and DURAIHEM, F. Z. Nonlinear dynamics of a bio-inspired 2-DOF low-frequency X-shaped vibration isolator with m-to-n layers driven harmonically under simultaneous primary and 1:1 internal resonances. Chaos, Solitons & Fractals, 190, 115786 (2025)
[45]	SUI, G., HOU, S., ZHANG, X., SHAN, X., HOU, C., SONG, H., HOU, W., and LI, J. A bio-inspired spider-like structure isolator for low-frequency vibration. Applied Mathematics and Mechanics (English Edition), 44(8), 1263–1286 (2023) https://doi.org/10.1007/s10483-023-3020-9
[46]	PU, H., LIU, J., WANG, M., DING, J., SUN, Y., PENG, Y., and LUO, J. Bio-inspired quasi-zero stiffness vibration isolator with quasilinear negative stiffness in full stroke. Journal of Sound and Vibration, 574, 118240 (2024)
[47]	WU, Z., JING, X., BIAN, J., LI, F., and ALLEN, R. Vibration isolation by exploring bio-inspired structural nonlinearity. Bioinspiration & Biomimetics, 10, 056015 (2015)
[48]	PAN, H., JING, X., SUN, W., and LI, Z. Analysis and design of a bioinspired vibration sensor system in noisy environment. ASME Transactions on Mechatronics, 23(2), 845–855 (2018)
[49]	WANG, Y., JING, X., DAI, H., and LI, F. M. Subharmonics and ultra-subharmonics of a bio-inspired nonlinear isolation system. International Journal of Mechanical Sciences, 152, 167–184 (2019)
[50]	WANG, Y., JING, X., and GUO, Y. Nonlinear analysis of a bio-inspired vertically asymmetric isolation system under different structural constraints. Nonlinear Dynamics, 95, 445–464 (2019)
[51]	JIANG, G., JING, X., and GUO, Y. A novel bio-inspired multi-joint anti-vibration structure and its nonlinear HSLDS properties. Mechanical Systems and Signal Processing, 138, 106552 (2020)
[52]	YAN, G., ZOU, H., WANG, S., ZHAO, L., WU, Z., and ZHANG, W. Bio-inspired vibration isolation: methodology and design. Applied Mechanics Reviews, 73, 020801 (2021)
[53]	OU, H., SUN, X., WU, Q., CHEN, Z., CHEN, Z., CHEN, Q., and HU, L. A novel bio-inspired kangaroo leg structure for low-frequency vibration isolation. Nonlinear Dynamics, 112, 1797–1814 (2024)
[54]	SAEED, N. A., MOHAMED, M. S., and ELAGAN, S. K. Periodic, quasi-periodic, and chaotic motions to diagnose a crack on a horizontally supported nonlinear rotor system. Symmetry, 12, 2059 (2020)
[55]	SAEED, N. A., MOATIMID, G. M., ELSABAA, F. M. F., and ELLABBAN, Y. Y. Time-delayed control to suppress a nonlinear system vibration utilizing the multiple scales homotopy approach. Archive of Applied Mechanics, 91, 1193–1215 (2021)
[56]	EISSA, M., KAMEL, M., SAEED, N. A., EL-GANAINI, W., and EL-GOHARY, H. Time-delayed positive-position and velocity feedback controller to suppress the lateral vibrations in nonlinear Jeffcott-rotor system. Menoufia Journal of Electronic Engineering Research, 27, 261–278 (2018)
[57]	SAEED, N. A., MOHAMED, M. S., ELAGAN, S. K., and AWREJCEWICZ, J. Integral resonant controller to suppress the nonlinear oscillations of a two-degree-of-freedom rotor active magnetic bearing system. Processes, 10, 271 (2022)
[58]	SAEED, N. A., MOATIMID, G. M., ELSABAA, F. M., ELLABBAN, Y. Y., ELAGAN, S. K., and MOHAMED, M. S. Time-delayed nonlinear integral resonant controller to eliminate the nonlinear oscillations of a parametrically excited system. IEEE Access, 9, 74836–74854 (2021)
[59]	EL-SHOURBAGY, S. M., SAEED, N. A., KAMEL, M., RASLAN, K. R., ABOUEL NASR, E., and AWREJCEWICZ, J. On the performance of a nonlinear position-velocity controller to stabilize rotor-active magnetic-bearings system. Symmetry, 13, 2069 (2021)
[60]	SAEED, N. A., EL-BENDARY, S., SAYED, M., MOHAMED, M., and ELAGAN, S. On the oscillatory behaviours and rub-impact forces of a horizontally supported asymmetric rotor system under position-velocity feedback controller. Latin American Journal of Solids and Structures, 18(2), e349, (2021)
[61]	SAEED, N. A., AWREJCEWICZ, J., HAFEZ, S. T., HOU, L., and ABOUDAIF, M. K. Stability, bifurcation, and vibration control of a discontinuous nonlinear rotor model under rub-impact effect. Nonlinear Dynamics, 111, 20661–20697 (2023)
[62]	PAN, H., JING, X., SUN, W., and GAO, H. A bioinspired dynamics-based adaptive tracking control for nonlinear suspension systems. IEEE Transactions on Control Systems Technology, 26(3), 903–914 (2018)
[63]	SUN, K., TANG, J., YANG, Y., JIANG, B., LI, Y., and CAO, D. Active control of quasi-zero-stiffness vibration isolator with variable load. International Journal of Structural Stability and Dynamics, 24(21), 2450243 (2024)
[64]	WEI, C. H-infinity optimal control based on output feedback for nonlinear two-degree-of-freedom vibration isolator with quasi-zero stiffness. Acta Mechanica, 235, 6365–6378 (2024)
[65]	WU, J., HONG, L., and JIANG, J. A robust and efficient stability analysis of periodic solutions based on harmonic balance method and Floquet-Hill formulation. Mechanical Systems and Signal Processing, 173, 109057 (2023)