Abstract
Existing quasi-zero stiffness (QZS) isolators are reviewed. In terms of their advantages, a novel X-shape QZS isolator combined with the cam-roller-spring mechanism (CRSM) is proposed. Different from the existing X-shape isolators, oblique springs are used to enhance the negative stiffness of the system. Meanwhile, the CRSM is used to eliminate the gravity of the loading mass, while the X-shape structure leaves its static position. The existing QZS isolators are demonstrated and classified according to their nonlinearity mechanisms and classical shapes. It is shown that the oblique spring can realize negative stiffness based on the simplest mechanism. The X-shape has a strong capacity of loading mass, while the CRSM can achieve a designed restoring force at any position. The proposed isolator combines all these advantages together. Based on the harmonic balance method (HBM) and the simulation, the displacement transmissibilities of the proposed isolator, the X-shape isolators just with oblique springs, and the X-shape isolators in the traditional form are studied. The results show that the proposed isolator has the lowest beginning isolation frequency and the smallest maximum displacement transmissibility. However, it still has some disadvantages similar to the existing QZS isolators. This means that its parameters should be designed carefully so as to avoid becoming a bistable system, in which there are two potential wells in the potential energy curve and thus the isolation performance will be worsened.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
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, 678–689 (2007)
IKEGAMI, R. E. A. Zero-G ground test simulation methods. Proceedings of the 11th Aerospace Testing Seminar, Institute of Environmental Science, Manhattan Beach (1988)
WOODARD, S. E. and HOUSNER, J. M. The nonlinear behavior of a passive zero-spring-rate suspension system. 29th Structures, Structural Dynamics and Materials Conference, Reston (1998)
LACOSTE, L. LaCoste and Romberg straight-line gravity meter. Geophysics, 48, 606–610 (1983)
IBRAHIM, R. A. Recent advances in nonlinear passive vibration isolators. Journal of Sound and Vibration, 314, 371–452 (2008)
GUO, L. C., 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)
NIU, F., MENG, L. S., WU, W. J., SUN, J. G., SU, W. H., MENG, G., and RAO, Z. S. Recent advances in quasi-zero-stiffness vibration isolation systems. Applied Mechanics and Materials, 397–400, 295–303 (2013)
ZHANG, J. Z., LI, D., CHEN, M. J., and DONG, S. An ultra-low frequency parallel connection nonlinear isolator for precision instruments. Key Engineering Materials, 257–258, 231–236 (2004)
CARRELLA, A., BRENNAN, M. J., KOVACIC, I., and WATERS, T. P. On the force transmissibility of a vibration isolator with quasi-zero-stiffness. Journal of Sound and Vibration, 322, 707–717 (2009)
CARRELLA, A., BRENNAN, M. J., WATERS, T. P., and LOPES, V., JR. Force and displacement transmissibility of a nonlinear isolator with high-static-low-dynamic-stiffness. International Journal of Mechanical Sciences, 55, 22–29 (2012)
TANG, B. and BRENNAN, M. J. On the shock performance of a nonlinear vibration isolator with high-static-low-dynamic-stiffness. International Journal of Mechanical Sciences, 81, 207–214 (2014)
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)
ABOLFATHI, A., BRENNAN, M. J., WATERS, T. P., and TANG, B. On the effects of mistuning a force-excited system containing a quasi-zero-stiffness vibration isolator. Journal of Vibration and Acoustics, 137, 044502 (2015)
PENG, Z. K., LANG, Z. Q., ZHAO, L., BILLINGS, S. A., TOMLINSON, G. R., and GUO, P. The force transmissibility of MDOF structures with a non-linear viscous damping device. International Journal of Non-Linear Mechanics, 46, 1305–1314 (2011)
WANG, Y., LI, S. M., CHENG, C., and SU, Y. Q. Adaptive control of a vehicle-seat-human coupled model using quasi-zero-stiffness vibration isolator as seat suspension. Journal of Mechanical Science and Technology, 32, 2973–2985 (2018)
XU, D. L., ZHANG, Y. Y., ZHOU, J. X., and LOU, J. J. On the analytical and experimental assessment of the performance of a quasi-zero-stiffness isolator. Journal of Vibration and Control, 20, 2314–2325 (2014)
ZHAO, F., JI, J. C., YE, K., and LUO, Q. T. Increase of quasi-zero stiffness region using two pairs of oblique springs. Mechanical Systems and Signal Processing, 144, 106975 (2020)
ZHAO, F., JI, J. C., LUO, Q. T., CAO, S. Q., CHEN, L. M., and DU, W. L. An improved quasi-zero stiffness isolator with two pairs of oblique springs to increase isolation frequency band. Nonlinear Dynamics 104, 349–365 (2021)
ZHAO, F., JI, J. C., YE, K., and LUO, Q. T. An innovative quasi-zero stiffness isolator with three pairs of oblique springs. International Journal of Mechanical Sciences, 192, 106093 (2021)
LU, Z. Q., BRENNAN, M. J., YANG, T. J., LI, X. H., and LIU, Z. G. An investigation of a two-stage nonlinear vibration isolation system. Journal of Sound and Vibration, 332, 1456–1464 (2013)
LU, Z. Q., YANG, T. J., BRENNAN, M. J., LI, X. H., and LIU, Z. G. On the performance of a two-stage vibration isolation system which has geometrically nonlinear stiffness. Journal of Vibration and Acoustics, 136, 064501 (2014)
WANG, Y., LI, S. M., NEILD, S. A., and JIANG, J. Z. Comparison of the dynamic performance of nonlinear one and two degree-of-freedom vibration isolators with quasi-zero stiffness. Nonlinear Dynamics, 88, 635–654 (2017)
DENG, T. C., WEN, G. L., DING, H., LU, Z. Q., and CHEN, L. Q. A bio-inspired isolator based on characteristics of quasi-zero stiffness and bird multi-layer neck. Mechanical Systems Signal Processing, 145, 106967 (2020)
XU, J. and SUN, X. T. A multi-directional vibration isolator based on quasi-zero-stiffness structure and time-delayed active control. International Journal of Mechanical Sciences, 100, 126–135 (2015)
ZHU, G. N., LIU, J., CAO, Q. J., CHENG, Y. F., LU, Z. C., and ZHU, Z. B. A two degree of freedom stable quasi-zero stiffness prototype and its applications in aseismic engineering. SCIENCE CHINA Technological Sciences, 63, 496–505 (2020)
TOBIAS, S. A. Design of small isolator units for the suppression of low frequency vibration. Journal of Mechanical Engineering Science, 26, 280–292 (1959)
PLATUS, D. L. Negative-stiffness-mechanism vibration isolation systems. Optics and Metrology, 1619, 44–54 (1991)
LIU, X. T., HUANG, X. C., and HUA, H. X. On the characteristics of a quasi-zero stiffness isolator using Euler buckled beam as negative stiffness corrector. Journal of Sound and Vibration, 332, 3359–3376 (2013)
YANG, J., XIONG, Y. P., and XING, J. T. Dynamics and power flow behaviour of a nonlinear vibration isolation system with a negative stiffness mechanism. Journal of Sound and Vibration, 332, 167–183 (2013)
XU, D. L., YU, Q. P., ZHOU, J. X., and BISHOP, S. R. Theoretical and experimental analyses of a nonlinear magnetic vibration isolator with quasi-zero-stiffness characteristic. Journal of Sound and Vibration, 332, 3377–3389 (2013)
JIANG, Y. L., SONG, C. S., DING, C. M., and XU, B. H. Design of magnetic-air hybrid quasi-zero stiffness vibration isolation system. Journal of Sound and Vibration, 477, 115346 (2020)
LAN, C. C., YANG, S. A., and WU, Y. S. Design and experiment of a compact quasi-zero-stiffness isolator capable of a wide range of loads. Journal of Sound and Vibration, 333, 4843–4858 (2014)
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)
AHN, H. J. Performance limit of a passive vertical isolator using a negative stiffness mechanism. Journal of Mechanical Science and Technology, 22, 2357–2364 (2008)
ZHANG, W. and ZHAO, J. B. Analysis on nonlinear stiffness and vibration isolation performance of scissor-like structure with full types. Nonlinear Dynamics, 86, 17–36 (2016)
CHENG, C., LI, S. M., WANG, Y., and JIANG, X. X. Force and displacement transmissibility of a quasi-zero stiffness vibration isolator with geometric nonlinear damping. Nonlinear Dynamics, 87, 2267–2279 (2017)
CHENG, C., LI, S. M., WANG, Y., and JIANG, X. X. Resonance of a quasi-zero stiffness vibration system under base excitation with load mismatch. International Journal of Structural Stability and Dynamics, 18, 1850002 (2018)
WANG, Y., LI, H. X., CHENG, C., DING, H., and CHEN, L. Q. Dynamic performance analysis of a mixed-connected inerter-based quasi-zero stiffness vibration isolator. Structural Control & Health Monitoring, 27, e2604 (2020)
SUN, X. T., JING, X. J., XU, J., and CHENG, L. Vibration isolation via a scissor-like structured platform. Journal of Sound and Vibration, 333, 2404–2420 (2014)
YAN, B., WANG, Z. H., MA, H. Y., BAO, H. H., WANG, K., and WU, C. Y. A novel lever-type vibration isolator with eddy current damping. Journal of Sound and Vibration, 494, 115862 (2021)
LIU, C. C., JING, X. J., and LI, F. M. Vibration isolation using a hybrid lever-type isolation system with an X-shape supporting structure. International Journal of Mechanical Sciences, 98, 169–177 (2015)
LIU, C. C., JING, X. J., and CHEN, Z. B. Band stop vibration suppression using a passive X-shape structured lever-type isolation system. Mechanical Systems and Signal Processing, 68–69, 342–353 (2016)
SUN, X. T. and JING, X. J. Analysis and design of a nonlinear stiffness and damping system with a scissor-like structure. Mechanical Systems and Signal Processing, 66–67, 723–742 (2016)
SUN, X. T. and JING, X. J. A nonlinear vibration isolator achieving high-static-low-dynamic stiffness and tunable anti-resonance frequency band. Mechanical Systems and Signal Processing, 80, 166–188 (2016)
JING, X. J., ZHANG, L. L., JIANG, G. Q., FENG, X., GUO, Y. Q., and XU, Z. D. Critical factors in designing a class of X-shaped structures for vibration isolation. Engineering Structures, 199, 109659 (2019)
WU, Z. J., JING, X. J., SUN, B., and LI, F. M. A 6DOF passive vibration isolator using X-shape supporting structures. Journal of Sound and Vibration, 380, 90–111 (2016)
FENG, X. and JING, X. J. Human body inspired vibration isolation: beneficial nonlinear stiffness, nonlinear damping & nonlinear inertia. Mechanical Systems and Signal Processing, 117, 786–812 (2019)
SUN, X. T. and JING, X. J. Multi-direction vibration isolation with quasi-zero stiffness by employing geometrical nonlinearity. Mechanical Systems and Signal Processing, 62–63, 149–163 (2015)
BIAN, J. and JING, X. J. 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)
GUO, L. C., KHIU, A., FAN, R. L., and WANG, X. Analysis of a passive scissor-like structure isolator with quasi-zero stiffness for a seating system vibration-isolation application. International Journal of Vehicle Design, 82, 224–240 (2020)
YAN, G., ZOU, H. X., WANG, S., ZHAO, L. C., GAO, Q. H., TAN, T., and ZHAN, W. M. Large stroke quasi-zero stiffness vibration isolator using three-link mechanism. Journal of Sound and Vibration, 478, 115344 (2020)
DAI, H. H., JING, X. J., SUN, C., WANG, Y., and YUE, X. K. Accurate modeling and analysis of a bio-inspired isolation system: with application to on-orbit capture. Mechanical Systems and Signal Processing, 109, 111–133 (2018)
DAI, H. H., JING, X. J., WANG, Y., YUE, X. K., and YUAN, J. P. Post-capture vibration suppression of spacecraft via a bio-inspired isolation system. Mechanical Systems and Signal Processing, 105, 214–240 (2018)
WANG, X., YUE, X. K., DAI, H. H., and YUAN, J. P. Vibration suppression for post-capture spacecraft via a novel bio-inspired Stewart isolation system. Acta Astronautica, 168, 1–22 (2020)
WANG, X., YUE, X. K., WEN, H. W., and YUAN, J. P. Hybrid passive/active vibration control of a loosely connected spacecraft system. Computer Modeling in Engineering & Sciences, 122, 61–87 (2020)
WU, Z. J., JING, X. J., BIAN, J., LI, F. M., and ALLEN, R. Vibration isolation by exploring bio-inspired structural nonlinearity. Bioinspiration & Biomimetics, 10, 056015 (2015)
HU, F. Z. and JING, X. J. A 6-DOF passive vibration isolator based on Stewart structure with X-shaped legs. Nonlinear Dynamics, 91, 157–185 (2018)
WANG, Y. and JING, X. J. Nonlinear stiffness and dynamical response characteristics of an asymmetric X-shaped structure. Mechanical Systems and Signal Processing, 125, 142–169 (2019)
WANG, Y., JING, X. J., and GUO, Y. Q. Nonlinear analysis of a bio-inspired vertically asymmetric isolation system under different structural constraints. Nonlinear Dynamics, 95, 445–464 (2019)
ALABUZHEV, P. M. and RIVIN, E. I. Vibration Protecting and Measuring Systems with Quasizero Stiffness, Hemisphere Publishing Corporation, New York (1989)
ZHOU, J. X., WANG, X. L., XU, D. L., 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)
WANG, X. L., ZHOU, J. X., XU, D. L., OUYANG, H. J., and DUAN, Y. Force transmissibility of a two-stage vibration isolation system with quasi-zero stiffness. Nonlinear Dynamics, 87, 633–646 (2017)
ZHOU, J. X., XIAO, Q. Y., XU, D. L., OUYANG, H. J., and LI, Y. L. A novel quasi-zero-stiffness strut and its applications in six-degree-of-freedom vibration isolation platform. Journal of Sound and Vibration, 394, 59–74 (2017)
ZHOU, J. X., XU, D. L., and BISHOP, S. A torsion quasi-zero stiffness vibration isolator. Journal of Sound and Vibration, 338, 121–133 (2015)
WANG, K., ZHOU, J. X., and XU, D. L. Sensitivity analysis of parametric errors on the performance of a torsion quasi-zero-stiffness vibration isolator. International Journal of Mechanical Sciences, 134, 336–346 (2017)
SUN, M. N., DONG, Z. X., SONG, G. Q., SUN, X. W., and LIU, W. J. A vibration isolation system using the negative stiffness corrector formed by cam-roller mechanisms with quadratic polynomial trajectory. Applied Sciences-Basel, 10, 3573 (2020)
ZHANG, Q. L., XIA, S. Y., XU, D. L., and PENG, Z. K. A torsion-translational vibration isolator with quasi-zero stiffness. Nonlinear Dynamics, 99, 1467–1488 (2020)
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)
YAO, Y. H., LI, H. G., LI, Y., and WANG, X. J. Analytical and experimental investigation of a high-static-low-dynamic stiffness isolator with cam-roller-spring mechanism. International Journal of Mechanical Sciences, 186, 134–142 (2020)
YAO, Y. H., WANG, X. J., and LI, H. G. Design and analysis of a high-static-low-dynamic stiffness isolator using the cam-roller-spring mechanism. Journal of Vibration and Acoustics, 142, 1–24 (2020)
YE, K., JI, J. C., and BROWN, T. Design of a quasi-zero stiffness isolation system for supporting different loads. Journal of Sound and Vibration, 471, 115198 (2020)
ROBERTSON, W. S., KIDNER, M. R. F., CAZZOLATO, B. S., and ZANDER, A. C. Theoretical design parameters for a quasi-zero stiffness magnetic spring for vibration isolation. Journal of Sound and Vibration, 326, 88–103 (2009)
WU, W. J., CHEN, X. D., and SHAN, Y. H. Analysis and experiment of a vibration isolator using a novel magnetic spring with negative stiffness. Journal of Sound and Vibration, 333, 2958–2970 (2014)
ZHENG, Y. S., ZHANG, X. N., LUO, Y. J., YAN, B., and MA, C. C. Design and experiment of a high-static-low-dynamic stiffness isolator using a negative stiffness magnetic spring. Journal of Sound and Vibration, 360, 31–52 (2016)
DONG, G. X., ZHANG, X. N., LUO, Y. J., ZHANG, Y. H., and XIE, S. L. Analytical study of the low frequency multi-direction isolator with high-static-low-dynamic stiffness struts and spatial pendulum. Mechanical Systems and Signal Processing, 110, 521–539 (2018)
DONG, G. X., ZHANG, X. N., XIE, S. L., YAN, B., and LUO, Y. J. Simulated and experimental studies on a high-static-low-dynamic stiffness isolator using magnetic negative stiffness spring. Mechanical Systems and Signal Processing, 86, 188–203 (2017)
ZHENG, Y. S., ZHANG, X. N., LUO, Y. J., ZHANG, Y. H., and XIE, S. L. Analytical study of a quasi-zero stiffness coupling using a torsion magnetic spring with negative stiffness. Mechanical Systems and Signal Processing, 100, 135–151 (2018)
ZHOU, J. X., WANG, K., XU, D. L., OUYANG, H. J., and FU, Y. M. Vibration isolation in neonatal transport by using a quasi-zero-stiffness isolator. Journal of Vibration and Control, 24, 3278–3291 (2018)
WANG, Q., ZHOU, J. X., XU, D. L., and OUYANG, H. J. Design and experimental investigation of ultra-low frequency vibration isolation during neonatal transport. Mechanical Systems and Signal Processing, 139, 106633 (2020)
ZHU, T., CAZZOLATO, B., ROBERTSON, W. S. P., and ZANDER, A. Vibration isolation using six degree-of-freedom quasi-zero stiffness magnetic levitation. Journal of Sound and Vibration, 358, 48–73 (2015)
KAMARUZAMAN, N. A., ROBERTSON, W. S. P., GHAYESH, M. H., CAZZOLATO, B. S., and ZANDER, A. C. Six degree of freedom quasi-zero stiffness magnetic spring with active control: theoretical analysis of passive versus active stability for vibration isolation. Journal of Sound and Vibration, 502, 116086 (2021)
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)
ZHENG, Y. S., LI, Q. P., YAN, B., LUO, Y. J., and ZHANG, X. N. A Stewart isolator with high-static-low-dynamic stiffness struts based on negative stiffness magnetic springs. Journal of Sound and Vibration, 422, 390–408 (2018)
YUAN, S. J., SUN, Y., WANG, M., DING, J. H., ZHAO, J. L., HUANG, Y. N., PENG, Y., XIE, S. R., LUO, J., PU, H. Y., LIU, F. Q., BAI, L., and YANG, X. D. Tunable negative stiffness spring using maxwell normal stress. International Journal of Mechanical Sciences, 193, 106127 (2021)
PU, H. Y., YUAN, S. J., PENG, Y., MENG, K., ZHAO, J. L., XIE, R. Q., HUANG, Y. N., SUN, Y., YANG, Y., XIE, S. R., LUO, J., and CHEN, X. D. Multi-layer electromagnetic spring with tunable negative stiffness for semi-active vibration isolation. Mechanical Systems and Signal Processing, 121, 942–960 (2019)
ARAKI, Y., KIMURA, K., ASAI, T., MASUI, T., OMORI, T., and KAINUMA, R. Integrated mechanical and material design of quasi-zero-stiffness vibration isolator with superelastic Cu-Al-Mn shape memory alloy bars. Journal of Sound and Vibration, 358, 74–83 (2015)
MENG, L. S., SUN, J. G., and WU, W. J. Theoretical design and characteristics analysis of a quasi-zero stiffness isolator using a disk spring as negative stiffness element. Shock and Vibration, 2015, 813763 (2015)
ZHOU, X. H., SUN, X., ZHAO, D. X., YANG, X., and TANG, K. H. The design and analysis of a novel passive quasi-zero stiffness vibration isolator. Journal of Vibration Engineering & Technologies, 9, 225–245 (2021)
YAN, B., YU, N., MA, H. Y., and WU, C. Y. A theory for bistable vibration isolators. Mechanical Systems and Signal Processing, 167, 108507 (2022)
ISHIDA, S., SUZUKI, K., and SHIMOSAKA, H. Design and experimental analysis of origami-inspired vibration isolator with quasi-zero-stiffness characteristic. Journal of Vibration and Acoustics, 139, 051004 (2017)
SADEGHI, S. and LI, S. Y. Fluidic origami cellular structure with asymmetric quasi-zero stiffness for low-frequency vibration isolation. Smart Materials and Structures, 28, 065006 (2019)
LIU, S. W., PENG, G. L., and JIN, K. Design and characteristics of a novel QZS vibration isolation system with origami-inspired corrector. Nonlinear Dynamics, 106, 255–277 (2021)
LI, Y. L. and XU, D. L. Vibration attenuation of high dimensional quasi-zero stiffness floating raft system. International Journal of Mechanical Sciences, 126, 186–195 (2017)
DING, H. and CHEN, L. Q. Nonlinear vibration of a slightly curved beam with quasi-zero-stiffness isolators. Nonlinear Dynamics, 95, 2367–2382 (2019)
DING, H., JI, J. C., and CHEN, L. Q. Nonlinear vibration isolation for fluid-conveying pipes using quasi-zero stiffness characteristics. Mechanical Systems and Signal Processing, 121, 675–688 (2019)
DING, H., LU, Z. Q., and CHEN, L. Q. Nonlinear isolation of transverse vibration of pre-pressure beams. Journal of Sound and Vibration, 442, 738–751 (2019)
BOUNA, H. S., NBENDJO, B. R. N., and WOAFO, P. Isolation performance of a quasi-zero stiffness isolator in vibration isolation of a multi-span continuous beam bridge under pier base vibrating excitation. Nonlinear Dynamics, 100, 1125–1141 (2020)
LUO, A. C. J. and HUANG, J. Analytical dynamics of period-m flows and chaos in nonlinear systems. International Journal of Bifurcation and Chaos, 22, 1250093 (2012)
LUO, A. C. J. and HUANG, J. Analytical solutions for asymmetric periodic motions to chaos in a hardening Duffing oscillator. Nonlinear Dynamics, 72, 417–438 (2013)
KOVACIC, I., BRENNAN, M. J., and LINETON, B. Effect of a static force on the dynamic behaviour of a harmonically excited quasi-zero stiffness system. Journal of Sound and Vibration, 325, 870–883 (2009)
LUO, A. C. J. and HUANG, J. Approximate solutions of periodic motions in nonlinear systems via a generalized harmonic balance. Journal of Vibration and Control, 18, 1661–1674 (2012)
Author information
Authors and Affiliations
Corresponding authors
Additional information
Project supported by the National Natural Science Foundation of China (No. 12002195), the National Science Fund for Distinguished Young Scholars of China (No. 12025204), the Program of Shanghai Municipal Education Commission of China (No. 2019-01-07-00-09-E00018), and the Pujiang Project of Shanghai Science and Technology Commission of China (No. 20PJ1404000)
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Mao, X., Yin, M., Ding, H. et al. Modeling, analysis, and simulation of X-shape quasi-zero-stiffness-roller vibration isolators. Appl. Math. Mech.-Engl. Ed. 43, 1027–1044 (2022). https://doi.org/10.1007/s10483-022-2871-6
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10483-022-2871-6