Abstract
The transmissibility of the forced resonance for the nonlinear vibration isolation system (VIS) coupled with quasi-zero stiffness (QZS) and quadratic damping under base excitation are investigated. By utilizing the averaging method, the approximate analytical solutions of primary resonance (PR) and 1/3 subharmonic resonance (SR) for the nonlinear vibration isolator with QZS and quadratic damping are acquired. Employing Lyapunov's first method, the stability conditions of steady-state solutions for the nonlinear VIS with QZS and quadratic damping are determined. According to the derived conditions for the existence of subharmonic resonance, it is proved that when the considered nonlinear VIS has subharmonic resonance, it only exists within a certain excitation frequency range. The accuracy of the approximate analytical solutions for the amplitude-frequency response, force transmissibility, and relative displacement transmissibility of the PR and SR of the nonlinear VIS is confirmed by comparing them with the numerical results. The effects of QZS and quadratic damping on transmissibility of both force and relative displacement of nonlinear VIS have been discussed. The analysis results indicate that by choosing the appropriate QZS parameter or quadratic damping coefficient, the subharmonic resonance of the nonlinear VIS under a certain base excitation can be completely eliminated. When the amplitude of the base excitation increases to the extent that the system exhibits significant resonance behavior, for the same coefficient value, the nonlinear VIS coupled with QZS and quadratic damping can achieve smaller initial vibration isolation frequency and better amplitude suppression effect than that with linear damping.
Similar content being viewed by others
References
Yang, J., Xiong, Y.P., Xing, J.T.: Dynamics and power flow behaviour of a nonlinear vibration isolation system with a negative stiffness mechanism. J. Sound Vib. 332(1), 167–183 (2013)
Lu, Z.Q., Gu, D.H., Ding, H., Lacarbonara, W., Chen, L.Q.: Nonlinear vibration isolation via a circular ring. Mech. Syst. Signal Process. 136, 106490 (2020)
Ding, H., Lu, Z.Q., Chen, L.Q.: Nonlinear isolation of transverse vibration of pre-pressure beams. J. Sound Vib. 442, 738–751 (2019)
Smirnov, V., Mondrus, V.: Comparison of linear and nonlinear vibration isolation system under random excitation. Procedia Eng. 153, 673–678 (2016)
Yang, T., Cao, Q., Hao, Z.: A novel nonlinear mechanical oscillator and its application in vibration isolation and energy harvesting. Mech. Syst. Signal Process. 155, 107636 (2021)
Niu, M.Q., Chen, L.Q.: Nonlinear vibration isolation via a compliant mechanism and wire ropes. Nonlinear Dyn. (2021). https://doi.org/10.1007/s11071-021-06588-9
Santhosh, B.: Dynamics and performance evaluation of an asymmetric nonlinear vibration isolation mechanism. J. Braz. Soc. Mech. Sci. Eng. 40, 169 (2018)
Ibrahim, R.A.: Recent advances in nonlinear passive vibration isolators. J. Sound Vib. 314(3–5), 371–452 (2008)
Lu, Z.Q., Chen, L.Q.: Some recent progresses in nonlinear passive isolations of vibrations. Chin. J. Theor. Appl. Mech. 49(3), 550–564 (2017)
Ji, J.C., Luo, Q., Ye, K.: Vibration control based metamaterials and origami structures: a state-of-the-art review. Mech. Syst. Signal Process. 161, 107945 (2021)
Jing, X.: The X-structure/mechanism approach to beneficial nonlinear design in engineering. Appl. Math. Mech. (English Edition) 43(7), 979–1000 (2022)
Carrella, A., Brennan, M.J., Waters, T.P.: Static analysis of a passive vibration isolator with quasi-zero-stiffness characteristic. J. Sound Vib. 301(3–5), 678–689 (2007)
Carrella, A., Brennan, M.J., Kovacic, I., Waters, T.P.: On the force transmissibility of a vibration isolator with quasi-zero-stiffness. J. Sound Vib. 322(4–5), 707–717 (2009)
Zhou, X., Sun, X., Zhao, D., Yang, X., Tang, K.: The design and analysis of a novel passive quasi-zero stiffness vibration isolator. J. Vib. Eng. Technol. 9, 225–245 (2021)
Wen, G., He, J., Liu, J., Lin, Y.: Design, analysis and semi-active control of a quasi-zero stiffness vibration isolation system with six oblique springs. Nonlinear Dyn. 106, 309–321 (2021)
Shaw, A.D., Gatti, G., Gonçalves, P.J.P., Tang, B., Brennan, M.J.: Design and test of an adjustable quasi-zero stiffness device and its use to suspend masses on a multi-modal structure. Mech. Syst. Signal Process. 152, 107354 (2021)
Wang, K., Zhou, J., Chang, Y., Ouyang, H., Xu, D., Yang, Y.: A nonlinear ultra-low-frequency vibration isolator with dual quasi-zero-stiffness mechanism. Nonlinear Dyn. 101, 755–773 (2020)
Suman, S., Balaji, P.S., Selvakumar, K., Kumaraswamidhas, L.A.: Nonlinear vibration control device for a vehicle suspension using negative stiffness mechanism. J. Vib. Eng. Technol. 9, 957–966 (2021)
Chen, T., Zheng, Y., Song, L., Gao, X., Li, Z.: Design of a new quasi-zero-stiffness isolator system with nonlinear positive stiffness configuration and its novel features. Nonlinear Dyn. 111, 5141–5163 (2023)
Zhao, F., Ji, J., Ye, K., Luo, Q.: An innovative quasi-zero stiffness isolator with three pairs of oblique springs. Int. J. Mech. Sci. 192, 106093 (2021)
Robertson, W.S., Kidner, M.R.F., Cazzolato, B.S., Zander, A.C.: Theoretical design parameters for a quasi-zero stiffness magnetic spring for vibration isolation. J. Sound Vib. 326(1–2), 88–103 (2009)
Zheng, Y., Zhang, X., Luo, Y., Yan, B., Ma, C.: Design and experiment of a high-static–low-dynamic stiffness isolator using a negative stiffness magnetic spring. J. Sound Vib. 360, 31–52 (2016)
Yuan, J., Jin, G., Ye, T., Chen, Y., Bai, J.: Theoretical modeling and analysis of a quasi-zero-stiffness vibration isolator equipped with extensible and axially magnetized negative stiffness modules. J. Vib. Control (2023). https://doi.org/10.1177/10775463221140437
Yuan, S., Sun, Y., Zhao, J., Meng, K., Wang, M., Pu, H., Peng, Y., Luo, J., Xie, S.: A tunable quasi-zero stiffness isolator based on a linear electromagnetic spring. J. Sound Vib. 482, 115449 (2020)
Wang, M., Su, P., Liu, S., Chai, K., Wang, B., Lu, J.: Design and analysis of electromagnetic quasi-zero stiffness vibration isolator. J. Vib. Eng. Technol. 11, 153–164 (2023)
Ma, Z., Zhou, R., Yang, Q., Lee, H.P., Chai, K.: A semi-active electromagnetic quasi-zero-stiffness vibration isolator. Int. J. Mech. Sci. 252, 108357 (2023)
An, J., Chen, G., Deng, X., Xi, C., Wang, T., He, H.: Analytical study of a pneumatic quasi-zero-stiffness isolator with mistuned mass. Nonlinear Dyn 108, 3297–3312 (2022)
Xu, X., Liu, H., Jiang, X., et al.: Uncertainty analysis and optimization of quasi-zero stiffness air suspension based on polynomial chaos method. Chin. J. Mech. Eng. 35, 93 (2022)
Wang, Q., Zhou, J., Wang, K., Xu, D., Wen, G.: Design and experimental study of a compact quasi-zero-stiffness isolator using wave springs. Sci. China Technol. Sci. 64, 2255–2271 (2021)
Jiang, Y., Song, C., Ding, C., Xu, B.: Design of magnetic-air hybrid quasi-zero stiffness vibration isolation system. J. Sound Vib. 477, 115346 (2020)
Zhang, Z., Zhang, Y.W., Ding, H.: Vibration control combining nonlinear isolation and nonlinear absorption. Nonlinear Dyn. 100, 2121–2139 (2020)
Liu, Y., Ji, W., Xu, L., Gu, H., Song, C.: Dynamic characteristics of quasi-zero stiffness vibration isolation system for coupled dynamic vibration absorber. Arch Appl Mech 91, 3799–3818 (2021)
Zeng, Y., Ding, H., Du, R.H., Chen, L.Q.: A suspension system with quasi-zero stiffness characteristics and inerter nonlinear energy sink. J. Vib. Control 28(1–2), 143–158 (2022)
Yan, B., Yu, N., Wu, C.: A state-of-the-art review on low-frequency nonlinear vibration isolation with electromagnetic mechanisms. Appl. Math. Mech.-Engl. Ed. 43, 1045–1062 (2022)
Ma, H., Yan, B.: Nonlinear damping and mass effects of electromagnetic shunt damping for enhanced nonlinear vibration isolation. Mech. Syst. Signal Process. 146, 107010 (2021)
Huang, D., Xu, W., Xie, W., Liu, Y.: Dynamical properties of a forced vibration isolation system with real-power nonlinearities in restoring and damping forces. Nonlinear Dyn. 81, 641–658 (2015)
Ho, C., Lang, Z.Q., Billings, S.A.: Design of vibration isolators by exploiting the beneficial effects of stiffness and damping nonlinearities. J. Sound Vib. 333(12), 2489–2504 (2014)
Dong, G., Zhang, Y., Luo, Y., Xie, S., Zhang, X.: Enhanced isolation performance of a high-static–low-dynamic stiffness isolator with geometric nonlinear damping. Nonlinear Dyn. 93, 2339–2356 (2018)
Gao, X., Teng, H.D.: Dynamics and isolation properties for a pneumatic near-zero frequency vibration isolator with nonlinear stiffness and damping. Nonlinear Dyn. 102, 2205–2227 (2020)
Lv, Q., Yao, Z.: Analysis of the effects of nonlinear viscous damping on vibration isolator. Nonlinear Dyn. 79, 2325–2332 (2015)
Cheng, C., Li, S., Wang, Y., Jiang, X.: Force and displacement transmissibility of a quasi-zero stiffness vibration isolator with geometric nonlinear damping. Nonlinear Dyn. 87, 2267–2279 (2017)
Liu, Y., Xu, L., Song, C., Gu, H., Ji, W.: Dynamic characteristics of a quasi-zero stiffness vibration isolator with nonlinear stiffness and damping. Arch. Appl. Mech. 89, 1743–1759 (2019)
Peng, Z.K., Meng, G., Lang, Z.Q., Zhang, W.M., Chu, F.L.: Study of the effects of cubic nonlinear damping on vibration isolations using harmonic balance method. Int. J. Non-Linear Mech. 47(10), 1073–1080 (2012)
Lu, Z., Brennan, M., Ding, H., Chen, L.: High-static-low-dynamic-stiffness vibration isolation enhanced by damping nonlinearity. Sci. China Technol. Sci. 62, 1103–1110 (2019)
Liu, C., Yu, K.: Superharmonic resonance of the quasi-zero-stiffness vibration isolator and its effect on the isolation performance. Nonlinear Dyn. 100, 95–117 (2020)
Wang, R.: Random vibrations of nonlinearly damped locomotive and rolling stock. J. Southwest Jiaotong Univ. 03, 101–112 (1985)
Fay, T.H.: Quadratic damping. Int. J. Math. Educ. Sci. Technol. 43(6), 789–803 (2012)
Guan, J., Zuo, J., Zhao, W., Gomi, N., Zhao, X.: Study on hydraulic dampers using a foldable inverted spiral origami structure. Vibration 5, 711–731 (2022)
Niu, J., Zhang, W., Shen, Y., Wang, J.: Subharmonic resonance of quasi-zero-stiffness vibration isolation system with dry friction damper. Chin. J. Theor. Appl. Mech. 55(4), 1092–1101 (2022)
Pierre, C., Ferri, A.A., Dowell, E.H.: Multi-harmonic analysis of dry friction damped systems using an incremental harmonic balance method. J. Appl. Mech. 52(4), 958–964 (1985)
Funding
This work was supported by the National Natural Science Foundation of China (Grant Nos. 12272241, 12202286) and Natural Science Foundation of Hebei Province (Grant No. A2021210012).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Niu, J., Zhang, W. & Zhang, X. Resonance analysis of vibration isolation system with quasi-zero stiffness and quadratic damping under base excitation. Acta Mech 234, 6377–6394 (2023). https://doi.org/10.1007/s00707-023-03714-z
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00707-023-03714-z