Applied Mathematics and Mechanics >
Nonlinear vibration of quasi-zero stiffness structure with piezoelectric harvester and RL-load: intra-well and inter-well oscillation modes under 1:1 internal resonance
Received date: 2025-04-10
Revised date: 2025-06-23
Online published: 2025-07-28
Supported by
Project supported by the National Key R&D Program of China (No. 2023YFE0125900)
Copyright
This study explores the nonlinear dynamics of a quasi-zero stiffness (QZS) vibration isolator coupled with a piezoelectric energy harvester connected to an RL-resonant circuit. The model of the system is formulated with the Lagrangian mechanics, representing a two-degree-of-freedom nonlinear electromechanical system subject to harmonic base excitation under a 1:1 internal resonance condition. The model is normalized, and the conditions dictating monostable and bistable oscillation modes are identified. The bifurcation characteristics of the coupled system are analyzed in both oscillation modes by means of harmonic balance and continuation methods. The vibration isolation performance, with and without the coupled harvester, is evaluated in terms of displacement transmissibility to assess its dual functionalities for vibration isolation and energy harvesting. Analytical results demonstrate that integrating a piezoelectric harvester into a monostable QZS isolator under 1:1 internal resonance does not compromise its vibration isolation capability while enabling efficient energy harvesting at extremely low-frequency base excitation. Furthermore, the system’s response under strong base excitation is investigated exclusively for energy harvesting in both monostable and bistable modes, leading to optimal structural parameter design. The conditions for intra-well and inter-well periodic oscillation modes, as well as chaotic responses, are analyzed analytically and validated numerically through stability charts, basins of attraction, bifurcation diagrams, time histories, and Poincaré maps. This work provides a comprehensive understanding of the oscillation dynamics of QZS isolators and offers valuable insights for optimizing their geometric parameters to function as high-performance vibration isolators and/or energy harvesters.
N. A. SAEED , Y. Y. ELLABBAN , Lei HOU , Haiming YI , Shun ZHONG , F. Z. DURAIHEM , O. M. OMARA . Nonlinear vibration of quasi-zero stiffness structure with piezoelectric harvester and RL-load: intra-well and inter-well oscillation modes under 1:1 internal resonance[J]. Applied Mathematics and Mechanics, 2025 , 46(8) : 1451 -1474 . DOI: 10.1007/s10483-025-3285-8
| [1] | IBRAHIM, R. Recent advances in nonlinear passive vibration isolators. Journal of Sound and Vibration, 314(3-5), 371–452 (2008) |
| [2] | LIU, C., JING, X., DALEY, S., and LI, F. Recent advances in micro-vibration isolation. Mechanical Systems and Signal Processing, 56-57, 55–80 (2015) |
| [3] | ZENG, R., WEN, G., ZHOU, J., YIN, S., WANG, Q., and WU, X. Experimental investigation of a non-smooth quasi-zero-stiffness isolator. Acta Mechanica Sinica, 39(6), 522415 (2023) |
| [4] | LING, P., MIAO, L., YE, B., YOU, J., ZHANG, W., and YAN, B. Ultra-low frequency vibration isolation of a novel click-beetle-inspired structure with large quasi-zero stiffness region. Journal of Sound and Vibration, 558, 117756 (2023) |
| [5] | LIU, C., ZHANG, W., YU, K., LIU, T., and ZHENG, Y. Quasi-zero-stiffness vibration isolation: designs, improvements and applications. Engineering Structures, 301, 117282 (2024) |
| [6] | WANG, K., ZHOU, J., TAN, D., LI, Z., LIN, Q., and XU, D. A brief review of metamaterials for opening low-frequency band gaps. Applied Mathematics and Mechanics (English Edition), 43(7), 1125–1144 (2022) https://doi.org/10.1007/s10483-022-2870-9 |
| [7] | SAEED, N. A. and EISSA, M. Bifurcation analysis of a transversely cracked nonlinear Jeffcott rotor system at different resonance cases. International Journal of Acoustics and Vibration, 24(2), 284–302 (2019) |
| [8] | 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) |
| [9] | 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) |
| [10] | 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) |
| [11] | 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) |
| [12] | 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) |
| [13] | 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) |
| [14] | 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) |
| [15] | ERTURK, A. and INMAN, D. J. Piezoelectric Energy Harvesting, John Wiley & Sons, Hoboken (2011) |
| [16] | OTTMAN, G., HOFMANN, H., BHATT, A., and LESIEUTRE, G. Adaptive piezoelectric energy harvesting circuit for wireless, remote power supply. IEEE Transactions on Power Electronics, 17(5), 669–676 (2002) |
| [17] | COOK-CHENNAULT, K. A., THAMBI, N., and SASTRY, A. M. Powering MEMS portable devices — a review of non-regenerative and regenerative power supply systems with special emphasis on piezoelectric energy harvesting systems. Smart Materials and Structures, 17(4), 043001 (2008) |
| [18] | ABDELKAREEM, M. A. A., XU, L., ALI, M. K. A., ELAGOUZ, A., MI, J., GUO, S., LIU, Y., and ZUO, L. Vibration energy harvesting in automotive suspension system: a detailed review. Applied Energy, 229, 672–699 (2018) |
| [19] | PUTRA, T., HUSAINI, and IKBAL, M. Automotive suspension component behaviors driven on flat and rough road surfaces. Heliyon, 7(7), e07528 (2021) |
| [20] | ZHOU, S., SONG, G., WANG, R., REN, Z., and WEN, B. Nonlinear dynamic analysis for coupled vehicle-bridge vibration system on nonlinear foundation. Mechanical Systems and Signal Processing, 87, 259–278 (2017) |
| [21] | ZUO, J., DONG, L., YANG, F., GUO, Z., WANG, T., and ZUO, L. Energy harvesting solutions for railway transportation: a comprehensive review. Renewable Energy, 202, 56–87 (2023) |
| [22] | GHOLIKHANI, M., ROSHANI, H., DESSOUKY, S., and PAPAGIANNAKIS, A. T. A critical review of roadway energy harvesting technologies. Applied Energy, 261, 114388 (2020) |
| [23] | LEI, S., GE, Y., LI, Q., and THOMPSON, D. J. Frequency-domain method for non-stationary stochastic vibrations of train-bridge coupled system with time-varying characteristics. Mechanical Systems and Signal Processing, 183, 109637 (2023) |
| [24] | CAHILL, P., HAZRA, B., KAROUMI, R., MATHEWSON, A., and PAKRASHI, V. Vibration energy harvesting-based monitoring of an operational bridge undergoing forced vibration and train passage. Mechanical Systems and Signal Processing, 106, 265–283 (2018) |
| [25] | YANG, K., ABDELKEFI, A., LI, X., MAO, Y., DAI, L., and WANG, J. Stochastic analysis of a galloping-random wind energy harvesting performance on a buoy platform. Energy Conversion and Management, 238, 114174 (2021) |
| [26] | BAO, B., ZHOU, S., and WANG, Q. Interplay between internal resonance and nonlinear magnetic interaction for multi-directional energy harvesting. Energy Conversion and Management, 244, 114465 (2021) |
| [27] | ABOHAMER, M. K., AWREJCEWICZ, J., STAROSTA, R., AMER, T. S., and BEK, M. A. Influence of the motion of a spring pendulum on energy-harvesting devices. Applied Sciences, 11(18), 8658 (2021) |
| [28] | AMER, T. S., WAHBA, A. M., ABOLILA, A. F., and GALAL, A. A. Optimizing stability and characteristics of a vibrating rigid body pendulum with energy harvesting device. Journal of Low Frequency Noise, Vibration and Active Control, 44(2), 893–937 (2025) |
| [29] | ABOHAMER, M. K., AWREJCEWICZ, J., and AMER, T. S. Modeling of the vibration and stability of a dynamical system coupled with an energy harvesting device. Alexandria Engineering Journal, 63, 377–397 (2023) |
| [30] | GU, Y., LIU, W., ZHAO, C., and WANG, P. A goblet-like non-linear electromagnetic generator for planar multi-directional vibration energy harvesting. Applied Energy, 266, 114846 (2020) |
| [31] | ZHAO, L., ZOU, H., ZHAO, Y., WU, Z., LIU, F., WEI, K., and ZHANG, W. Hybrid energy harvesting for self-powered rotor condition monitoring using maximal utilization strategy in structural space and operation process. Applied Energy, 314, 118983 (2022) |
| [32] | LAI, Z., WANG, S., ZHU, L., ZHANG, G., WANG, J., YANG, K., and YURCHENKO, D. A hybrid piezo-dielectric wind energy harvester for high-performance vortex-induced vibration energy harvesting. Mechanical Systems and Signal Processing, 150, 107212 (2021) |
| [33] | SUN, W., JIANG, Z., XU, X., HAN, Q., and CHU, F. Harmonic balance analysis of output characteristics of free-standing mode triboelectric nanogenerators. International Journal of Mechanical Sciences, 207, 106668 (2021) |
| [34] | ZOU, H., ZHAO, L., WANG, Q., GAO, Q., YAN, G., WEI, K., and ZHANG, W. A self-regulation strategy for triboelectric nanogenerator and self-powered wind-speed sensor. Nano Energy, 95, 106990 (2022) |
| [35] | AMER, T. S., ARAB, A., and GALAL, A. A. On the influence of an energy harvesting device on a dynamical system. Journal of Low Frequency Noise, Vibration and Active Control, 43(2), 669–705 (2024) |
| [36] | AMER, T. S., ABDELHFEEZ, S. A., ELBAZ, R. F., and ABOHAMER, M. K. Investigation of the dynamical analysis, stability, and bifurcation for a connected damped oscillator with a piezoelectric harvester. Journal of Vibration Engineering & Technologies, 13, 155 (2025) |
| [37] | FANG, S., ZHOU, S., YURCHENKO, D., YANG, T., and LIAO, W. H. Multistability phenomenon in signal processing, energy harvesting, composite structures, and metamaterials: a review. Mechanical Systems and Signal Processing, 166, 108419 (2022) |
| [38] | CHEN, K., DING, X., TIAN, L., SHEN, H., SONG, R., BIAN, Y., and YANG, Q. An M-shaped buckled beam for enhancing nonlinear energy harvesting. Mechanical Systems and Signal Processing, 188, 110066 (2023) |
| [39] | LAN, C., LIAO, Y., HU, G., and TANG, L. Equivalent impedance and power analysis of monostable piezoelectric energy harvesters. Journal of Intelligent Material Systems and Structures, 31(14), 1697–1715 (2020) |
| [40] | DULIN, S., LIN, K., SERDUKOVA, L., KUSKE, R., and YURCHENKO, D. Improving the performance of a two-sided vibro-impact energy harvester with asymmetric restitution coefficients. International Journal of Mechanical Sciences, 217, 106983 (2022) |
| [41] | FAN, K., LIU, J., WEI, D., ZHANG, D., ZHANG, Y., and TAO, K. A cantilever-plucked and vibration-driven rotational energy harvester with high electric outputs. Energy Conversion and Management, 244, 114504 (2021) |
| [42] | TANG, L., YANG, Y., and SOH, C. K. Toward broadband vibration-based energy harvesting. Journal of Intelligent Material Systems and Structures, 21(18), 1867–1897 (2010) |
| [43] | CAO, D., WANG, J., GUO, X., LAI, S., and SHEN, Y. Recent advancement of flow-induced piezoelectric vibration energy harvesting techniques: principles, structures, and nonlinear designs. Applied Mathematics and Mechanics (English Edition), 43(7), 959–978 (2022) https://doi.org/10.1007/s10483-022-2867-7 |
| [44] | ZHOU, S., CAO, J., and LIN, J. Theoretical analysis and experimental verification for improving energy harvesting performance of nonlinear monostable energy harvesters. Nonlinear Dynamics, 86(3), 1599–1611 (2016) |
| [45] | FAN, K., TAN, Q., LIU, H., ZHANG, Y., and CAI, M. Improved energy harvesting from low-frequency small vibrations through a monostable piezoelectric energy harvester. Mechanical Systems and Signal Processing, 117, 594–608 (2019) |
| [46] | KUMAR, A., ALI, S. F., and AROCKIARAJAN, A. Exploring the benefits of an asymmetric monostable potential function in broadband vibration energy harvesting. Applied Physics Letters, 112(23), 233901 (2018) |
| [47] | REZAEI, M., TALEBITOOTI, R., LIAO, W. H., and FRISWELL, M. I. Integrating PZT layer with tuned mass damper for simultaneous vibration suppression and energy harvesting considering exciter dynamics: an analytical and experimental study. Journal of Sound and Vibration, 546, 117413 (2023) |
| [48] | LI, X., LIU, K., XIONG, L., and TANG, L. Development and validation of a piecewise linear nonlinear energy sink for vibration suppression and energy harvesting. Journal of Sound and Vibration, 503, 116104 (2021) |
| [49] | LU, Z., ZHAO, L., DING, H., and CHEN, L. A dual-functional metamaterial for integrated vibration isolation and energy harvesting. Journal of Sound and Vibration, 509, 116251 (2021) |
| [50] | BUKHARI, M. and BARRY, O. Simultaneous energy harvesting and vibration control in a nonlinear metastructure: a spectro-spatial analysis. Journal of Sound and Vibration, 473, 115215 (2020) |
| [51] | FANG, S., CHEN, K., ZHAO, B., LAI, Z., ZHOU, S., and LIAO, W. H. Simultaneous broadband vibration isolation and energy harvesting at low frequencies with quasi-zero stiffness and nonlinear monostability. Journal of Sound and Vibration, 553, 117684 (2023) |
| [52] | LI, Y., BAKER, E., REISSMAN, T., SUN, C., and LIU, W. K. Design of mechanical metamaterials for simultaneous vibration isolation and energy harvesting. Applied Physics Letters, 111(25), 251903 (2017) |
| [53] | LIU, C., ZHAO, R., YU, K., LEE, H. P., and LIAO, B. Simultaneous energy harvesting and vibration isolation via quasi-zero-stiffness support and radially distributed piezoelectric cantilever beams. Applied Mathematical Modelling, 100, 152–169 (2021) |
| [54] | 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) |
| [55] | 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) |
| [56] | SHAHRAEENI, M., SOROKIN, V., MACE, B., and ILANKO, S. Effect of damping nonlinearity on the dynamics and performance of a quasi-zero-stiffness vibration isolator. Journal of Sound and Vibration, 526, 116822 (2022) |
| [57] | SAEED, N. A., AWREJCEWICZ, J., ALKASHIF, M. A., and MOHAMED, M. S. 2D and 3D visualization for the static bifurcations and nonlinear oscillations of a self-excited system with time-delayed controller. Symmetry, 14, 621 (2022) |
| [58] | SUN, K. H., LIU, X., and ZHU, C. X. The 0-1 test algorithm for chaos and its applications. Chinese Physics B, 19, 110510 (2010) |
/
| 〈 |
|
〉 |
Email Alert
RSS