Skip to main content
SpringerLink
Log in

Nonlinear dynamics of a track nonlinear energy sink

  • Research
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

A track nonlinear energy sink (TNES) has been proven as an effective control strategy. However, most researches focus on numerical or experimental aspects, with relatively little analytical study of intrinsic dynamic characteristics of the TNES. This study aims to investigate the nonlinear behaviors of a harmonically excited linear structure coupled with the TNES and reveal the vibration reduction performance of the TNES from the perspective of analysis. Firstly, the motion form of the TNES system is qualitatively analyzed by the global bifurcation, the time history, the Fourier spectrum, the phase trajectory, and the Poincare map. Secondly, the amplitude-frequency response for the periodic steady-state motion is quantitatively analyzed by combining the harmonic balance with the pseudo-arc length extension method and validated through numerical solutions. Then, the stability and bifurcation feature of the periodic steady-state solution are revealed by leveraging the Floquet theorem. Finally, the damping efficiency is explored. The results demonstrate that the complex nonlinearity of the TNES system can result in the coexistence of the periodic, quasi-periodic, and chaotic motion. Saddle-node bifurcations and Hopf bifurcations are discovered in the approximate analytical solutions. Strongly modulated responses (SMR) caused by Hopf bifurcation can greatly improve the damping efficiency of the TNES. In addition, a proper increase in mass ratio can suppress the adverse effects of frequency islands or saddle-node bifurcation curves near the resonance region. This research provides the necessary theoretical basis for optimizing and designing the TNES.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22
Fig. 23
Fig. 24
Fig. 25
Fig. 26

Similar content being viewed by others

Data availability

The data that support the findings of this study are available from the corresponding author [H. Ding], upon reasonable request.

References

  1. Ma, J.J., Liu, Z.T., Wang, C.S., Liu, F.J., Li, D., Guo, Y., Cai, J.C.: Nonlinear energy sink vibration suppression and parameter optimization of a finite-length beam on the elastic medium based on the modified Winkler theory. Nonlinear Dyn. 112, 59–80 (2024). https://doi.org/10.1007/s11071-023-09015-3

    Article  Google Scholar 

  2. Wang, T., Tang, Y., Qian, X.Y., Ding, Q., Yang, T.Z.: Enhanced nonlinear performance of nonlinear energy sink under large harmonic excitation using acoustic black hole effect. Nonlinear Dyn. 111, 12871–12898 (2023). https://doi.org/10.1007/s11071-023-08511-w

    Article  Google Scholar 

  3. Wang, Y.F., Kang, H.J., Cong, Y.Y., Guo, T.D., Fu, T.: Vibration suppression of a cable-stayed beam by a nonlinear energy sink. Nonlinear Dyn. 111, 14829–14849 (2023). https://doi.org/10.1007/s11071-023-08651-z

    Article  Google Scholar 

  4. Tehrani, G.G., Dardel, M.: Mitigation of nonlinear oscillations of a Jeffcott rotor system with an optimized damper and nonlinear energy sink. Int. J. Non-Linear Mech. 98, 122–136 (2018). https://doi.org/10.1016/j.ijnonlinmec.2017.10.011

    Article  Google Scholar 

  5. Wang, J., Zhang, X., Liu, Y., Qin, Z., Ma, L., Chu, F.: Rotor vibration control via integral magnetorheological damper. Int. J. Mech. Sci. 252, 108362 (2023). https://doi.org/10.1016/j.ijmecsci.2023.108362

    Article  Google Scholar 

  6. Pacheco, D.R.Q., Marques, F.D., Ferreira, A.J.M.: Panel flutter suppression with nonlinear energy sinks: numerical modeling and analysis. Int. J. Non-Linear Mech. 106, 108–114 (2018). https://doi.org/10.1016/j.ijnonlinmec.2018.08.009

    Article  Google Scholar 

  7. Zhang, S., Yang, Y., Li, Y., Wang, F., Ge, Y., Yang, Y.: Research on vibration suppression of spacecraft flexible appendage based on nonlinear energy sink. J. Vib. Eng. Technol. 11, 449–459 (2022). https://doi.org/10.1007/s42417-022-00587-9

    Article  Google Scholar 

  8. Luo, H., Tang, Z., Zhu, H.: High-performance isolation systems with rate-independent linear damping for seismic protection of high-rise buildings. Soil Dyn. Earthq. Eng. 171, 107976 (2023). https://doi.org/10.1016/j.soildyn.2023.107976

    Article  Google Scholar 

  9. Xiang, Y., Tan, P., He, H., Chen, Q., Ikeda, Y.: Seismic optimization for hysteretic damping-tuned mass damper (HD-TMD) subjected to white-noise excitation. Struct. Control. Health Monit. 2023, 1–21 (2023). https://doi.org/10.1155/2023/1465042

    Article  Google Scholar 

  10. Araz, O., Elias, S., Kablan, F.: Seismic-induced vibration control of a multi-story building with double tuned mass dampers considering soil-structure interaction. Soil Dyn. Earthq. Eng. 166, 107765 (2023). https://doi.org/10.1016/j.soildyn.2023.107765

    Article  Google Scholar 

  11. Salvi, J., Pioldi, F., Rizzi, E.: Optimum tuned mass dampers under seismic soil-structure interaction. Soil Dyn. Earthq. Eng. 114, 576–597 (2018). https://doi.org/10.1016/j.soildyn.2018.07.014

    Article  Google Scholar 

  12. Liu, S., Li, H., Xie, J., Yang, S., Li, P.: Comparative study on seismic performance of structure with TMD and particle TMD considering SSI effects. J. Build. Eng. 77, 107446 (2023). https://doi.org/10.1016/j.jobe.2023.107446

    Article  Google Scholar 

  13. Rathi, A.K., Chakraborty, A.: Reliability-based performance optimization of TMD for vibration control of structures with uncertainty in parameters and excitation. Struct. Control. Health Monit. 24, e1857 (2017). https://doi.org/10.1002/stc.1857

    Article  Google Scholar 

  14. Vakakis, A.F.: Inducing passive nonlinear energy sinks in vibrating systems. J. Vib. Acoust. 123, 324–332 (2001). https://doi.org/10.1115/1.1368883

    Article  Google Scholar 

  15. Vakakis, A.F., Gendelman, O.V.: Energy pumping in nonlinear mechanical oscillators: part II—resonance capture. J. Appl. Mech. 68, 42–48 (2001). https://doi.org/10.1115/1.1345525

    Article  MathSciNet  Google Scholar 

  16. Gaidai, O., Gu, Y., Xing, Y., Wang, J., Yurchenko, D.: Numerical optimisation of a classical stochastic system for targeted energy transfer. Theor. Appl. Mech. Lett. 13, 100422 (2023). https://doi.org/10.1016/j.taml.2022.100422

    Article  Google Scholar 

  17. Gendelman, O.V., Gourdon, E., Lamarque, C.H.: Quasiperiodic energy pumping in coupled oscillators under periodic forcing. J. Sound Vib. 294, 651–662 (2006). https://doi.org/10.1016/j.jsv.2005.11.031

    Article  Google Scholar 

  18. Mcfarland, D.M., Kerschen, G., Kowtko, J.J., Lee, Y.S., Bergman, L.A., Vakakis, A.F.: Experimental investigation of targeted energy transfers in strongly and nonlinearly coupled oscillators. J. Acoust. Soc. Am. 118, 791–799 (2005). https://doi.org/10.1121/1.1944649

    Article  Google Scholar 

  19. Wang, T., Ding, Q.: Targeted energy transfer analysis of a nonlinear oscillator coupled with bistable nonlinear energy sink based on nonlinear normal modes. J. Sound Vib. 556, 117727 (2023). https://doi.org/10.1016/j.jsv.2023.117727

    Article  Google Scholar 

  20. Dou, J., Yao, H., Li, H., Cao, Y., Liang, S.: Vibration suppression of multi-frequency excitation using cam-roller nonlinear energy sink. Nonlinear Dyn. 111, 11939–11964 (2023). https://doi.org/10.1007/s11071-023-08477-9

    Article  Google Scholar 

  21. Zhang, S., Zhou, J., Ding, H., Wang, K., Xu, D.: Fractional nonlinear energy sinks. Appl. Math. Mech. 44, 711–726 (2023). https://doi.org/10.1007/s10483-023-2984-9

    Article  MathSciNet  Google Scholar 

  22. Ding, H., Shao, Y.: NES cell. Appl. Math. Mech. 43, 1793–1804 (2022). https://doi.org/10.1007/s10483-022-2934-6

    Article  MathSciNet  Google Scholar 

  23. Li, S.B., Ding, H.: Effective damping zone of nonlinear energy sinks. Nonlinear Dyn. 111, 18605–18629 (2023). https://doi.org/10.1007/s11071-023-08874-0

    Article  Google Scholar 

  24. Geng, X.F., Ding, H., Jing, X.J., Mao, X.Y., Wei, K.X., Chen, L.Q.: Dynamic design of a magnetic-enhanced nonlinear energy sink. Mech. Syst. Signal Process. 185, 109813 (2023). https://doi.org/10.1016/j.ymssp.2022.109813

    Article  Google Scholar 

  25. Geng, X.F., Ding, H.: Theoretical and experimental study of an enhanced nonlinear energy sink. Nonlinear Dyn. 104, 3269–3291 (2021). https://doi.org/10.1007/s11071-021-06553-6

    Article  Google Scholar 

  26. Zeng, Y.C., Ding, H.: A tristable nonlinear energy sink. Int. J. Mech. Sci. 238, 107839 (2023). https://doi.org/10.1016/j.ijmecsci.2022.107839

    Article  Google Scholar 

  27. Sui, P., Shen, Y., Wang, X.: Study on response mechanism of nonlinear energy sink with inerter and grounded stiffness. Nonlinear Dyn. 111, 7157–7179 (2023). https://doi.org/10.1007/s11071-022-08226-4

    Article  Google Scholar 

  28. Aghayari, J., Bab, S., Safarpour, P., Rahi, A.: A novel modal vibration reduction of a disk-blades of a turbine using nonlinear energy sinks on the disk. Mech. Mach. Theory 155, 104048 (2021). https://doi.org/10.1016/j.mechmachtheory.2020.104048

    Article  Google Scholar 

  29. Liu, C., Wang, D.: Dynamic analysis of micro-shock absorbers. J. Vib. Eng. Technol. 11, 3029–3038 (2023). https://doi.org/10.1007/s42417-022-00728-0

    Article  Google Scholar 

  30. Zang, J., Chen, L.Q.: Complex dynamics of a harmonically excited structure coupled with a nonlinear energy sink. Acta Mech. Sin. 33, 801–822 (2017). https://doi.org/10.1007/s10409-017-0671-x

    Article  MathSciNet  Google Scholar 

  31. Wu, Z., Seguy, S., Paredes, M.: Basic constraints for design optimization of cubic and bistable nonlinear energy sink. J. Vib. Acoust. (2021). https://doi.org/10.1115/1.4051548

    Article  Google Scholar 

  32. Luongo, A., Zulli, D.: Dynamic analysis of externally excited NES-controlled systems via a mixed Multiple Scale/ Harmonic Balance algorithm. Nonlinear Dyn. 70, 2049–2061 (2012). https://doi.org/10.1007/s11071-012-0597-6

    Article  MathSciNet  Google Scholar 

  33. Franzini, G.R., Maciel, V.S.F., Vernizzi, G.J., Zulli, D.: Simultaneous passive suppression and energy harvesting from galloping using a bistable piezoelectric nonlinear energy sink. Nonlinear Dyn. 111, 22215–22236 (2023). https://doi.org/10.1007/s11071-023-08888-8

    Article  Google Scholar 

  34. Parseh, M., Dardel, M., Ghasemi, M.H.: Investigating the robustness of nonlinear energy sink in steady state dynamics of linear beams with different boundary conditions. Commun. Nonlinear Sci. Numer. Simul. 29, 50–71 (2015). https://doi.org/10.1016/j.cnsns.2015.04.020

    Article  MathSciNet  Google Scholar 

  35. Parseh, M., Dardel, M., Ghasemi, M.H.: Performance comparison of nonlinear energy sink and linear tuned mass damper in steady-state dynamics of a linear beam. Nonlinear Dyn. 81, 1981–2002 (2015). https://doi.org/10.1007/s11071-015-2120-3

    Article  MathSciNet  Google Scholar 

  36. Taghipour, J., Dardel, M.: Steady state dynamics and robustness of a harmonically excited essentially nonlinear oscillator coupled with a two-DOF nonlinear energy sink. Mech. Syst. Signal Process. 62–63, 164–182 (2015). https://doi.org/10.1016/j.ymssp.2015.03.018

    Article  Google Scholar 

  37. Zang, J., Zhang, Y.W.: Responses and bifurcations of a structure with a lever-type nonlinear energy sink. Nonlinear Dyn. 98, 889–906 (2019). https://doi.org/10.1007/s11071-019-05233-w

    Article  Google Scholar 

  38. Zeng, Y.C., Ding, H., Du, R.H., Chen, L.Q.: Micro-amplitude vibration suppression of a bistable nonlinear energy sink constructed by a buckling beam. Nonlinear Dyn. 108, 3185–3207 (2022). https://doi.org/10.1007/s11071-022-07378-7

    Article  Google Scholar 

  39. Wang, J.J., Wierschem, N.E., Spencer, B.F., Lu, X.L.: Experimental study of track nonlinear energy sinks for dynamic response reduction. Eng. Struct. 94, 9–15 (2015). https://doi.org/10.1016/j.engstruct.2015.03.007

    Article  Google Scholar 

  40. Wang, J.J., Wierschem, N.E., Wang, B., Spencer, B.F.: Multi-objective design and performance investigation of a high-rise building with track nonlinear energy sinks. Struct. Design Tall Spec. Build. 29, e1692 (2019). https://doi.org/10.1002/tal.1692

    Article  Google Scholar 

  41. Wang, J.J., Wierschem, N.E., Spencer, B.F., Lu, X.L.: Track nonlinear energy sink for rapid response reduction in building structures. J. Eng. Mech. 141, 04014104 (2015). https://doi.org/10.1061/(Asce)Em.1943-7889.0000824

    Article  Google Scholar 

  42. Wang, J.J., Wierschem, N.E., Spencer, B.F., Lu, X.L.: Numerical and experimental study of the performance of a single-sided vibro-impact track nonlinear energy sink. Earthq. Eng. Struct. Dyn. 45, 635–652 (2015). https://doi.org/10.1002/eqe.2677

    Article  Google Scholar 

  43. Wang, J.J., Li, H.B., Wang, B., Liu, Z.B., Zhang, C.: Development of a two-phased nonlinear mass damper for displacement mitigation in base-isolated structures. Soil Dyn. Earthq. Eng. 123, 435–448 (2019). https://doi.org/10.1016/j.soildyn.2019.05.007

    Article  Google Scholar 

  44. Wang, J.J., Zhang, C., Li, H.B., Liu, Z.B.: Experimental and numerical studies of a novel track bistable nonlinear energy sink with improved energy robustness for structural response mitigation. Eng. Struct. 237, 112184 (2021). https://doi.org/10.1016/j.engstruct.2021.112184

    Article  Google Scholar 

  45. Wang, J.J., Zhang, C., Li, H.B., Liu, Z.B.: A vertical-vibro-impact-enhanced track bistable nonlinear energy sink for robust and comprehensive control of structures. Struct. Control. Health Monit. 29, e2931 (2022). https://doi.org/10.1002/stc.2931

    Article  Google Scholar 

  46. Wang, J.J., Zheng, Y.Q.: Development and robustness investigation of track-based asymmetric nonlinear energy sink for impulsive response mitigation. Eng. Struct. 286, 116127 (2023). https://doi.org/10.1016/j.engstruct.2023.116127

    Article  Google Scholar 

  47. Zuo, H., Zhu, S.: Development of novel track nonlinear energy sinks for seismic performance improvement of offshore wind turbine towers. Mech. Syst. Signal Process. 172, 108975 (2022). https://doi.org/10.1016/j.ymssp.2022.108975

    Article  Google Scholar 

  48. Dou, J., Yao, H., Li, H., Li, J., Jia, R.: A track nonlinear energy sink with restricted motion for rotor systems. Int. J. Mech. Sci. 259, 108631 (2023). https://doi.org/10.1016/j.ijmecsci.2023.108631

    Article  Google Scholar 

  49. Wang, J.J., Wang, B., Wierschem, N.E., Spencer, B.F.: Dynamic analysis of track nonlinear energy sinks subjected to simple and stochastice excitations. Earthq. Eng. Struct. Dyn. 49, 863–883 (2020). https://doi.org/10.1002/eqe.3268

    Article  Google Scholar 

  50. Li, S.B., Li, J., Zhu, H.J., Lai, S.K.: Dynamical analysis and numerical verification of a non-smooth nonlinear energy sink. Int. J. Non-Linear Mech. 151, 104381 (2023). https://doi.org/10.1016/j.ijnonlinmec.2023.104381

    Article  Google Scholar 

  51. Ma, K., Du, J.T., Liu, Y., Chen, X.M.: Torsional vibration attenuation of a closed-loop engine crankshaft system via the tuned mass damper and nonlinear energy sink under multiple operating conditions. Mech. Syst. Signal Process. 207, 110941 (2024). https://doi.org/10.1016/J.Ymssp.2023.110941

    Article  Google Scholar 

  52. Zhao, Y.H., Du, J.T., Chen, Y.L., Liu, Y.: Nonlinear dynamic behavior analysis of an elastically restrained double-beam connected through a mass-spring system that is nonlinear. Nonlinear Dyn. 111, 8947–8971 (2023). https://doi.org/10.1007/s11071-023-08351-8

    Article  Google Scholar 

  53. Zheng, Z.Q., Chang, Z.Y., Zhao, L.: Mitigating deepwater jacket offshore platform vibration under wave and earthquake loadings utilizing nonlinear energy sinks. Ocean Eng. 283, 115096 (2023). https://doi.org/10.1016/j.oceaneng.2023.115096

    Article  Google Scholar 

  54. Wu, Z.H., Seguy, S., Paredes, M.: Basic constraints for design optimization of cubic and bistable nonlinear energy sink. J. Vib. Acoust. 144, 021003 (2022). https://doi.org/10.1115/1.4051548

    Article  Google Scholar 

  55. Li, T., Lamarque, C.H., Seguy, S., Berlioz, A.: Chaotic characteristic of a linear oscillator coupled with vibro-impact nonlinear energy sink. Nonlinear Dyn. 91, 2319–2330 (2017). https://doi.org/10.1007/s11071-017-4015-y

    Article  Google Scholar 

  56. Li, T., Seguy, S., Berlioz, A.: On the dynamics around targeted energy transfer for vibro-impact nonlinear energy sink. Nonlinear Dyn. 87, 1453–1466 (2016). https://doi.org/10.1007/s11071-016-3127-0

    Article  Google Scholar 

  57. Grinberg, I., Lanton, V., Gendelman, O.V.: Response regimes in linear oscillator with 2DOF nonlinear energy sink under periodic forcing. Nonlinear Dyn. 69, 1889–1902 (2012). https://doi.org/10.1007/s11071-012-0394-2

    Article  MathSciNet  Google Scholar 

  58. Tan, T., Wang, Z., Zhang, L., Liao, W.H., Yan, Z.: Piezoelectric autoparametric vibration energy harvesting with chaos control feature. Mech. Syst. Signal Process. 161, 107989 (2021). https://doi.org/10.1016/j.ymssp.2021.107989

    Article  Google Scholar 

  59. Moslemi, A., Homaeinezhad, M.R.: Effects of viscoelasticity on the stability and bifurcations of nonlinear energy sinks. Appl. Math. Mech. 44, 141–158 (2022). https://doi.org/10.1007/s10483-023-2944-9

    Article  MathSciNet  Google Scholar 

  60. Zhang, Y., Kong, X., Yue, C., Guo, J.: Characteristic analysis and design of nonlinear energy sink with cubic damping considering frequency detuning. Nonlinear Dyn. 111, 15817–15836 (2023). https://doi.org/10.1007/s11071-023-08673-7

    Article  Google Scholar 

  61. Huang, L., Yang, X.D.: Dynamics of a novel 2-DOF coupled oscillators with geometry nonlinearity. Nonlinear Dyn. 111, 18753–18777 (2023). https://doi.org/10.1007/s11071-023-08809-9

    Article  Google Scholar 

  62. Wang, X., Geng, X.F., Mao, X.Y., Ding, H., Jing, X.J., Chen, L.Q.: Theoretical and experimental analysis of vibration reduction for piecewise linear system by nonlinear energy sink. Mech. Syst. Signal Process. 172, 109001 (2022). https://doi.org/10.1016/j.ymssp.2022.109001

    Article  Google Scholar 

  63. Hou, L., Chen, H.Z., Che, Y.S., Lu, K., Liu, Z.S.: Bifurcation and stability analysis of a nonlinear rotor system subjected to constant excitation and rub-impact. Mech. Syst. Signal Process. 125, 65–78 (2019). https://doi.org/10.1016/j.ymssp.2018.07.019

    Article  Google Scholar 

  64. Li, H., Li, A., Kong, X.: Design criteria of bistable nonlinear energy sink in steady-state dynamics of beams and plates. Nonlinear Dyn. 103(2), 1475–1497 (2021). https://doi.org/10.1007/s11071-020-06178-1

    Article  Google Scholar 

  65. Qiu, D., Li, T., Seguy, S., Paredes, M.: Efficient targeted energy transfer of bistable nonlinear energy sink: application to optimal design. Nonlinear Dyn. 92(2), 443–461 (2018). https://doi.org/10.1007/s11071-018-4067-7

    Article  Google Scholar 

  66. Starosvetsky, Y., Gendelman, O.V.: Attractors of harmonically forced linear oscillator with attached nonlinear energy sink. II: optimization of a nonlinear vibration absorber. Nonlinear Dyn. 51(1–2), 47–57 (2007). https://doi.org/10.1007/s11071-006-9168-z

    Article  Google Scholar 

  67. Narimani, A., Golnaraghi, M.F., Jazar, G.N.: Sensitivity analysis of the frequency response of a piecewise linear system in a frequency island. J. Vib. Control 10, 175–198 (2004). https://doi.org/10.1177/1077546304032993

    Article  Google Scholar 

Download references

Funding

The authors gratefully acknowledge the support of the National Science Fund for Distinguished Young Scholars (No. 12025204).

Author information

Authors and Affiliations

  1. Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai Institute of Applied Mathematics and Mechanics, School of Mechanics and Engineering Science, Shanghai University, Shanghai, 200444, China

    Hao-Bo Li, Hu Ding, Tien-Chong Chang & Li-Qun Chen

  2. Shaoxing Institute of Technology, Shanghai University, Shaoxing, 312074, China

    Hu Ding

Authors
  1. Hao-Bo Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hu Ding
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Tien-Chong Chang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Li-Qun Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

Hao-Bo Li: Investigation, Writing—original draft. Hu Ding: Conceptualization, Funding acquisition, Writing—review & editing. Tien-Chong Chang: Writing—review, Funding acquisition. Li-Qun Chen: Writing—editing.

Corresponding authors

Correspondence to Hu Ding or Tien-Chong Chang.

Ethics declarations

Conflict of interest

The authors declare that they have no known competing financial interest or personal relationships that could have appeared to influence the work reported in this paper.

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.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, HB., Ding, H., Chang, TC. et al. Nonlinear dynamics of a track nonlinear energy sink. Nonlinear Dyn (2024). https://doi.org/10.1007/s11071-024-09683-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11071-024-09683-9

Keywords

Search

Navigation