Skip to main content
SpringerLink
Log in

Two-dimensional inerter-enhanced nonlinear energy sink

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

Multiple-direction vibration widely exists in the environment, which is harmful to instruments and people. Hence, the high demand for vibration isolators with exceptional efficiency is thus evident. Although the proposed nonlinear energy sink was proven to be highly utility, most of the developed such devices, even with some inerter enhanced NESs, are limited to one-dimensional vibration suppression. In this paper, we propose a two-dimensional inerter-enhanced NES (2D IE-NES) which is applicable for multiple-directional low-frequency vibration suppression. The Lagrangian method is employed to derive the dynamic equations of the 2D IE-NES. Then these differential equations are solved by means of the Runge–Kutta method. Numerical results show that the novel 2D IE-NES configuration can isolate vibration more efficiently than the traditional NES under instantaneous shock, constant periodic and stochastic excitation. The role of the inerters in the vibration decaying process is highlighted in energy flow. This paper provides a new platform for the low-frequency multiple-direction vibration isolator.

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

Similar content being viewed by others

Data Availability

Enquiries about data availability should be directed to the authors.

References

  1. Cobb, R.G., Sullivan, J.M., Das, A., Davis, L.P., Hyde, T.T., Davis, T., Rahman, Z.H., Spanos, J.T.: Vibration isolation and suppression system for precision payloads in space. Smart Mater. Struct. 8, 798–812 (1999)

    Google Scholar 

  2. Sullivan, L.A., Fuentes, R.J., Babuska, V., Erwin R.S., Anderson E.H.: On-orbit active vibration isolation: the satellite ultraquiet isolation technologies experiment (SUITE), AIAA (2003)

  3. Aglietti, G.S., Langley, R.S., Gabriel, S.B.: Model building and verification for active control of microvibrations with probabilistic assessment of the effects of uncertainties. Proc. Inst. Mech. Eng. Part C Mech. Eng. Sci. 218, 389–399 (2004)

    Google Scholar 

  4. Ikago, K., Saito, K., Inoue, N.: Seismic control of single-degree-of-freedom structure using tuned viscous mass damper. Earthq. Eng. Struct. Dyn. 41, 453–474 (2012)

    Google Scholar 

  5. Ricciardelli, F., Vickery, B.J.: Tuned vibration absorbers with dry friction damping. Earthq. Eng. Struct. Dyn. 28, 707–724 (1999)

    Google Scholar 

  6. Xu, Q., Niu, J.K., Yao, H.L., Zhao, L.C., Wen, B.C.: Nonlinear dynamic behavior and stability of a rotor/seal system with the dynamic vibration absorber. Adv. Mech. Eng. 11, 1–17 (2019)

    Google Scholar 

  7. Ibrahim, R.A.: Recent advances in nonlinear passive vibration isolators. J. Sound Vib. 314, 371–452 (2008)

    Google Scholar 

  8. Zhao, F., Cao, S.Q., Luo, Q.T., Li, L.Q., Ji, J.C.: Practical design of the QZS isolator with one pair of oblique bars by considering pre-compression and low-dynamic stiffness. Nonlinear Dyn. 108, 3313–3330 (2022)

    Google Scholar 

  9. Zhao, F., Ji, J.C., Luo, Q.T., Cao, S.Q., Chen, L.M., Du, W.L.: An improved quasi-zero stiffness isolator with two pairs of oblique springs to increase isolation frequency band. Nonlinear Dyn. 104, 349–365 (2021)

    Google Scholar 

  10. Zhao, F., Ji, J.C., Ye, K., Luo, Q.T.: An innovative quasi-zero stiffness isolator with three pairs of oblique springs. Int. J. Mech. Sci. 192, 106903 (2021)

    Google Scholar 

  11. Vakakis, A.F.: Inducing passive nonlinear energy sinks in vibrating systems. J. Vib. Acoust. 123, 324–332 (2001)

    Google Scholar 

  12. Lee, Y.S., Vakakis, A.F. Bergman, L.A., McFarland, D.M., Kerschen, G., Nuceca, F., Tsakirtzis, S., Panagopoulos, P.N.: Passive non-linear targeted energy transfer and its applications to vibration absorption: a review. Proc. IMechE Part K J. Multi-body Dyn 222, 77–134 (2008)

  13. Vakakis, A.F., Gendelman, O.V., Bergman, L.A., McFarland, D.M., Kerschen, G., Lee, Y.S.: Nonlinear Targeted Energy Transfer in Mechanical and Structural Systems, pp. 161–229. Springer, Berlin (2008)

    Google Scholar 

  14. Gendelman, O.V.: Targeted energy transfer in systems with external and self-excitation. Proc. Inst. Mech. Eng. Part C J. Mech. Eng. Sci. 225, 2007–2043 (2011)

    Google Scholar 

  15. Li, X., Ding, H., Chen, L.Q.: Effects of weights on vibration suppression via a nonlinear energy sink under vertical stochastic excitations. Mech. Syst. Signal Process. 173, 109073 (2022)

    Google Scholar 

  16. Chen, L.Q., Li, X., Lu, Z.Q., Zhang, Y.W., Ding, H.: Dynamic effects of weights on vibration reduction by a nonlinear energy sink moving vertically. J. Sound Vib. 451, 99–119 (2019)

    Google Scholar 

  17. Li, X., Mojahed, A., Chen, L.Q., Bergman, L.A., Vakakis, A.F.: Shock response mitigation of a large-scale structure by modal energy redistribution facilitated by a strongly nonlinear absorber. Acta Mech. Sin. 38, 121464 (2022)

    MathSciNet  Google Scholar 

  18. Haris, A., Motato, E., Theodossiades, S., Rahnejat, H., Kelly, P., Vakakis, A.F., Bergman, L.A., McFarland, D.M.: A study on torsional vibration attenuation in automotive drivetrains using absorbers with smooth and non-smooth nonlinearities. Appl. Math. Model. 46, 674–690 (2017)

    MathSciNet  Google Scholar 

  19. Geng, X.F., Ding, H., Mao, X.Y., Chen, L.Q.: Nonlinear energy sink with limited vibration amplitude. Mech. Syst. Signal Process. 156, 107625 (2021)

    Google Scholar 

  20. Dang, W.H., Wang, Z.H., Chen, L.Q., Yang, T.Z.: A high-efficient nonlinear energy sink with a one-way energy converter. Nonlinear Dyn. 109, 2247–2261 (2022)

    Google Scholar 

  21. Youssef, B., Leine, R.I.: A complete set of design rules for a vibro-impact NES based on a multiple scales approximation of a nonlinear mode. J. Sound Vib. 501, 116043 (2021)

    Google Scholar 

  22. Qiu, D., Seguy, S., Paredes, M.: Design criteria for optimally tuned vibro-impact nonlinear energy sink. J. Sound Vib. 442, 497–513 (2019)

    Google Scholar 

  23. Farid, M., Gendelman, O.V., Babitsky, V.I.: Dynamics of a hybrid vibro-impact nonlinear energy sink. J. Appl. Math. Mec. 101, 341–352 (2019)

    Google Scholar 

  24. Wang, J.J., Wierschem, N., Spencer, B.F., Lu, X.L.: Experimental study of track nonlinear energy sinks for dynamic response reduction. Eng. Struct. 94, 9–15 (2015)

    Google Scholar 

  25. Wang, J.J., Wierschem, N., Wang, B., Spencer, B.F.: Multi-objective design and performance investigation of a high-rise building with track nonlinear energy sinks. Sturct. Des. Tall Spec. 29, 1692–1715 (2019)

    Google Scholar 

  26. Gendelman, O.V., Sapsis, T., Vakakis, A.F., Bergman, L.A.: Enhanced passive targeted energy transfer in strongly nonlinear mechanical oscillators. J. Sound Vib. 330, 1–8 (2011)

    Google Scholar 

  27. 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)

    MathSciNet  Google Scholar 

  28. Bellizzi, S., Chung, K.W., Sampaio, R.: Response regimes of a linear oscillator with a nonlinear energy sink involving an active damper with delay. Nonlinear Dyn. 97, 1667–1684 (2019)

    Google Scholar 

  29. Yang, T.Z., Hou, S., Qin, Z.H., Ding, Q., Chen, L.Q.: A dynamic reconfigurable nonlinear energy sink. J. Sound Vib. 494, 115629 (2021)

    Google Scholar 

  30. Zhang, M.J., Wu, T., Øiseth, O.: Vortex-induced vibration control of a flexible circular cylinder using a nonlinear energy sink. J. Wind Eng. Ind. Aerod. 229, 105163 (2022)

    Google Scholar 

  31. Jin, Y., Yang, T.Z.: Enhanced vibration suppression and energy harvesting in fluid-conveying pipes. Appl. Math. Mech.-Engl. 44, 1487–1496 (2023)

    MathSciNet  Google Scholar 

  32. Fasihi, A., Shahgholi, M., Ghahremani, S.: The effects of nonlinear energy sink and piezoelectric energy harvester on aeroelastic instability of an airfoil. J. Vib. Control 28, 1418–1432 (2021)

    MathSciNet  Google Scholar 

  33. Fang, Z.W., Zhang, Y.W., Li, X., Ding, H., Chen, L.Q.: Integration of a nonlinear energy sink and a giant magnetostrictive energy harvester. J. Sound Vib. 391, 35–49 (2017)

    Google Scholar 

  34. Li, X., Zhang, Y.W., Ding, H., Chen, L.Q.: Dynamics and evaluation of a nonlinear energy sink integrated by a piezoelectric energy harvester under a harmonic excitation. J. Vib. Control 25, 851–867 (2019)

    MathSciNet  Google Scholar 

  35. Ebrahimzade, N., Dardel, M., Shafaghat, R.: Investigating the aeroelastic behaviors of rotor blades with nonlinear energy sinks. AIAA J. 56, 2856–5869 (2018)

    Google Scholar 

  36. Tehrani, G.G., Dardel, M.: Vibration mitigation of a flexible bladed rotor dynamic system with passive dynamic absorbers. Commun. Nonlinear Sci. Numer. Simul. 69, 1–30 (2018)

    MathSciNet  Google Scholar 

  37. Sun, Y.H., Zhang, Y.W., Ding, H., Chen, L.Q.: Nonlinear energy sink for a flywheel system vibration reduction. J. Sound Vib. 429, 305–324 (2018)

    Google Scholar 

  38. Vakakis, A.F., Georgiades, F.: Dynamics of a linear beam with an attached local nonlinear energy sink. Commun. Nonlinear Sci. Numer. Simul. 12, 643–651 (2007)

    Google Scholar 

  39. Zhang, W.Y., Niu, M.Q., Chen, L.Q.: Vibration reduction of a timoshenko beam with multiple parallel nonlinear energy sinks. Appl. Sci. 12, 9008 (2022)

    Google Scholar 

  40. Duan, Z.L., Cui, J.G., Chen, L.Q., Yang, T.Z.: Nonlinear mechanical Roton. ASME. J. Appl. Mech. 90(3), 031010 (2023). https://doi.org/10.1115/1.4056583

  41. Yang, T.Z., Duan, Z.L., Meng, X.B., Liu, S.L., Chen, L.Q.: Roton enabled mechanical diode at extremely low frequency. ASME. J. Appl. Mech. 91(1), 011010 (2023). https://doi.org/10.1115/1.4063143

  42. Inoue, N., Ikago, K.: Displacement control design of buildings: design method of long-period seismic isolation buildings against earthquake. Maruzen, Tokyo (2012)

    Google Scholar 

  43. Smith, M.C.: Synthesis of mechanical networks: the inerter. IEEE Trans. Automat. Contr. 47, 1648–1662 (2002)

    MathSciNet  Google Scholar 

  44. Liang, Q.G., Li, L.Y.: Optimal design for a novel inerter-based clutching tuned mass damper system. J. Vib. Control 26, 2050–2059 (2020)

    MathSciNet  Google Scholar 

  45. Zhang, Z., Ding, H., Zhang, Y.W., Chen, L.Q.: Vibration suppression of an elastic beam with boundary inerter-enhanced nonlinear energy sinks. Acta Mech. Sin. 37, 387–401 (2021)

    MathSciNet  Google Scholar 

  46. Wang, J.J., Wang, B., Zhang, C., Liu, Z.B.: Effectiveness and robustness of an asymmetric nonlinear energy sink-inerter for dynamic response mitigation. Earthq. Eng. Struct. Dyn. 50, 1628–1650 (2021)

    Google Scholar 

  47. Aly, A.M.: Proposed robust tuned mass damper for response mitigation in buildings exposed to multidirectional wind. Struct. Des. Tall Spec. Build. 23, 664–691 (2014)

    Google Scholar 

  48. Shuai, C.G., Li, B.Y., Ma, J.G.: A novel multi-directional vibration isolation system with high-static-low-dynamic stiffness. Acta Mech. 233, 5199–5214 (2022)

    Google Scholar 

  49. Wang, Y., Wang, P.L., Meng, H.D., Chen, L.Q.: Nonlinear vibration and dynamic performance analysis of the inerter-based multi-directional vibration isolator. Arch. Appl. Mech. 92, 3597–3629 (2022)

    Google Scholar 

  50. Saeed, A.S., Al-Shudeifat, M.A., Cantwell, W.J., Vakakis, A.F.: Two-dimensional nonlinear energy sink for effective passive seismic mitigation. Commun. Nonlinear Sci. Numer. Simul. 99, 105787 (2021)

    MathSciNet  Google Scholar 

  51. Li, W.K., Wierschem, N.E., Li, X.H., Yang, T.J., Brennan, M.J.: Numerical study of a single-sided vibro-impact track nonlinear energy sink considering horizontal and vertical dynamics. J. Vib. Acoust. 141, 061013 (2019)

    Google Scholar 

  52. Wang, J.J., Wierschem, N., Spencer, B.F., Jr., 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 (2016)

    Google Scholar 

  53. Wang, Y., Xu, B.B., Dai, J.G., Chen, L.Q.: Enhanced dynamic performance of a half-vehicle system using inerter-based nonlinear energy sink. 1–24 (2023)

  54. Lee, Y.S., Kerschen, G., Vakakis, A.F., Panagopoulos, P., Bergman, L.A., Mc Farland, D.M.: Complicated dynamics of a linear oscillator with an essentially nonlinear local attachment. Physica D 204, 41–69 (2005)

    MathSciNet  Google Scholar 

  55. Gendelman, O.V., Manevitch, L.I., Vakakis, A.F., Closkey, R.M.: Energy pumping in nonlinear mechanical oscillators: part I—dynamics of the underlying hamiltonian systems. J. Appl. Mech. 68, 34–41 (2001)

    MathSciNet  Google Scholar 

  56. Zhao, Y.P.: An improved energy envelope stochastic averaging method and its application to a nonlinear oscillator. J. Vib. Control 23, 119–130 (2015)

    MathSciNet  Google Scholar 

Download references

Funding

This work was supported by National Science Foundation of China grants (Grant Nos. 12232014 and 12072221), Central Universities (Grant No. 2013017), and the Ten Thousand Talents Program.

Author information

Authors and Affiliations

  1. School of Mechanical Engineering and Automation, Northeastern University, Shenyang, 110819, China

    Tianzhi Yang & Wenhu Dang

  2. School of Science, Harbin Institute of Technology, Shenzhen, 518055, China

    Liqun Chen

Authors
  1. Tianzhi Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Wenhu Dang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Liqun Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tianzhi Yang.

Ethics declarations

Conflict of interest

We declare no competing interests.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix

Appendix

$$\begin{aligned} & M\left( {\ddot{x}_{pr} - A_{u} \left( {2\pi f_{u} } \right)^{2} \cos (2\pi f_{u} t)} \right) \\ & \quad - \frac{{k_{n} (x_{nr} - x_{pr} )\left( {\sqrt {(l + y_{nr} - y_{pr} )^{2} + (x_{nr} - x_{pr} )^{2} } - l} \right)}}{{\sqrt {(l + y_{nr} - y_{pr} )^{2} + (x_{nr} - x_{pr} )^{2} } }} \\ & \quad - \frac{{k_{n} (l + x_{nr} - x_{pr} )\left( {\sqrt {(l + x_{nr} - x_{pr} )^{2} + (y_{nr} - y_{pr} )^{2} } - l} \right)}}{{\sqrt {(l + x_{nr} - x_{pr} )^{2} + (y_{nr} - y_{pr} )^{2} } }} \\ & \quad + k_{p} x_{pr} \left( {1 - \frac{l}{{\sqrt {(l + y_{pr} )^{2} + x_{pr}^{2} } }}} \right) \\ & \quad + \frac{{k_{p} (l + x_{pr} )\left( {\sqrt {(l + x_{pr} )^{2} + y_{pr}^{2} } - l} \right)}}{{\sqrt {(l + x_{pr} )^{2} + y_{pr}^{2} } }} \\ & \quad - \frac{{l + x_{nr} - x_{pr} }}{{4\left( {(l + x_{nr} - x_{pr} )^{2} + (y_{nr} - y_{pr} )^{2} } \right)^{2} }} \\ & \quad \left( \begin{array}{l} 2c_{n} \left( {(l + x_{nr} - x_{pr} )^{2} + (y_{nr} - y_{pr} )^{2} } \right)\left( {2(l + x_{nr} - x_{pr} )\left( {\dot{x}_{nr} - \dot{x}_{pr} } \right) + 2(y_{nr} - y_{pr} )\left( {\dot{y}_{nr} - \dot{y}_{pr} } \right)} \right) \hfill \\ \quad - b_{n} \left( \begin{array}{l} \left( {2(l + x_{nr} - x_{pr} )\left( {\dot{x}_{nr} - \dot{x}_{pr} } \right) + 2(y_{nr} - y_{pr} )\left( {\dot{y}_{nr} - \dot{y}_{pr} } \right)} \right)^{2} - 4\left( {(l + x_{nr} - x_{pr} )^{2} + (y_{nr} - y_{pr} )^{2} } \right) \hfill \\ \left( {(l + x_{nr} - x_{pr} )\left( {\ddot{x}_{nr} - \ddot{x}_{pr} } \right) + \left( {\dot{x}_{nr} - \dot{x}_{pr} } \right)^{2} + (y_{nr} - y_{pr} )\left( {\ddot{y}_{nr} - \ddot{y}_{pr} } \right) + \left( {\dot{y}_{nr} - \dot{y}_{pr} } \right)^{2} } \right) \hfill \\ \end{array} \right) \hfill \\ \end{array} \right) \\ & \quad + \frac{{l + x_{pr} }}{{\sqrt {(l + x_{pr} )^{2} + y_{pr}^{2} } }} \\ & \quad \left( \begin{array}{l} \frac{{b_{n} }}{{4\left( {(l + x_{nr} - x_{pr} )^{2} + (y_{nr} - y_{pr} )^{2} } \right)^{3/2} }}\left( \begin{array}{l} 4\left( \begin{array}{l} (l + x_{nr} - x_{pr} )^{2} \hfill \\\quad + (y_{nr} - y_{pr} )^{2} \hfill \\ \end{array} \right)\left( \begin{array}{l} (l + x_{nr} - x_{pr} )\left( {\ddot{x}_{nr} - \ddot{x}_{pr} } \right) \hfill \\ \quad+ \left( {\dot{x}_{nr} - \dot{x}_{pr} } \right)^{2} + (y_{nr} - y_{pr} ) \hfill \\ \left( {\ddot{y}_{nr} - \ddot{y}_{pr} } \right) + \left( {\dot{y}_{nr} - \dot{y}_{pr} } \right)^{2} \hfill \\ \end{array} \right) \hfill \\ - \left( \begin{array}{l} 2(l + x_{nr} - x_{pr} )\left( {\dot{x}_{nr} - \dot{x}_{pr} } \right) \hfill \\\quad + 2(y_{nr} - y_{pr} )\left( {\dot{y}_{nr} - \dot{y}_{pr} } \right) \hfill \\ \end{array} \right)^{2} \hfill \\ \end{array} \right) \hfill \\\quad + \frac{{c_{n} \left( {2(l + x_{nr} - x_{pr} )\left( {\dot{x}_{nr} - \dot{x}_{pr} } \right) + 2(y_{nr} - y_{pr} )\left( {\dot{y}_{nr} - \dot{y}_{pr} } \right)} \right)}}{{2\sqrt {(l + x_{nr} - x_{pr} )^{2} + (y_{nr} - y_{pr} )^{2} } }} + \frac{{c_{p} \left( {(l + x_{pr} )\dot{x}_{pr} + y_{pr} \dot{y}_{pr} } \right)}}{{\sqrt {(l + x_{pr} )^{2} + y_{pr}^{2} } }} \hfill \\ \end{array} \right) \\ & \quad - \frac{{x_{nr} - x_{pr} }}{{4\left( {(l + y_{nr} - y_{pr} )^{2} + (x_{nr} - x_{pr} )^{2} } \right)^{2} }} \\ & \quad \times \left( \begin{array}{l} 2c_{n} \left( \begin{array}{l} (l + y_{nr} - y_{pr} )^{2} \hfill \\ \quad + (x_{nr} - x_{pr} )^{2} \hfill \\ \end{array} \right)\left( \begin{array}{l} 2(l + y_{nr} - y_{pr} )\left( {\dot{y}_{nr} - \dot{y}_{pr} } \right) \hfill \\ \quad+ 2(x_{nr} - x_{pr} )\left( {\dot{x}_{nr} - \dot{x}_{pr} } \right) \hfill \\ \end{array} \right) \hfill \\ \quad- b_{n} \left( \begin{array}{l} \left( \begin{array}{l} 2(l + y_{nr} - y_{pr} )\left( {\dot{y}_{nr} - \dot{y}_{pr} } \right) \hfill \\\quad + 2(x_{nr} - x_{pr} )\left( {\dot{x}_{nr} - \dot{x}_{pr} } \right) \hfill \\ \end{array} \right)^{2} \hfill \\ \quad- 4\left( {(l + y_{nr} - y_{pr} )^{2} + (x_{nr} - x_{pr} )^{2} } \right) \hfill \\ \left( \begin{array}{l} (l + y_{nr} - y_{pr} )\left( {\ddot{y}_{nr} - \ddot{y}_{pr} } \right) \hfill \\ \quad + (x_{nr} - x_{pr} )\left( {\ddot{x}_{nr} - \ddot{x}_{pr} } \right) + \left( {\dot{x}_{nr} - \dot{x}_{pr} } \right)^{2} + \left( {\dot{y}_{nr} - \dot{y}_{pr} } \right)^{2} \hfill \\ \end{array} \right) \hfill \\ \end{array} \right) \hfill \\ \end{array} \right) \\ & \quad + \frac{{x_{pr} }}{{\sqrt {(l + y_{pr} )^{2} + x_{pr}^{2} } }} \\ \end{aligned}$$
$$\quad \left( \begin{array}{l} \frac{{b_{n} }}{{4\left( {(l + y_{nr} - y_{pr} )^{2} + (x_{nr} - x_{pr} )^{2} } \right)^{3/2} }}\left( \begin{array}{l} 4\left( \begin{array}{l} (l + y_{nr} - y_{pr} )^{2} \hfill \\ \quad+ (x_{nr} - x_{pr} )^{2} \hfill \\ \end{array} \right) \hfill \\ \left( \begin{array}{l} (l + y_{nr} - y_{pr} )\left( {\ddot{y}_{nr} - \ddot{y}_{pr} } \right) \hfill \\ \quad+ (x_{nr} - x_{pr} )\left( {\ddot{x}_{nr} - \ddot{x}_{pr} } \right) \hfill \\ \quad+ \left( {\dot{x}_{nr} - \dot{x}_{pr} } \right)^{2} + \left( {\dot{y}_{nr} - \dot{y}_{pr} } \right)^{2} \hfill \\ \end{array} \right) \hfill \\ \quad- \left( \begin{array}{l} 2(l + y_{nr} - y_{pr} )\left( {\dot{y}_{nr} - \dot{y}_{pr} } \right) \hfill \\ \quad + 2(x_{nr} - x_{pr} )\left( {\dot{x}_{nr} - \dot{x}_{pr} } \right) \hfill \\ \end{array} \right)^{2} \hfill \\ \end{array} \right) \hfill \\ \quad + \frac{{c_{n} \left( {2(l + y_{nr} - y_{pr} )\left( {\dot{y}_{nr} - \dot{y}_{pr} } \right) + 2(x_{nr} - x_{pr} )\left( {\dot{x}_{nr} - \dot{x}_{pr} } \right)} \right)}}{{2\sqrt {(l + y_{nr} - y_{pr} )^{2} + (x_{nr} - x_{pr} )^{2} } }} + \frac{{c_{p} \left( {(l + y_{pr} )\dot{y}_{pr} + x_{pr} \dot{x}_{pr} } \right)}}{{\sqrt {(l + y_{pr} )^{2} + x_{pr}^{2} } }} \hfill \\ \end{array} \right)$$
(A.1)
$$\begin{aligned} & - \frac{{k_{n} (y_{nr} - y_{pr} )\left( {\sqrt {(l + x_{nr} - x_{pr} )^{2} + (y_{nr} - y_{pr} )^{2} } - l} \right)}}{{\sqrt {(l + x_{nr} - x_{pr} )^{2} + (y_{nr} - y_{pr} )^{2} } }} \\ & - \frac{{k_{n} (l + y_{nr} - y_{pr} )\left( {\sqrt {(l + y_{nr} - y_{pr} )^{2} + (x_{nr} - x_{pr} )^{2} } - l} \right)}}{{\sqrt {(l + y_{nr} - y_{pr} )^{2} + (x_{nr} - x_{pr} )^{2} } }} \\ & + k_{p} y_{pr} \left( {1 - \frac{l}{{\sqrt {(l + x_{pr} )^{2} + y_{pr}^{2} } }}} \right) \\ & + \frac{{k_{p} (l + y_{pr} )\left( {\sqrt {(l + y_{pr} )^{2} + x_{pr}^{2} } - l} \right)}}{{\sqrt {(l + y_{pr} )^{2} + x_{pr}^{2} } }} \\ & - \frac{{y_{nr} - y_{pr} }}{{\sqrt {(l + x_{nr} - x_{pr} )^{2} + (y_{nr} - y_{pr} )^{2} } }} \\ & \quad \left( \begin{array}{l} b_{n} \left( \begin{array}{l} \frac{{2(l + x_{nr} - x_{pr} )\left( {\ddot{x}_{nr} - \ddot{x}_{pr} } \right) + 2\left( {\dot{x}_{nr} - \dot{x}_{pr} } \right)^{2} + 2(y_{nr} - y_{pr} )\left( {\ddot{y}_{nr} - \ddot{y}_{pr} } \right) + 2\left( {\dot{y}_{nr} - \dot{y}_{pr} } \right)^{2} }}{{2\sqrt {(l + x_{nr} - x_{pr} )^{2} + (y_{nr} - y_{pr} )^{2} } }} \hfill \\ - \frac{{\left( {2(l + x_{nr} - x_{pr} )\left( {\dot{x}_{nr} - \dot{x}_{pr} } \right) + 2(y_{nr} - y_{pr} )\left( {\dot{y}_{nr} - \dot{y}_{pr} } \right)} \right)^{2} }}{{4\left( {(l + x_{nr} - x_{pr} )^{2} + (y_{nr} - y_{pr} )^{2} } \right)^{3/2} }} \hfill \\ \end{array} \right) \hfill \\ + \frac{{c_{n} \left( {2(l + x_{nr} - x_{pr} )\left( {\dot{x}_{nr} - \dot{x}_{pr} } \right) + 2(y_{nr} - y_{pr} )\left( {\dot{y}_{nr} - \dot{y}_{pr} } \right)} \right)}}{{2\sqrt {(l + x_{nr} - x_{pr} )^{2} + (y_{nr} - y_{pr} )^{2} } }} \hfill \\ \end{array} \right) \\ & + \frac{{y_{pr} }}{{\sqrt {(l + x_{pr} )^{2} + y_{pr}^{2} } }} \\ & \quad \left( \begin{array}{l} b_{n} \left( \begin{array}{l} \frac{{2(l + x_{nr} - x_{pr} )\left( {\ddot{x}_{nr} - \ddot{x}_{pr} } \right) + 2\left( {\dot{x}_{nr} - \dot{x}_{pr} } \right)^{2} + 2(y_{nr} - y_{pr} )\left( {\ddot{y}_{nr} - \ddot{y}_{pr} } \right) + 2\left( {\dot{y}_{nr} - \dot{y}_{pr} } \right)^{2} }}{{2\sqrt {(l + x_{nr} - x_{pr} )^{2} + (y_{nr} - y_{pr} )^{2} } }} \hfill \\ - \frac{{\left( {2(l + x_{nr} - x_{pr} )\left( {\dot{x}_{nr} - \dot{x}_{pr} } \right) + 2(y_{nr} - y_{pr} )\left( {\dot{y}_{nr} - \dot{y}_{pr} } \right)} \right)^{2} }}{{4\left( {(l + x_{nr} - x_{pr} )^{2} + (y_{nr} - y_{pr} )^{2} } \right)^{3/2} }} \hfill \\ \end{array} \right) \hfill \\ \quad + \frac{{c_{n} \left( {2(l + x_{nr} - x_{pr} )\left( {\dot{x}_{nr} - \dot{x}_{pr} } \right) + 2(y_{nr} - y_{pr} )\left( {\dot{y}_{nr} - \dot{y}_{pr} } \right)} \right)}}{{2\sqrt {(l + x_{nr} - x_{pr} )^{2} + (y_{nr} - y_{pr} )^{2} } }} + \frac{{c_{p} \left( {2(l + x_{pr} )\dot{x}_{pr} + 2y_{pr} \dot{y}_{pr} } \right)}}{{2\sqrt {(l + x_{pr} )^{2} + y_{pr}^{2} } }} \hfill \\ \end{array} \right) \\ & - \frac{{l + y_{nr} - y_{pr} }}{{\sqrt {(l + y_{nr} - y_{pr} )^{2} + (x_{nr} - x_{pr} )^{2} } }} \\ & \quad \left( \begin{array}{l} b_{n} \left( \begin{array}{l} \frac{{2(l + y_{nr} - y_{pr} )\left( {\ddot{y}_{nr} - \ddot{y}_{pr} } \right) + 2(x_{nr} - x_{pr} )\left( {\ddot{x}_{nr} - \ddot{x}_{pr} } \right) + 2\left( {\dot{x}_{nr} - \dot{x}_{pr} } \right)^{2} + 2\left( {\dot{y}_{nr} - \dot{y}_{pr} } \right)^{2} }}{{2\sqrt {(l + y_{nr} - y_{pr} )^{2} + (x_{nr} - x_{pr} )^{2} } }} \hfill \\ - \frac{{\left( {2(l + y_{nr} - y_{pr} )\left( {\dot{y}_{nr} - \dot{y}_{pr} } \right) + 2(x_{nr} - x_{pr} )\left( {\dot{x}_{nr} - \dot{x}_{pr} } \right)} \right)^{2} }}{{4\left( {(l + y_{nr} - y_{pr} )^{2} + (x_{nr} - x_{pr} )^{2} } \right)^{3/2} }} \hfill \\ \end{array} \right) \hfill \\ \quad + \frac{{c_{n} \left( {2(l + y_{nr} - y_{pr} )\left( {\dot{y}_{nr} - \dot{y}_{pr} } \right) + 2(x_{nr} - x_{pr} )\left( {\dot{x}_{nr} - \dot{x}_{pr} } \right)} \right)}}{{2\sqrt {(l + y_{nr} - y_{pr} )^{2} + (x_{nr} - x_{pr} )^{2} } }} \hfill \\ \end{array} \right) \\ \end{aligned}$$
$$\begin{aligned} & + M\left( {\ddot{y}_{pr} - A_{v} \left( {2\pi f_{v} } \right)^{2} \cos (2\pi f_{v} t)} \right) + \frac{{l + y_{pr} }}{{\sqrt {(l + y_{pr} )^{2} + x_{pr}^{2} } }} \\ & \quad \left( \begin{array}{l} b_{n} \left( \begin{array}{l} \frac{{2(l + y_{nr} - y_{pr} )\left( {\ddot{y}_{nr} - \ddot{y}_{pr} } \right) + 2(x_{nr} - x_{pr} )\left( {\ddot{x}_{nr} - \ddot{x}_{pr} } \right) + 2\left( {\dot{x}_{nr} - \dot{x}_{pr} } \right)^{2} + 2\left( {\dot{y}_{nr} - \dot{y}_{pr} } \right)^{2} }}{{2\sqrt {(l + y_{nr} - y_{pr} )^{2} + (x_{nr} - x_{pr} )^{2} } }} \hfill \\ - \frac{{\left( {2(l + y_{nr} - y_{pr} )\left( {\dot{y}_{nr} - \dot{y}_{pr} } \right) + 2(x_{nr} - x_{pr} )\left( {\dot{x}_{nr} - \dot{x}_{pr} } \right)} \right)^{2} }}{{4\left( {(l + y_{nr} - y_{pr} )^{2} + (x_{nr} - x_{pr} )^{2} } \right)^{3/2} }} \hfill \\ \end{array} \right) \hfill \\ \quad + \frac{{c_{n} \left( {2(l + y_{nr} - y_{pr} )\left( {\dot{y}_{nr} - \dot{y}_{pr} } \right) + 2(x_{nr} - x_{pr} )\left( {\dot{x}_{nr} - \dot{x}_{pr} } \right)} \right)}}{{2\sqrt {(l + y_{nr} - y_{pr} )^{2} + (x_{nr} - x_{pr} )^{2} } }} + \frac{{c_{p} \left( {2(l + y_{pr} )\dot{y}_{pr} + 2x_{pr} \dot{x}_{pr} } \right)}}{{2\sqrt {(l + y_{pr} )^{2} + x_{pr}^{2} } }} \hfill \\ \end{array} \right) \\ \end{aligned}$$
(A.2)
$$\begin{aligned} & m\left( {\ddot{x}_{nr} - A_{u} \left( {2\pi f_{u} } \right)^{2} \cos \left( {2\pi f_{u} t} \right)} \right) + \frac{1}{4} \\ & \quad \left( \begin{array}{l} \frac{{l + x_{nr} - x_{pr} }}{{\left( {(l + x_{nr} - x_{pr} )^{2} + (y_{nr} - y_{pr} )^{2} } \right)^{2} }} \hfill \\ \left( \begin{array}{l} 2c_{n} \left( {(l + x_{nr} - x_{pr} )^{2} + (y_{nr} - y_{pr} )^{2} } \right)\left( {2(l + x_{nr} - x_{pr} )\left( {\dot{x}_{nr} - \dot{x}_{pr} } \right) + 2(y_{nr} - y_{pr} )\left( {\dot{y}_{nr} - \dot{y}_{pr} } \right)} \right) \hfill \\ - b_{n} \left( \begin{array}{l} \left( {2(l + x_{nr} - x_{pr} )\left( {\dot{x}_{nr} - \dot{x}_{pr} } \right) + 2(y_{nr} - y_{pr} )\left( {\dot{y}_{nr} - \dot{y}_{pr} } \right)} \right)^{2} \hfill \\ - 4\left( {(l + x_{nr} - x_{pr} )^{2} + (y_{nr} - y_{pr} )^{2} } \right) \hfill \\ \left( {(l + x_{nr} - x_{pr} )\left( {\ddot{x}_{nr} - \ddot{x}_{pr} } \right) + \left( {\dot{x}_{nr} - \dot{x}_{pr} } \right)^{2} + (y_{nr} - y_{pr} )\left( {\ddot{y}_{nr} - \ddot{y}_{pr} } \right) + \left( {\dot{y}_{nr} - \dot{y}_{pr} } \right)^{2} } \right) \hfill \\ \end{array} \right) \hfill \\ \end{array} \right) \hfill \\ \quad + \frac{{x_{nr} - x_{pr} }}{{\left( {(l + y_{nr} - y_{pr} )^{2} + (x_{nr} - x_{pr} )^{2} } \right)^{2} }} \hfill \\ \left( \begin{array}{l} 2c_{n} \left( {(l + y_{nr} - y_{pr} )^{2} + (x_{nr} - x_{pr} )^{2} } \right)\left( \begin{array}{l} 2(l + y_{nr} - y_{pr} )\left( {\dot{y}_{nr} - \dot{y}_{pr} } \right) \hfill \\ + 2(x_{nr} - x_{pr} )\left( {\dot{x}_{nr} - \dot{x}_{pr} } \right) \hfill \\ \end{array} \right) \hfill \\ - b_{n} \left( \begin{array}{l} \left( {2(l + y_{nr} - y_{pr} )\left( {\dot{y}_{nr} - \dot{y}_{pr} } \right) + 2(x_{nr} - x_{pr} )\left( {\dot{x}_{nr} - \dot{x}_{pr} } \right)} \right)^{2} \hfill \\ - 4\left( {(l + y_{nr} - y_{pr} )^{2} + (x_{nr} - x_{pr} )^{2} } \right) \hfill \\ \left( {(l + y_{nr} - y_{pr} )\left( {\ddot{y}_{nr} - \ddot{y}_{pr} } \right) + (x_{nr} - x_{pr} )\left( {\ddot{x}_{nr} - \ddot{x}_{pr} } \right) + \left( {\dot{x}_{nr} - \dot{x}_{pr} } \right)^{2} + \left( {\dot{y}_{nr} - \dot{y}_{pr} } \right)^{2} } \right) \hfill \\ \end{array} \right) \hfill \\ \end{array} \right) \hfill \\ \end{array} \right) \\ & \quad + \frac{{k_{n} (x_{nr} - x_{pr} )\left( {\sqrt {(l + y_{nr} - y_{pr} )^{2} + (x_{nr} - x_{pr} )^{2} } - l} \right)}}{{\sqrt {(l + y_{nr} - y_{pr} )^{2} + (x_{nr} - x_{pr} )^{2} } }} \\ & \quad + \frac{{k_{n} (l + x_{nr} - x_{pr} )\left( {\sqrt {(l + x_{nr} - x_{pr} )^{2} + (y_{nr} - y_{pr} )^{2} } - l} \right)}}{{\sqrt {(l + x_{nr} - x_{pr} )^{2} + (y_{nr} - y_{pr} )^{2} } }} \\ \end{aligned}$$
(A.3)
$$\begin{aligned} & m\left( {\ddot{y}_{nr} - A_{v} \left( {2\pi f_{v} } \right)^{2} \cos \left( {2\pi f_{v} t} \right)} \right) + \frac{1}{4} \\ & \quad \left( \begin{array}{l} \frac{{y_{nr} - y_{pr} }}{{\left( {(l + x_{nr} - x_{pr} )^{2} + (y_{nr} - y_{pr} )^{2} } \right)^{2} }} \hfill \\ \left( \begin{array}{l} 2c_{n} \left( {(l + x_{nr} - x_{pr} )^{2} + (y_{nr} - y_{pr} )^{2} } \right)\left( {2(l + x_{nr} - x_{pr} )\left( {\dot{x}_{nr} - \dot{x}_{pr} } \right) + 2(y_{nr} - y_{pr} )\left( {\dot{y}_{nr} - \dot{y}_{pr} } \right)} \right) \hfill \\ - b_{n} \left( \begin{array}{l} \left( {2(l + x_{nr} - x_{pr} )\left( {\dot{x}_{nr} - \dot{x}_{pr} } \right) + 2(y_{nr} - y_{pr} )\left( {\dot{y}_{nr} - \dot{y}_{pr} } \right)} \right)^{2} \hfill \\ - 4\left( {(l + x_{nr} - x_{pr} )^{2} + (y_{nr} - y_{pr} )^{2} } \right)\left( \begin{array}{l} (l + x_{nr} - x_{pr} )\left( {\ddot{x}_{nr} - \ddot{x}_{pr} } \right) \hfill \\ + \left( {\dot{x}_{nr} - \dot{x}_{pr} } \right)^{2} + (y_{nr} - y_{pr} ) \hfill \\ \left( {\ddot{y}_{nr} - \ddot{y}_{pr} } \right) + \left( {\dot{y}_{nr} - \dot{y}_{pr} } \right)^{2} \hfill \\ \end{array} \right) \hfill \\ \end{array} \right) \hfill \\ \end{array} \right) \hfill \\ \quad + \frac{{l + y_{nr} - y_{pr} }}{{\left( {(l + y_{nr} - y_{pr} )^{2} + (x_{nr} - x_{pr} )^{2} } \right)^{2} }} \hfill \\ \quad \left( \begin{array}{l} 2c_{n} \left( {(l + y_{nr} - y_{pr} )^{2} + (x_{nr} - x_{pr} )^{2} } \right)\left( {2(l + y_{nr} - y_{pr} )\left( {\dot{y}_{nr} - \dot{y}_{pr} } \right) + 2(x_{nr} - x_{pr} )\left( {\dot{x}_{nr} - \dot{x}_{pr} } \right)} \right) \hfill \\ - b_{n} \left( \begin{array}{l} \left( {2(l + y_{nr} - y_{pr} )\left( {\dot{y}_{nr} - \dot{y}_{pr} } \right) + 2(x_{nr} - x_{pr} )\left( {\dot{x}_{nr} - \dot{x}_{pr} } \right)} \right)^{2} \hfill \\ - 4\left( {(l + y_{nr} - y_{pr} )^{2} + (x_{nr} - x_{pr} )^{2} } \right)\left( \begin{array}{l} (l + y_{nr} - y_{pr} )\left( {\ddot{y}_{nr} - \ddot{y}_{pr} } \right) \hfill \\ + (x_{nr} - x_{pr} )\left( {\ddot{x}_{nr} - \ddot{x}_{pr} } \right) \hfill \\ + \left( {\dot{x}_{nr} - \dot{x}_{pr} } \right)^{2} + \left( {\dot{y}_{nr} - \dot{y}_{pr} } \right)^{2} \hfill \\ \end{array} \right) \hfill \\ \end{array} \right) \hfill \\ \end{array} \right) \hfill \\ \end{array} \right) \\ & \quad + \frac{{k_{n} (y_{nr} - y_{pr} )\left( {\sqrt {(l + x_{nr} - x_{pr} )^{2} + (y_{nr} - y_{pr} )^{2} } - l} \right)}}{{\sqrt {(l + x_{nr} - x_{pr} )^{2} + (y_{nr} - y_{pr} )^{2} } }} \\ & \quad + \frac{{k_{n} (l + y_{nr} - y_{pr} )\left( {\sqrt {(l + y_{nr} - y_{pr} )^{2} + (x_{nr} - x_{pr} )^{2} } - l} \right)}}{{\sqrt {(l + y_{nr} - y_{pr} )^{2} + (x_{nr} - x_{pr} )^{2} } }} \\ \end{aligned}$$
(A.4)

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

Yang, T., Dang, W. & Chen, L. Two-dimensional inerter-enhanced nonlinear energy sink. Nonlinear Dyn 112, 379–401 (2024). https://doi.org/10.1007/s11071-023-09056-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-023-09056-8

Keywords

Search

Navigation