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Comparison of a modified vibro-impact nonlinear energy sink with other kinds of NESs

  • Modelling and analysis of mechanical systems dynamics
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Abstract

Vibration mitigation is essential to many dynamical and engineering structures that are subjected to destructive vibration amplitudes induced by impulsive loading, seismic excitation, blasts, flutter, collisions, fluid–structure interaction and so on. Unprotected structures by vibration absorbers could be exposed to failure which lead to enormous losses in human lives, major equipment and economy. Employing the nonlinear targeted energy transfer (TET) concept in nonlinear vibration absorbers which are later called as nonlinear energy sinks has ignited a very rapid research interest since 2001. Up-to-date, considerable growth in the NESs field has taken place. Accordingly, various types of NESs have been introduced for vibration mitigation in variety of dynamical and structural engineering systems. The types of introduced NESs included, but not limited to, stiffness-based, rotating and the vibro-impact NESs. Among these common types of NESs, the most effective and efficient one is the single-sided vibro-impact (SSVI) nonlinear energy sink (NES). However, most of investigations has implemented a coefficient of restitution of 0.7 which closely corresponds to a steel-to-steel impact. Therefore, this paper is aimed to further improve the SSVI NES by including the coefficient of restitution in the performance optimization. Accordingly, significant improvement in the SSVI NES performance is obtained when the coefficient of restitution is found to be near 0.45. In addition, performance comparison between the enhanced SSVIe NES with several existing types of NESs is performed here where a nine-story large-scale structure is employed for this numerical comparison. Accordingly, the performance of the enhanced SSVIe NES of nearly 0.45 coefficient of restitution is found to be more robust to the initial impulsive energy levels and to its physical parameters variation than other kinds of existing NESs.

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References

  1. Gendelman OV (2001) Transition of energy to a nonlinear localized mode in a highly asymmetric system of two oscillators. Nonlinear Dyn 25(1):237–253. https://doi.org/10.1023/a:1012967003477

    Article  MathSciNet  MATH  Google Scholar 

  2. Vakakis AF (2001) Inducing passive nonlinear energy sinks in vibrating systems. J Vib Acoust 123(3):324–332. https://doi.org/10.1115/1.1368883

    Article  Google Scholar 

  3. Vakakis AF, Gendelman O (2000) Energy pumping in nonlinear mechanical oscillators: part II—resonance capture. J Appl Mech 68(1):42–48. https://doi.org/10.1115/1.1345525

    Article  MATH  Google Scholar 

  4. Gendelman O, Manevitch LI, Vakakis AF, M’Closkey R (2000) Energy pumping in nonlinear mechanical oscillators: part I—dynamics of the underlying hamiltonian systems. J Appl Mech 68(1):34–41. https://doi.org/10.1115/1.1345524

    Article  MATH  Google Scholar 

  5. Panagopoulos PN, Vakakis AF, Tsakirtzis S (2004) Transient resonant interactions of finite linear chains with essentially nonlinear end attachments leading to passive energy pumping. Int J Solids Struct 41(22):6505–6528. https://doi.org/10.1016/j.ijsolstr.2004.05.005

    Article  MATH  Google Scholar 

  6. McFarland DM, Bergman LA, Vakakis AF (2005) Experimental study of non-linear energy pumping occurring at a single fast frequency. Int J Non-Linear Mech 40(6):891–899. https://doi.org/10.1016/j.ijnonlinmec.2004.11.001

    Article  MATH  Google Scholar 

  7. Gourdon E, Lamarque CH (2005) Energy pumping with various nonlinear structures: numerical evidences. Nonlinear Dyn 40(3):281–307. https://doi.org/10.1007/s11071-005-6610-6

    Article  MathSciNet  MATH  Google Scholar 

  8. Vakakis AF, Gendelman OV, Bergman LA, McFarland DM, Kerschen G, Lee YS (2008) Nonlinear targeted energy transfer in mechanical and structural systems. Springer, Berlin

    MATH  Google Scholar 

  9. Lee YS, Kerschen G, Vakakis AF, Panagopoulos P, Bergman L, McFarland DM (2005) Complicated dynamics of a linear oscillator with a light, essentially nonlinear attachment. Physica D 204(1):41–69. https://doi.org/10.1016/j.physd.2005.03.014

    Article  MathSciNet  MATH  Google Scholar 

  10. Gourdon E, Alexander NA, Taylor CA, Lamarque CH, Pernot S (2007) Nonlinear energy pumping under transient forcing with strongly nonlinear coupling: theoretical and experimental results. J Sound Vib 300(3):522–551. https://doi.org/10.1016/j.jsv.2006.06.074

    Article  Google Scholar 

  11. Quinn DD, Gendelman O, Kerschen G, Sapsis TP, Bergman LA, Vakakis AF (2008) Efficiency of targeted energy transfers in coupled nonlinear oscillators associated with 1:1 resonance captures: Part I. J Sound Vib 311(3):1228–1248. https://doi.org/10.1016/j.jsv.2007.10.026

    Article  Google Scholar 

  12. Sapsis TP, Vakakis AF, Gendelman OV, Bergman LA, Kerschen G, Quinn DD (2009) Efficiency of targeted energy transfers in coupled nonlinear oscillators associated with 1:1 resonance captures: Part II, analytical study. J Sound Vib 325(1):297–320. https://doi.org/10.1016/j.jsv.2009.03.004

    Article  Google Scholar 

  13. Sapsis TP, Quinn DD, Vakakis AF, Bergman LA (2012) Effective stiffening and damping enhancement of structures with strongly nonlinear local attachments. J Vib Acoust 134(1):011016. https://doi.org/10.1115/1.4005005

    Article  Google Scholar 

  14. Nucera F, Lo Iacono F, McFarland DM, Bergman LA, Vakakis AF (2008) Application of broadband nonlinear targeted energy transfers for seismic mitigation of a shear frame: experimental results. J Sound Vib 313(1):57–76. https://doi.org/10.1016/j.jsv.2007.11.018

    Article  Google Scholar 

  15. Nucera F, McFarland DM, Bergman LA, Vakakis AF (2010) Application of broadband nonlinear targeted energy transfers for seismic mitigation of a shear frame: computational results. J Sound Vib 329(15):2973–2994. https://doi.org/10.1016/j.jsv.2010.01.020

    Article  Google Scholar 

  16. Hubbard SA, McFarland DM, Bergman LA, Vakakis AF (2010) Targeted energy transfer between a model flexible wing and nonlinear energy sink. Journal of Aircraft 47(6):1918–1931. https://doi.org/10.2514/1.c001012

    Article  Google Scholar 

  17. Yang K, Zhang YW, Ding H, Yang TZ, Li Y, Chen LQ (2017) Nonlinear energy sink for whole-spacecraft vibration reduction. J Vib Acoust 139(2):021011. https://doi.org/10.1115/1.4035377

    Article  Google Scholar 

  18. Andersen D, Starosvetsky Y, Vakakis A, Bergman L (2012) Dynamic instabilities in coupled oscillators induced by geometrically nonlinear damping. Nonlinear Dyn 67(1):807–827. https://doi.org/10.1007/s11071-011-0028-0

    Article  MathSciNet  Google Scholar 

  19. Andersen DK, Vakakis AF, Bergman LA (2011) Dynamics of a system of coupled oscillators with geometrically nonlinear damping. Nonlinear Modeling and Applications 2:1–7

    Google Scholar 

  20. Lee YS, Vakakis AF, Bergman LA, McFarland DM, Kerschen G (2008) Enhancing the robustness of aeroelastic instability suppression using multi-degree-of-freedom nonlinear energy sinks. AIAA Journal 46(6):1371–1394. https://doi.org/10.2514/1.30302

    Article  Google Scholar 

  21. Quinn DD, Hubbard S, Wierschem N, Al-Shudeifat MA, Ott RJ, Luo J, Spencer BF, McFarland DM, Vakakis AF, Bergman LA (2012) Equivalent modal damping, stiffening and energy exchanges in multi-degree-of-freedom systems with strongly nonlinear attachments. Proceedings of the Institution of Mechanical Engineers, Part K: Journal of Multi-body Dynamics 226(2):122–146. https://doi.org/10.1177/1464419311432671

    Article  Google Scholar 

  22. Vakakis AF, AL-Shudeifat MA, Hasan MA (2014) Interactions of propagating waves in a one-dimensional chain of linear oscillators with a strongly nonlinear local attachment. Meccanica 49(10):2375–2397. https://doi.org/10.1007/s11012-014-0008-9

    Article  MATH  Google Scholar 

  23. Al-Shudeifat MA (2017) Nonlinear energy sinks with nontraditional kinds of nonlinear restoring forces. J Vib Acoust 139(2):024503–024505. https://doi.org/10.1115/1.4035479

    Article  Google Scholar 

  24. AL-Shudeifat MA (2014) Highly efficient nonlinear energy sink. Nonlinear Dyn 76(4):1905–1920. https://doi.org/10.1007/s11071-014-1256-x

    Article  MathSciNet  MATH  Google Scholar 

  25. Fang X, Wen J, Yin J, Yu D (2017) Highly efficient continuous bistable nonlinear energy sink composed of a cantilever beam with partial constrained layer damping. Nonlinear Dyn 87(4):2677–2695. https://doi.org/10.1007/s11071-016-3220-4

    Article  Google Scholar 

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

    Article  Google Scholar 

  27. Habib G, Romeo F (2017) The tuned bistable nonlinear energy sink. Nonlinear Dyn 89(1):179–196. https://doi.org/10.1007/s11071-017-3444-y

    Article  Google Scholar 

  28. Manevitch LI, Sigalov G, Romeo F, Bergman LA, Vakakis A (2013) Dynamics of a linear oscillator coupled to a bistable light attachment: analytical study. Journal of Applied Mechanics. doi 10(1115/1):4025150

    Google Scholar 

  29. Romeo F, Sigalov G, Bergman LA, Vakakis AF (2014) Dynamics of a linear oscillator coupled to a bistable light attachment: numerical study. J Comput Nonlinear Dyn. https://doi.org/10.1115/1.4027224

    Article  Google Scholar 

  30. AL-Shudeifat MA (2019) Nonlinear energy sinks with piecewise-linear nonlinearities. J Comput Nonlinear Dyn 14(12):124501. https://doi.org/10.1115/1.4045052

    Article  Google Scholar 

  31. AL-Shudeifat MA, Wierschem NE, Bergman LA, Vakakis AF (2017) Numerical and experimental investigations of a rotating nonlinear energy sink. Meccanica 52(4):763–779. https://doi.org/10.1007/s11012-016-0422-2

    Article  Google Scholar 

  32. Blanchard AB, Gendelman OV, Bergman LA, Vakakis AF (2016) Capture into slow-invariant-manifold in the fluid–structure dynamics of a sprung cylinder with a nonlinear rotator. J Fluids Struct 63:155–173. https://doi.org/10.1016/j.jfluidstructs.2016.03.009

    Article  Google Scholar 

  33. Gendelman OV, Sigalov G, Manevitch LI, Mane M, Vakakis AF, Bergman LA (2011) Dynamics of an eccentric rotational nonlinear energy sink. J Appl Mech 79(1):011012. https://doi.org/10.1115/1.4005402

    Article  Google Scholar 

  34. Sigalov G, Gendelman OV, AL-Shudeifat MA, Manevitch LI, Vakakis AF, Bergman LA (2012) Resonance captures and targeted energy transfers in an inertially-coupled rotational nonlinear energy sink. Nonlinear Dyn 69(4):1693–1704. https://doi.org/10.1007/s11071-012-0379-1

    Article  MathSciNet  Google Scholar 

  35. Sigalov G, Gendelman OV, AL-Shudeifat MA, Manevitch LI, Vakakis AF, Bergman LA (2012) Alternation of regular and chaotic dynamics in a simple two-degree-of-freedom system with nonlinear inertial coupling. Chaos: An Interdisciplinary Journal of Nonlinear Science 22(1):013118. https://doi.org/10.1063/1.3683480

    Article  MathSciNet  MATH  Google Scholar 

  36. Saeed AS, AL-Shudeifat MA, Vakakis AF (2019) Rotary-oscillatory nonlinear energy sink of robust performance. Int J Non-Linear Mech 117:103249

    Article  Google Scholar 

  37. Saeed AS, AL-Shudeifat MA, Vakakis AF, Cantwell WJ (2020) Rotary-impact nonlinear energy sink for shock mitigation: analytical and numerical investigations. Arch Appl Mech 90(3):495–521

    Article  Google Scholar 

  38. Felix JLP, Balthazar JM, Dantas MJH (2009) On energy pumping, synchronization and beat phenomenon in a nonideal structure coupled to an essentially nonlinear oscillator. Nonlinear Dyn 56(1–2):1–11. https://doi.org/10.1007/s11071-008-9374-y

    Article  MathSciNet  MATH  Google Scholar 

  39. Gendelman OV (2012) Analytic treatment of a system with a vibro-impact nonlinear energy sink. J Sound Vib 331(21):4599–4608. https://doi.org/10.1016/j.jsv.2012.05.021

    Article  Google Scholar 

  40. Li T, Gourc E, Seguy S, Berlioz A (2017) Dynamics of two vibro-impact nonlinear energy sinks in parallel under periodic and transient excitations. Int J Non-Linear Mech 90:100–110. https://doi.org/10.1016/j.ijnonlinmec.2017.01.010

    Article  Google Scholar 

  41. Li T, Lamarque CH, Seguy S, Berlioz A (2018) Chaotic characteristic of a linear oscillator coupled with vibro-impact nonlinear energy sink. Nonlinear Dyn 91(4):2319–2330. https://doi.org/10.1007/s11071-017-4015-y

    Article  Google Scholar 

  42. Li T, Seguy S, Berlioz A (2016) Dynamics of cubic and vibro-impact nonlinear energy sink: analytical, numerical, and experimental analysis. J Vib Acoust 138(3):031010. https://doi.org/10.1115/1.4032725

    Article  Google Scholar 

  43. Li T, Seguy S, Berlioz A (2017) On the dynamics around targeted energy transfer for vibro-impact nonlinear energy sink. Nonlinear Dyn 87(3):1453–1466. https://doi.org/10.1007/s11071-016-3127-0

    Article  Google Scholar 

  44. Li T, Seguy S, Berlioz A (2017) Optimization mechanism of targeted energy transfer with vibro-impact energy sink under periodic and transient excitation. Nonlinear Dyn 87(4):2415–2433. https://doi.org/10.1007/s11071-016-3200-8

    Article  Google Scholar 

  45. Gendelman OV, Alloni A (2015) Dynamics of forced system with vibro-impact energy sink. J Sound Vib 358:301–314. https://doi.org/10.1016/j.jsv.2015.08.020

    Article  Google Scholar 

  46. Gourc E, Michon G, Seguy S, Berlioz A (2015) Targeted energy transfer under harmonic forcing with a vibro-impact nonlinear energy sink: analytical and experimental developments. J Vib Acoust 137(3):031008. https://doi.org/10.1115/1.4029285

    Article  Google Scholar 

  47. Pennisi G, Stephan C, Gourc E, Michon G (2017) Experimental investigation and analytical description of a vibro-impact NES coupled to a single-degree-of-freedom linear oscillator harmonically forced. Nonlinear Dyn 88(3):1769–1784. https://doi.org/10.1007/s11071-017-3344-1

    Article  Google Scholar 

  48. Karayannis I, Vakakis AF, Georgiades F (2008) Vibro-impact attachments as shock absorbers. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 222(10):1899–1908. https://doi.org/10.1243/09544062jmes864

    Article  Google Scholar 

  49. Lee YS, Nucera F, Vakakis AF, McFarland DM, Bergman LA (2009) Periodic orbits, damped transitions and targeted energy transfers in oscillators with vibro-impact attachments. Physica D 238(18):1868–1896. https://doi.org/10.1016/j.physd.2009.06.013

    Article  MATH  Google Scholar 

  50. Nucera F, Vakakis AF, McFarland DM, Bergman LA, Kerschen G (2007) Targeted energy transfers in vibro-impact oscillators for seismic mitigation. Nonlinear Dyn 50(3):651–677. https://doi.org/10.1007/s11071-006-9189-7

    Article  MATH  Google Scholar 

  51. Georgiadis F, Vakakis AF, McFarland DM, Bergman L (2005) Shock isolation through passive energy pumping caused by nonsmooth nonlinearities. International Journal of Bifurcation and Chaos 15(6):1989–2001. https://doi.org/10.1142/s0218127405013101

    Article  Google Scholar 

  52. Al-Shudeifat MA, Wierschem N, Quinn DD, Vakakis AF, Bergman LA, Spencer BF (2013) Numerical and experimental investigation of a highly effective single-sided vibro-impact non-linear energy sink for shock mitigation. Int J Non-Linear Mech 52:96–109. https://doi.org/10.1016/j.ijnonlinmec.2013.02.004

    Article  Google Scholar 

  53. Li W, Wierschem NE, Li X, Yang T (2018) On the energy transfer mechanism of the single-sided vibro-impact nonlinear energy sink. J Sound Vib 437:166–179. https://doi.org/10.1016/j.jsv.2018.08.057

    Article  Google Scholar 

  54. Luo J, Wierschem NE, Hubbard SA, Fahnestock LA, Quinn DD, McFarland DM, Spencer BF, Vakakis AF, Bergman LA (2014) Large-scale experimental evaluation and numerical simulation of a system of nonlinear energy sinks for seismic mitigation. Eng Struct 77:34–48. https://doi.org/10.1016/j.engstruct.2014.07.020

    Article  Google Scholar 

  55. Wierschem NE, Hubbard SA, Luo J, Fahnestock LA, Spencer BF, McFarland DM, Quinn DD, Vakakis AF, Bergman LA (2017) Response attenuation in a large-scale structure subjected to blast excitation utilizing a system of essentially nonlinear vibration absorbers. J Sound Vib 389:52–72. https://doi.org/10.1016/j.jsv.2016.11.003

    Article  Google Scholar 

  56. Al-Shudeifat MA, Vakakis AF, Bergman LA (2015) Shock mitigation by means of low- to high-frequency nonlinear targeted energy transfers in a large-scale structure. J Comput Nonlinear Dyn 11(2):021006. https://doi.org/10.1115/1.4030540

    Article  Google Scholar 

  57. Saeed AS, AL-Shudeifat MA (2019) A comparison of the common types of nonlinear energy sinks. In: 15th international conference dynamical systems—theory and applications DSTA, Lodz, Poland, 2–5 Dec 2019

  58. Nelder JA, Mead R (1965) A simplex method for function minimization. Comput J 7(4):308–313. https://doi.org/10.1093/comjnl/7.4.308

    Article  MathSciNet  MATH  Google Scholar 

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  1. Aerospace Engineering, Khalifa University of Science and Technology, 127788, Abu Dhabi, UAE

    Mohammad A. AL-Shudeifat & Adnan S. Saeed

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Appendix

Appendix

$${\mathbf{M}} = \left[ {\begin{array}{*{20}c} {\text{1037} - m} & \text{0} & \text{0} & \text{0} & \text{0} & \text{0} & \text{0} & \text{0} & \text{0} \\ \text{0} & {\text{1074}} & \text{0} & \text{0} & \text{0} & \text{0} & \text{0} & \text{0} & \text{0} \\ \text{0} & \text{0} & {\text{1075}} & \text{0} & \text{0} & \text{0} & \text{0} & \text{0} & \text{0} \\ \text{0} & \text{0} & \text{0} & {\text{1075}} & \text{0} & \text{0} & \text{0} & \text{0} & \text{0} \\ \text{0} & \text{0} & \text{0} & \text{0} & {\text{1075}} & \text{0} & \text{0} & \text{0} & \text{0} \\ \text{0} & \text{0} & \text{0} & \text{0} & \text{0} & {\text{1075}} & \text{0} & \text{0} & \text{0} \\ \text{0} & \text{0} & \text{0} & \text{0} & \text{0} & \text{0} & {\text{1075}} & \text{0} & \text{0} \\ \text{0} & \text{0} & \text{0} & \text{0} & \text{0} & \text{0} & \text{0} & {\text{1075}} & \text{0} \\ \text{0} & \text{0} & \text{0} & \text{0} & \text{0} & \text{0} & \text{0} & \text{0} & {\text{1098}} \\ \end{array} } \right]{\text{kg}}$$

where m is the NES mass.

The stiffness matrix of the linear nine-story structure is given by:

$$K = \left[ {\begin{array}{*{20}c} { 3} . 6 9 6 2& { - 3} . 7 5 4 4 &{ 0} . 0 3 7 5& { 0} . 0 0 3 5 &{ 0} . 0 0 3 2 &{ 0} . 0 0 3 2 &{ 0} . 0 0 3 2 &{ 0} . 0 0 3 2 &{ 0} . 0 0 3 0\hfill \\ { - 3} . 7 5 4 4 &{ 7} . 7 4 3 0 &{ - 4} . 0 5 3 4 &{ 0} . 0 6 4 6 &{ - 0} . 0 0 0 6 &{ 0} . 0 0 0 2 &{ 0} . 0 0 0 2 &{ 0} . 0 0 0 2 &{ 0} . 0 0 0 2\hfill \\ { 0} . 0 3 7 5 &{ - 4} . 0 5 3 4& { 8} . 2 1 4 1 &{ - 4} . 2 6 5 7&{ 0} . 0 6 7 8 &{ - 0} . 0 0 0 7 &{ 0} . 0 0 0 1 &{ 0} . 0 0 0 1 &{ 0} . 0 0 0 1\hfill \\ { 0} . 0 0 3 5 &{ 0} . 0 6 4 6 &{ - 4} . 2 6 5 7 &{ 8} . 3 9 8 6& { - 4} . 2 6 8 0 &{ 0} . 0 6 7 7 &{ - 0} . 0 0 0 8 &{ 0} . 0 0 0 0 &{ 0} . 0 0 0 0\hfill \\ { 0} . 0 0 3 2 &{ - 0} . 0 0 0 6 &{ 0} . 0 6 7 8 &{ - 4} . 2 6 8 0 &{ 8} . 3 9 8 6 &{ - 4} . 2 6 8 0 &{ 0} . 0 6 7 7& { - 0} . 0 0 0 8 &{ 0} . 0 0 0 0\hfill \\ { 0} . 0 0 3 2 &{ 0} . 0 0 0 2& { - 0} . 0 0 0 7 &{ 0} . 0 6 7 7& { - 4} . 2 6 8 0 &{ 8} . 3 9 8 6& { - 4} . 2 6 8 0 &{ 0} . 0 6 7 7& { - 0} . 0 0 0 8\hfill \\ { 0} . 0 0 3 2& { 0} . 0 0 0 2& { 0} . 0 0 0 1& { - 0} . 0 0 0 8& { 0} . 0 6 7 7 &{ - 4} . 2 6 8 0 &{ 8} . 3 9 8 6 &{ - 4} . 2 6 8 0 &{ 0} . 0 6 7 7\hfill \\ { 0} . 0 0 3 2 &{ 0} . 0 0 0 2 &{ 0} . 0 0 0 1& { 0} . 0 0 0 0 &{ - 0} . 0 0 0 8& { 0} . 0 6 7 7 &{ - 4} . 2 6 8 0 &{ 8} . 3 9 8 9 &{ - 4} . 2 6 5 8\hfill \\ { 0} . 0 0 3 0 &{ 0} . 0 0 0 2 &{ 0} . 0 0 0 1 &{ 0} . 0 0 0 0 &{ 0} . 0 0 0 0& { - 0} . 0 0 0 8 &{ 0} . 0 6 7 7 &{ - 4} . 2 6 5 8 &{ 7} . 6 5 8 3\hfill \\ \end{array}} \right] \times 10^{6}\, {\text{N/m}}$$

Assuming proportional viscous damping distribution and modal damping ratios of \(\zeta = 0.01\), the damping matrix of the linear nine-story structure is given by:

$${\mathbf{C}} = \left[ {\begin{array}{*{20}c} { 0} . 8 7 4 9 &{ - 0} . 5 1 3 3 &{ - 0} . 1 4 1 8& { - 0} . 0 5 9 0 &{ - 0} . 0 3 1 0 &{ - 0} . 0 1 8 0 &{ - 0} . 0 1 0 7 &{ - 0} . 0 0 6 0 &{ - 0} . 0 0 2 8\hfill \\ { - 0} . 5 1 3 3 &{ 1} . 3 4 0 1 &{ - 0} . 5 3 3 0& { - 0} . 1 0 0 4& { - 0} . 0 4 4 1& { - 0} . 0 2 3 7 &{ - 0} . 0 1 3 7& { - 0} . 0 0 7 8 &{ - 0} . 0 0 3 9\hfill \\ { - 0} . 1 4 1 8 &{ - 0} . 5 3 3 0& { 1} . 6 4 6 0 &{ - 0} . 5 9 8 8 &{ - 0} . 1 1 7 9& { - 0} . 0 5 1 8 &{ - 0} . 0 2 7 3 &{ - 0} . 0 1 4 9 &{ - 0} . 0 0 7 2\hfill \\ { - 0} . 0 5 9 0& { - 0} . 1 0 0 4& { - 0} . 5 9 8 8 &{ 1} . 6 9 6 3 &{ - 0} . 5 8 3 4 &{ - 0} . 1 0 9 5& { - 0} . 0 4 6 1& { - 0} . 0 2 2 6& { - 0} . 0 1 0 4\hfill \\ { - 0} . 0 3 1 0 &{ - 0} . 0 4 4 1 &{ - 0} . 1 1 7 9 &{ - 0} . 5 8 3 4& { 1} . 7 0 4 7 &{ - 0} . 5 7 7 7& { - 0} . 1 0 4 7& { - 0} . 0 4 1 3 &{ - 0} . 0 1 7 3\hfill \\ { - 0} . 0 1 8 0& { - 0} . 0 2 3 7 &{ - 0} . 0 5 1 8& { - 0} . 1 0 9 5 &{ - 0} . 5 7 7 7& { 1} . 7 0 9 5 &{ - 0} . 5 7 2 9& { - 0} . 0 9 9 2 &{ - 0} . 0 3 4 4\hfill \\ { - 0} . 0 1 0 7& { - 0} . 0 1 3 7 &{ - 0} . 0 2 7 3 &{ - 0} . 0 4 6 1 &{ - 0} . 1 0 4 7 &{ - 0} . 5 7 2 9& { 1} . 7 1 5 1 &{ - 0} . 5 6 5 3 &{ - 0} . 0 8 8 8\hfill \\ { - 0} . 0 0 6 0 &{ - 0} . 0 0 7 8 &{ - 0} . 0 1 4 9 &{ - 0} . 0 2 2 6& { - 0} . 0 4 1 3 &{ - 0} . 0 9 9 2& { - 0} . 5 6 5 3 &{ 1} . 7 2 7 3 &{ - 0} . 5 4 9 8\hfill \\ { - 0} . 0 0 2 8& { - 0} . 0 0 3 9 &{ - 0} . 0 0 7 2& { - 0} . 0 1 0 4& { - 0} . 0 1 7 3& { - 0} . 0 3 4 4 &{ - 0} . 0 8 8 8 &{ - 0} . 5 4 9 8 &{ 1} . 7 4 5 0\hfill \\ \end{array}} \right] \times 10^{3}\, {\text{N}} \, {\text{s/m}}$$

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AL-Shudeifat, M.A., Saeed, A.S. Comparison of a modified vibro-impact nonlinear energy sink with other kinds of NESs. Meccanica 56, 735–752 (2021). https://doi.org/10.1007/s11012-020-01193-3

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