Advertisement

Nonlinear Dynamics

, Volume 95, Issue 3, pp 2367–2382 | Cite as

Nonlinear vibration of a slightly curved beam with quasi-zero-stiffness isolators

  • Hu Ding
  • Li-Qun ChenEmail author
Original Paper

Abstract

Bending vibration of isolated structures has always been neglected when the vibration isolation was studied. Isolated structures have usually been treated as discrete systems. In this study, dynamics of a slightly curved beam supported by quasi-zero-stiffness systems are firstly presented. In order to achieve quasi-zero-stiffness, a nonlinear isolation system is implemented via three linear springs. A nonlinear dynamic model of the slightly curved beam with nonlinear isolations is established. It includes square nonlinearity, cubic nonlinearity, and nonlinear boundaries. Then, the mode functions and the frequencies of the curved beam with elastic boundaries are derived. The schemes of the finite difference method (FDM) and the Galerkin truncation method (GTM) are, respectively, proposed to obtain nonlinear responses of the curved beam with nonlinear boundaries. Numerical results demonstrate that both the GTM and the FDM yield accurate solutions for the nonlinear dynamics of curved structures with nonsimple boundaries. The multi-mode resonance characteristics of the curved beam affect the vibration isolation efficiency. The quasi-zero-stiffness isolators reduce the transmissibility of modal resonances and provide a promising future for isolating the bending vibration of the flexible structure. However, the initial curvature significantly increases the resonant frequency of the flexible structure, and thus the frequency range of the effective vibration isolation is narrower. Furthermore, the quadratic nonlinear terms in the curved beam make the dynamic phenomenon more complicated. Therefore, it is more challenging and necessary to investigate the isolation of the bending vibration of the initial curved structure.

Keywords

Curved beam Nonlinear vibration Quasi-zero-stiffness Nonlinear isolation 

Notes

Acknowledgements

The authors would wish to thank anonymous reviewers for those critical comments. The authors gratefully acknowledge the support of the National Natural Science Foundation of China (Nos. 11772181, 11832009, and 11572182), the Program of Shanghai Municipal Education Commission (No. 17SG38), and the Key Research Projects of Shanghai Science and Technology Commission (No. 18010500100).

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

References

  1. 1.
    Arena, A., Lacarbonara, W.: Nonlinear parametric modeling of suspension bridges under aeroelastic forces: torsional divergence and flutter. Nonlinear Dyn. 70(4), 2487–2510 (2012)MathSciNetGoogle Scholar
  2. 2.
    Song, M.T., Cao, D.Q., Zhu, W.D., Bi, Q.S.: Dynamic response of a cable-stayed bridge subjected to a moving vehicle load. Acta Mech. 227(10), 2925–2945 (2016)MathSciNetGoogle Scholar
  3. 3.
    Kang, H.J., Zhao, Y.Y., Zhu, H.P.: Out-of-plane free vibration analysis of a cable-arch structure. J. Sound Vib. 332(4), 907–921 (2013)Google Scholar
  4. 4.
    Arena, A., Pacitti, A., Lacarbonara, W.: Nonlinear response of elastic cables with flexural-torsional stiffness. Int. J. Solids Struct. 87, 267–277 (2016)Google Scholar
  5. 5.
    Cornil, M.B., Capolungo, L., Qu, J.M., Jairazbhoy, V.A.: Free vibration of a beam subjected to large static deflection. J. Sound Vib. 303(3–5), 723–740 (2007)Google Scholar
  6. 6.
    Zhang, W., Cao, D.X.: Studies on bifurcation and chaos of a string-beam coupled system with two degrees-of-freedom. Nonlinear Dyn. 45(1–2), 131–147 (2006)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Liu, X.L., Shangguan, W.B., Jing, X.J., Ahmed, W.: Vibration isolation analysis of clutches based on trouble shooting of vehicle accelerating noise. J. Sound Vib. 382, 84–99 (2016)Google Scholar
  8. 8.
    Sun, X.T., Zhang, S., Xu, J.: Parameter design of a multi-delayed isolator with asymmetrical nonlinearity. Int. J. Mech. Sci. 138, 398–408 (2018)Google Scholar
  9. 9.
    Ibrahim, R.A.: Recent advances in nonlinear passive vibration isolators. J. Sound Vib. 314(3–5), 371–452 (2008)Google Scholar
  10. 10.
    Niu, F., Meng, L.S., Wu, W.J., Sun, J.G., Su, W.H., Meng, G., Rao, Z.S.: Recent advances in quasi-zero-stiffness vibration isolation systems. Appl. Mech. Mater. 397–400, 295–303 (2013)Google Scholar
  11. 11.
    Liu, C.C., Jing, X.J., Daley, S., Li, F.M.: Recent advances in micro-vibration isolation. Mech. Syst. Signal Pr. 56–57, 55–80 (2015)Google Scholar
  12. 12.
    Lacarbonara, W., Nayfeh, A.H., Kreider, W.: Experimental validation of reduction methods for nonlinear vibrations of distributed-parameter systems: analysis of a buckled beam. Nonlinear Dyn. 17(2), 95–117 (1998)zbMATHGoogle Scholar
  13. 13.
    Lacarbonara, W.: Buckling and post-buckling of non-uniform non-linearly elastic rods. Int. J. Mech. Sci. 50(8), 1316–1325 (2008)zbMATHGoogle Scholar
  14. 14.
    Adam, C., Ziegler, F.: Moderately large forced oblique vibrations of elastic-viscoplastic deteriorating slightly curved beams. Arch. Appl. Mech. 67(6), 375–392 (1997)zbMATHGoogle Scholar
  15. 15.
    Smelova-Reynolds, T., Dowell, E.H.: The role of higher modes in the chaotic motion of the buckled beam. Int. J. Nonlinear Mech. 31(6), 931–939 (1996)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Huang, J.L., Su, K.L.R., Lee, Y.Y.R., Chen, S.H.: Various bifurcation phenomena in a nonlinear curved beam subjected to base harmonic excitation. Int. J. Bifurc. Chaos. 28(7), 1830023 (2018)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Sinir, B.G.: Bifurcation and chaos of slightly curved pipes. Math. Comput. Appl. 15(3), 490–502 (2010)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Li, Y.D., Yang, Y.R.: Nonlinear vibration of slightly curved pipe with conveying pulsating fluid. Nonlinear Dyn. 88(4), 2513–2529 (2017)Google Scholar
  19. 19.
    Messaris, G.A.T., Karahalios, G.T.: Unsteady fluid flow in a slightly curved annular pipe: the impact of the annulus on the flow physics. Phys. Fluids 29(2), 021903 (2017)Google Scholar
  20. 20.
    Czerwinski, A., Luczko, J.: Non-planar vibrations of slightly curved pipes conveying fluid in simple and combination parametric resonances. J. Sound Vib. 413, 270–290 (2018)Google Scholar
  21. 21.
    Owoseni, O.D., Orolu, K.O., Oyediran, A.A.: Dynamics of slightly curved pipe conveying hot pressurized fluid resting on linear and nonlinear viscoelastic foundations. J. Vib. Acoust. 140(2), 021005 (2018)Google Scholar
  22. 22.
    Oz, H.R., Pakdemirli, M., Ozkaya, E., Yilmaz, M.: Non-linear vibrations of a slightly curved beam resting on a non-linear elastic foundation. J. Sound Vib. 212(2), 295–309 (1998)Google Scholar
  23. 23.
    Ozkaya, E., Sarigul, M., Boyaci, H.: Nonlinear transverse vibrations of a slightly curved beam resting on multiple springs. Int. J. Acoust. Vib. 21(4), 379–393 (2016)Google Scholar
  24. 24.
    Ozkaya, E., Sarigul, M., Boyaci, H.: Nonlinear transverse vibrations of a slightly curved beam carrying a concentrated mass. Acta Mech. Sin. 25(6), 871–882 (2009)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Oz, H.R.: In-plane vibrations of cracked slightly curved beams. Struct. Eng. Mech. 36(6), 679–695 (2010)Google Scholar
  26. 26.
    Emam, S.A., Nayfeh, A.H.: Nonlinear responses of buckled beams to subharmonic-resonance excitations. Nonlinear Dyn. 35(2), 105–122 (2004)zbMATHGoogle Scholar
  27. 27.
    Lee, Y.Y., Huang, J.L., Hui, C.K., Ng, C.F.: Sound absorption of a quadratic and cubic nonlinearly vibrating curved panel absorber. Appl. Math. Model. 36(11), 5574–5588 (2012)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Mbong, T.L.M.D., Siewe, M.S., Tchawoua, C.: Controllable parametric excitation effect on linear and nonlinear vibrational resonances in the dynamics of a buckled beam. Commun. Nonlinear Sci. 54, 377–388 (2018)MathSciNetGoogle Scholar
  29. 29.
    Tomasiello, S.: A DQ based approach to simulate the vibrations of buckled beams. Nonlinear Dyn. 50(1–2), 37–48 (2007)zbMATHGoogle Scholar
  30. 30.
    Nayfeh, A.H., Emam, S.A.: Exact solution and stability of postbuckling configurations of beams. Nonlinear Dyn. 54(4), 395–408 (2008)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Susanto, K.: Vibration analysis of piezoelectric laminated slightly curved beams using distributed transfer function method. Int. J. Solids Struct. 46(6), 1564–1573 (2009)zbMATHGoogle Scholar
  32. 32.
    Li, X., Zhang, Y.W., Ding, H., Chen, L.Q.: Integration of a nonlinear energy sink and a piezoelectric energy harvester. Appl. Math. Mech. Engl. 38(7), 1019–1030 (2017)MathSciNetGoogle Scholar
  33. 33.
    Zhang, Y.W., Fang, B., Chen, Y.: Vibration isolation performance evaluation of the discrete whole-spacecraft vibration isolation platform for flexible spacecrafts. Meccanica 47(5), 1185–1195 (2012)zbMATHGoogle Scholar
  34. 34.
    Virgin, L.N., Santillan, S.T., Plaut, R.H.: Vibration isolation using extreme geometric nonlinearity. J. Sound Vib. 315(3), 721–731 (2008)Google Scholar
  35. 35.
    Li, S., Fang, B., Yang, T.Z., Zhang, Y.W., Tan, L.J., Huang, W.H.: Dynamics of vibration isolation system obeying fractional differentiation. Aircr. Eng. Aerosp. Technol. 84(2), 103–108 (2012)Google Scholar
  36. 36.
    Jiang, J.F., Cao, D.Q., Chen, H.T., Zhao, K.: The vibration transmissibility of a single degree of freedom oscillator with nonlinear fractional order damping. Int. J. Syst. Sci. 48(11), 2379–2393 (2017)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Ho, C., Zhu, Y.P., Lang, Z.Q., Billings, S.A., Kohiyama, M., Wakayama, S.: Nonlinear damping based semi-active building isolation system. J. Sound. Vib. 424, 302–317 (2018)Google Scholar
  38. 38.
    Hu, F.Z., Jing, X.J.: A 6-DOF passive vibration isolator based on Stewart structure with X-shaped legs. Nonlinear Dyn. 91(1), 157–185 (2018)Google Scholar
  39. 39.
    Yu, H.J., Sun, X.T., Xu, J., Zhang, S.: Transition sets analysis based parametrical design of nonlinear metal rubber isolator. Int. J. Nonlinear Mech. 96, 93–105 (2017)Google Scholar
  40. 40.
    Lu, Z.Q., Yang, T.J., Brennan, M.J., Li, X.H., Liu, Z.G.: On the performance of a two-stage vibration isolation system which has geometrically nonlinear stiffness. J. Vib. Acoust. 136(6), 064501 (2014)Google Scholar
  41. 41.
    Lu, Z.Q., Yang, T.J., Brennan, M.J., Liu, Z.G., Chen, L.Q.: Experimental investigation of a two-stage nonlinear vibration isolation system with high-static-low-dynamic stiffness. J. Appl. Mech. T ASME. 84(2), 021001 (2017)Google Scholar
  42. 42.
    Zheng, Y.S., Li, Q.P., Yan, B., Luo, Y.J., Zhang, X.N.: A Stewart isolator with high-static-low-dynamic stiffness struts based on negative stiffness magnetic springs. J. Sound Vib. 422, 390–408 (2018)Google Scholar
  43. 43.
    Sun, X.T., Shu, Z., Jian, X., Feng, W.: Dynamical analysis and realization of an adaptive isolator. J. Appl. Mech. T ASME. 85(1), 011002 (2018)Google Scholar
  44. 44.
    Shen, Y.J., Yang, S.P., Xing, H.J., Ma, H.X.: Design of single degree-of-freedom optimally passive vibration isolation system. J. Vib. Eng. Technol. 3(1), 25–36 (2015)Google Scholar
  45. 45.
    Liu, C.R., Xu, D.L., Zhou, J.X., Bishop, S.: On theoretical and experimental study of a two-degree-of-freedom anti-resonance floating vibration isolation system. J. Vib. Control. 21(10), 1886–1901 (2015)Google Scholar
  46. 46.
    Huang, X.C., Sun, J.Y., Hua, H.X., Zhang, Z.Y.: The isolation performance of vibration systems with general velocity-displacement-dependent nonlinear damping under base excitation: numerical and experimental study. Nonlinear Dyn. 85(2), 777–796 (2016)MathSciNetGoogle Scholar
  47. 47.
    Lu, Z.Q., Brennan, M.J., Yang, T.J., Li, X.H., Liu, Z.G.: An investigation of a two-stage nonlinear vibration isolation system. J. Sound Vib. 332(6), 1456–1464 (2013)Google Scholar
  48. 48.
    Lu, Z.Q., Brennan, M.J., Chen, L.Q.: On the transmissibilities of nonlinear vibration isolation system. J. Sound Vib. 375, 28–37 (2016)Google Scholar
  49. 49.
    Hao, Z.F., Cao, Q.J., Wiercigroch, M.: Nonlinear dynamics of the quasi-zero-stiffness SD oscillator based upon the local and global bifurcation analyses. Nonlinear Dyn. 87(2), 987–1014 (2017)Google Scholar
  50. 50.
    Li, Y.L., Xu, D.L.: Vibration attenuation of high dimensional quasi-zero stiffness floating raft system. Int. J. Mech. Sci. 126, 186–195 (2017)Google Scholar
  51. 51.
    Wang, X.L., Zhou, J.X., Xu, D.L., Ouyang, H.J., Duan, Y.: Force transmissibility of a two-stage vibration isolation system with quasi-zero stiffness. Nonlinear Dyn. 87(1), 633–646 (2017)Google Scholar
  52. 52.
    Zheng, Y.S., Zhang, X.N., Luo, Y.J., Zhang, Y.H., Xie, S.L.: Analytical study of a quasi-zero stiffness coupling using a torsion magnetic spring with negative stiffness. Mech. Syst. Signal Pr. 100, 135–151 (2018)Google Scholar
  53. 53.
    Zhang, Y.W., Fang, B., Zang, J.: Dynamic features of passive whole-spacecraft vibration isolation platform based on non-probabilistic reliability. J. Vib. Control. 21(1), 60–67 (2015)Google Scholar
  54. 54.
    Fan, Z.J., Lee, J.H., Kang, K.H., Kim, K.J.: The forced vibration of a beam with viscoelastic boundary supports. J. Sound Vib. 210(5), 673–682 (1998)zbMATHGoogle Scholar
  55. 55.
    Lv, B.L., Li, W.Y., Ouyang, H.J.: Moving force-induced vibration of a rotating beam with elastic boundary conditions. Int. J. Struct. Stab. Dy. 15(1), 1450035 (2015)MathSciNetzbMATHGoogle Scholar
  56. 56.
    Zhang, T., Ouyang, H., Zhang, Y.O., Lv, B.L.: Nonlinear dynamics of straight fluid-conveying pipes with general boundary conditions and additional springs and masses. Appl. Math. Model. 40(17–18), 7880–7900 (2016)MathSciNetGoogle Scholar
  57. 57.
    Wang, Y.R., Fang, Z.W.: Vibrations in an elastic beam with nonlinear supports at both ends. J. Appl. Mech. Tech. Phys. 56(2), 337–346 (2015)MathSciNetzbMATHGoogle Scholar
  58. 58.
    Mao, X.Y., Ding, H., Chen, L.Q.: Vibration of flexible structures under nonlinear boundary conditions. J. Appl. Mech. T ASME. 84(11), 111006 (2017)Google Scholar
  59. 59.
    Ding, H., Wang, S., Zhang, Y.-W.: Free and forced nonlinear vibration of a transporting belt with pulley support ends. Nonlinear Dyn. 92(4), 2037–2048 (2018)Google Scholar
  60. 60.
    Ding, H., Lim, C.W., Chen, L.Q.: Nonlinear vibration of a traveling belt with non-homogeneous boundaries. J. Sound Vib. 424, 78–93 (2018)Google Scholar
  61. 61.
    Tu, Y.Q., Zheng, G.T.: On the vibration isolation of flexible structures. J. Appl. Mech. T ASME. 74(3), 415–420 (2007)zbMATHGoogle Scholar
  62. 62.
    Ding, H., Dowell, E.H., Chen, L.Q.: Transmissibility of bending vibration of an elastic beam. J. Vib. Acoust. 140(3), 031007 (2018)Google Scholar
  63. 63.
    Ding, H., Zhu, M.H., Chen, L.Q.: Nonlinear vibration isolation of a viscoelastic beam. Nonlinear Dyn. 92(2), 325–349 (2018)Google Scholar
  64. 64.
    Mayoof, F.N., Hawwa, M.A.: Chaotic behavior of a curved carbon nanotube under harmonic excitation. Chaos Soliton Fract. 42(3), 1860–1867 (2009)Google Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Shanghai Institute of Applied Mathematics and MechanicsShanghai UniversityShanghaiChina
  2. 2.Harbin Institute of TechnologyShenzhenChina

Personalised recommendations