Nonlinear Dynamics

, Volume 95, Issue 3, pp 2543–2554 | Cite as

Nonlinear vibration of a beam with asymmetric elastic supports

  • Hu DingEmail author
  • Yi Li
  • Li-Qun Chen
Original Paper


Under the conditions of horizontal placement and only considering geometric nonlinearity, depending on the boundary constraints, primary resonances of an elastic beam exhibit either hardening or softening nonlinear behavior. In this paper, the conversion of softening nonlinear characteristics to hardening characteristics is studied by using the multi-scale perturbation method. Therefore, in a local sense, the condition is established for the resonance of the elastic beam exhibits only linear characteristics by finding the balance between asymmetric elastic support and geometric nonlinearity. A viscoelastic beam supported by vertical springs is proposed with nonrotatable left boundary and freely rotatable right end. In order to truncate the continuous system, natural frequencies and modes of the proposed asymmetric beam are analyzed. The steady-state responses of the beam excited by a distributed harmonic force are, respectively, obtained by an approximate analytical method and a numerical approach. Under the condition that the beam is placed horizontally, the transition from the cantilever state to the clamped–pinned state is demonstrated by constructing different asymmetry support conditions. The resonance peak of the first-order primary resonance is used to demonstrate the transition from softening nonlinear characteristics to the hardening characteristics. This research shows that the transformation from softening characteristics to hardening characteristics caused by asymmetric elastic support and geometric nonlinearity exists only in the first-order mode resonance.


Geometric nonlinearity Elastic beam Asymmetric elastic boundaries Nonlinear characteristics 



The authors gratefully acknowledge the support of the National Natural Science Foundation of China [Grant Numbers 11772181, 11422214], the “Dawn” Program of Shanghai Education Commission (Grant Number 17SG38) and the Innovation Program of Shanghai Municipal Education Commission [Grant Number 2017-01-07-00-09-E00019].

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interests.


  1. 1.
    Cao, D.X., Zhang, W.: Global bifurcations and chaotic dynamics for a string-beam coupled system. Chaos Soliton Fract. 37(3), 858–875 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Zhao, Y.Y., Kang, H.J.: In-plane free vibration analysis of cable–arch structure. J. Sound Vib. 312(3), 363–379 (2008)CrossRefGoogle Scholar
  3. 3.
    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)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Ding, H., Zu, J.W.: Steady-state responses of pulley–belt systems with a one-way clutch and belt bending stiffness. J. Vib. Acoust. 136(4), 041006 (2014)CrossRefGoogle Scholar
  5. 5.
    Silva, C.J., Daqaq, M.F.: On estimating the effective nonlinearity of structural modes using approximate modal shapes. J. Vib. Control. 20(11), 1751–1764 (2014)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Alhazza, K.A., Nayfeh, A.H., Daqaq, M.F.: On utilizing delayed feedback for active-multimode vibration control of cantilever beams. J. Sound Vib. 319, 735–752 (2009)CrossRefGoogle Scholar
  7. 7.
    Arafat, H.N., Nayfeh, A.H., Chin, C.M.: Nonlinear nonplanar dynamics of parametrically excited cantilever beams. Nonlinear Dyn. 15(1), 31–61 (1998)CrossRefzbMATHGoogle Scholar
  8. 8.
    Ding, H., Zhang, G.C., Chen, L.Q., Yang, S.P.: Forced vibrations of supercritically transporting viscoelastic beams. J. Vib. Acoust. 134(5), 051007 (2012)CrossRefGoogle Scholar
  9. 9.
    Ding, H., Dowell, E.H., Chen, L.Q.: Transmissibility of bending vibration of an elastic beam. J. Vib. Acoust. 140(3), 031007 (2018)CrossRefGoogle Scholar
  10. 10.
    Mahmoodi, S.N., Jahli, N., Khadem, S.E.: An experimental investigation of nonlinear vibration and frequency response analysis of cantilever viscoelastic beams. J. Sound Vib. 311(3–5), 1409–1419 (2008)CrossRefGoogle Scholar
  11. 11.
    Pratiher, B., Dwivedy, S.K.: Nonlinear vibrations and frequency response analysis of a cantilever beam under periodically varying magnetic field. Mech. Based Des. Struct. 39(3), 378–391 (2011)CrossRefzbMATHGoogle Scholar
  12. 12.
    Pratiher, B., Dwivedy, S.K.: Nonlinear vibration of a magneto-elastic cantilever beam with tip mass. J. Vib. Acoust. 131(2), 021011 (2009)CrossRefGoogle Scholar
  13. 13.
    Aureli, M., Pagano, C., Porfiri, M.: Nonlinear finite amplitude torsional vibrations of cantilevers in viscous fluids. J. Appl. Phys. 111(12), 124915 (2012)CrossRefGoogle Scholar
  14. 14.
    Pratiher, B.: Vibration control of a transversely excited cantilever beam with tip mass. Arch. Appl. Mech. 82(1), 31–42 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Abdelkefi, A., Yan, Z.M., Hajj, M.R.: Modeling and nonlinear analysis of piezoelectric energy harvesting from transverse galloping. Smart. Mater. Struct. 22(2), 025016 (2013)CrossRefGoogle Scholar
  16. 16.
    Caruntu, D.I., Martinez, I., Knecht, M.W.: Reduced order model analysis of frequency response of alternating current near half natural frequency electrostatically actuated MEMS cantilevers. J. Comput. Nonlinear Dyn. 8(3), 031011 (2012)CrossRefGoogle Scholar
  17. 17.
    Caruntu, D.I., Martinez, I.: Reduced order model of parametric resonance of electrostatically actuated MEMS cantilever resonators. Int. J. Nonlinear Mech. 66, 28–32 (2014)CrossRefGoogle Scholar
  18. 18.
    Singh, S.S., Pal, P., Pandey, A.K.: Mass sensitivity of nonuniform microcantilever beams. J. Vib. Acoust. 138(6), 064502 (2016)CrossRefGoogle Scholar
  19. 19.
    Farokhi, H., Ghayesh, M.H., Gholipour, A.: Dynamics of functionally graded micro-cantilevers. Int. J. Eng. Sci. 115, 117–130 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Nayfeh, A.H., Arafat, H.N.: Nonlinear response of cantilever beams to combination and subcombination resonances. Shock Vib. 5(5–6), 277–288 (1998)CrossRefGoogle Scholar
  21. 21.
    Anderson, T.J., Nayfeh, A.H., Balachandran, B.: Experimental verification of the importance of the nonlinear curvature in the response of a cantilever beam. J. Vib. Acoust. 118(1), 21–27 (1996)CrossRefGoogle Scholar
  22. 22.
    Yabuno, H., Nayfeh, A.H.: Nonlinear normal modes of a parametrically excited cantilever beam. Nonlinear Dyn. 25(1–3), 65–77 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Silva, C.J., Daqaq, M.F.: Nonlinear flexural response of a slender cantilever beam of constant thickness and linearly-varying width to a primary resonance excitation. J. Sound Vib. 389, 438–453 (2017)CrossRefGoogle Scholar
  24. 24.
    Azrar, L., Benamar, R., White, R.G.: A semi-analytical approach to the non-linear dynamic response problem of beams at large vibration amplitudes, part ii: multimode approach to the steady state forced periodic response. J. Sound Vib. 255(1), 1–41 (2002)CrossRefGoogle Scholar
  25. 25.
    Wang, Y.Q., Zu, J.W.: Analytical analysis for vibration of longitudinally moving plate submerged in infinite liquid domain. Appl. Math. Mech. Engl. 38(5), 625–646 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Wielentejczyk, P., Lewandowski, R.: Geometrically nonlinear, steady state vibration of viscoelastic beams. Int. J. Nonlinear Mech. 89, 177–186 (2017)CrossRefGoogle Scholar
  27. 27.
    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)CrossRefGoogle Scholar
  28. 28.
    Ghayesh, M.H., Kazemirad, S., Darabi, M.A., Woo, P.: Thermo-mechanical nonlinear vibration analysis of a spring-mass-beam system. Arch. Appl. Mech. 82(3), 317–331 (2012)CrossRefzbMATHGoogle Scholar
  29. 29.
    Mahmoudkhani, S., Haddadpour, H.: Nonlinear vibration of viscoelastic sandwich plates under narrow-band random excitations. Nonlinear Dyn. 74(1–2), 165–188 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Ding, H., Li, D.P.: Static and dynamic behaviors of belt-drive dynamic systems with a one-way clutch. Nonlinear Dyn. 78(2), 1553–1575 (2014)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Tang, Y.Q., Zhang, D.B., Rui, M., Wang, X., Zhu, D.C.: Dynamic stability of axially accelerating viscoelastic plates with longitudinally varying tensions. Appl. Math. Mech. Engl. 37(12), 1647–1668 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Ding, H., Huang, L.L., Mao, X.Y., Chen, L.Q.: Primary resonance of traveling viscoelastic beam under internal resonance. Appl. Math. Mech. Engl. 38(1), 1–14 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Yang, Z.X., Han, Q.K., Chen, Y.G., Jin, Z.H.: Nolinear harmonic response characteristics and experimental investigation of cantilever hard-coating plate. Nonlinear Dyn. 89(1), 27–38 (2017)CrossRefGoogle Scholar
  34. 34.
    Lenci, S., Clementi, F., Rega, G.: A comprehensive analysis of hardening/softening behaviour of shearable planar beams with whatever axial boundary constraint. Meccanica 51(11), 2589–2606 (2016)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Lenci, S., Rega, G.: Axial–transversal coupling in the free nonlinear vibrations of Timoshenko beams with arbitrary slenderness and axial boundary conditions. Proc. R. Soc. A Math. Phys. Eng. Sci. 472(2190), 20160057 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Yang, X.D., Zhang, W.: Nonlinear dynamics of axially moving beam with coupled longitudinal–transversal vibrations. Nonlinear Dyn. 78(4), 2547–2556 (2014)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Zhang, G.C., Chen, L.Q., Ding, H.: Forced vibration of tip-massed cantilever with nonlinear magnetic interactions. Int. J. Appl. Mech. 6(2), 1450015 (2014)CrossRefGoogle Scholar
  38. 38.
    Ghayesh, M.H., Farokhi, H., Gholipour, A., Hussain, S.: Complex motion characteristics of three-layered Timoshenko microarches. Microsyst. Technol. 23(8), 3731–3744 (2017)CrossRefzbMATHGoogle Scholar
  39. 39.
    Ding, H., Zhu, M.H., Chen, L.Q.: Nonlinear vibration isolation of a viscoelastic beam. Nonlinear Dyn. 92(2), 325–349 (2018)CrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Shanghai Institute of Applied Mathematics and MechanicsShanghai UniversityShanghaiChina
  2. 2.Shanghai Key Laboratory of Mechanics in Energy EngineeringShanghaiChina
  3. 3.Department of MechanicsShanghai UniversityShanghaiChina

Personalised recommendations