Advertisement

Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

On experiments in harmonically excited cantilever plates with 1:2 internal resonance

  • 61 Accesses

Abstract

This work presents some experimental results for resonant nonlinear response of hyperelastic plates for 1:2 internal resonance. Previously developed topology optimization methods are used to design and fabricate candidate resonant plates using 3-D printing. One such plate is subjected to harmonic transverse excitation with increasing amplitudes in a frequency range where 1:2 internal resonances are expected to be activated. While the fabricated structure exhibits coupled mode internal resonance activated response when subjected to higher levels of excitation, the plate also displays other interesting nonlinear behavior. These include nonlinear periodic as well as amplitude modulated motions of the directly excited mode and these motions super-imposed on the coupled mode response.

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

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

References

  1. 1.

    Nayfeh, A.H.: Nonlinear Interactions: analytical, computational, and experimental methods. Wiley, New York (2000)

  2. 2.

    Bajaj, A.K., Chang, S.I., Johnson, J.M.: Amplitude modulated dynamics of a resonantly excited autoparametric two degree-of-freedom system. Nonlinear Dyn 5(4), 433–457 (1994)

  3. 3.

    Balachandran, B., Nayfeh, A.H.: Nonlinear motions of beam-mass structure. Nonlinear Dyn 1(1), 39–61 (1990)

  4. 4.

    Vyas, A., Bajaj, A.K., Raman, A., Peroulis, D.: A microresonator design based on nonlinear 1: 2 internal resonance in flexural structural modes. J Microelectromech Syst 18(3), 744–762 (2009)

  5. 5.

    Tripathi, A., Bajaj, A.K.: Computational synthesis for nonlinear dynamics based design of planar resonant structures. J Vib Acoust 135(5), 051031:1–051031:13 (2013)

  6. 6.

    Chia, C.Y.: Nonlinear analysis of plates. McGraw-Hill, New York (1980)

  7. 7.

    Amabili, M.: Nonlinear vibrations and stability of shells and plates. Cambridge University Press, Cambridge (2008)

  8. 8.

    Malatkar, P.: Nonlinear vibrations of cantilever beams and plates. PhD Thesis, Virginia Polytechnic Institute and State University (2003)

  9. 9.

    Malatkar, P., Nayfeh, A.H.: Modal interactions in a cantilever plate: an experimental study. In: 45th AIAA/ASME/ASCE/AHS/ASC structures, structural dynamics and materials conference 19–22 April, Palm Springs, California (2004)

  10. 10.

    Ribeiro, P., Petyt, M.: Geometrical non-linear, steady-state, forced, periodic vibration of plate, part I: model and convergence study. J Sound Vib 226(5), 955–983 (1999)

  11. 11.

    Ribeiro, P., Petyt, M.: Geometrical non-linear, steady-state, forced, periodic vibration of plate, part II: stability study and analysis of multi-modal response. J Sound Vib 226(5), 985–1010 (1999)

  12. 12.

    Amabili, M.: Nonlinear vibrations of rectangular plates with different boundary conditions: theory and experiments. Comput Struct 82(31–32), 2587–2605 (2004)

  13. 13.

    Ribeiro, P., Petyt, M.: Non-linear free vibration of isotropic plates with internal resonance. Int J Non-Linear Mech 35(2), 263–278 (2000)

  14. 14.

    Amabili, M.: Theory and experiments for large-amplitude vibrations of rectangular plates with geometric imperfections. J Sound Vib 291(3–5), 539–565 (2006)

  15. 15.

    Amabili, M.: Nonlinear vibrations of viscoelastic rectangular plates. J Sound Vib 362, 142–156 (2016). https://doi.org/10.1016/j.jsv.2015.09.035

  16. 16.

    Zhang, J., Yang, X., Zhang, W.: Free vibrations and nonlinear responses for a cantilever honeycomb sandwich plate. Adv Mater Sci Eng 8162873, 1–12 (2018). https://doi.org/10.1155/2018/8162873

  17. 17.

    Daqaq, M.F., Masana, R., Erturk, A., Quinn, D.D.: On the role of nonlinearities in vibratory energy harvesting: a critical review and discussion. Appl Mech Rev 66(4), 040801 (2014)

  18. 18.

    Tripathi, A., Bajaj, A.K.: Design for 1:2 internal resonances in in-plane vibrations of plates with hyperelastic material. J Vib Acoust 136(6), 061005:1–10 (2014)

  19. 19.

    Tripathi, A., Bajaj, A.K.: Topology optimization and internal resonances in transverse vibrations of hyperelastic plates. Int J Solids Struct 81, 311–328 (2016). https://doi.org/10.1016/j.ijsolstr.2015.11.029

  20. 20.

    Qui, J., Sun, K., Rudas, I.J., Gao, H.: Command filter-based adaptive NN control for MIMO nonlinear systems with full-state constraints and actuator hysteresis. IEEE Trans Cybern (2019). https://doi.org/10.1109/TCYB.2019.2944761

  21. 21.

    Rivlin, R.S.: Large elastic deformations of isotropic materials IV: further developments of the general theory. Proc R Soc A 241(835), 379–397 (1948)

  22. 22.

    Nayfeh, A.H., Balachandran, B.: Applied nonlinear dynamics: analytical, computational, and experimental methods, vol. 1. Wiley, New York (1995)

  23. 23.

    Saetta, E., Rega, G.: Unified 2D continuous and reduced order modeling of thermomechanically coupled laminated plate of nonlinear vibrations. Meccanica 49(8), 1723–1749 (2014)

Download references

Acknowledgements

This work was supported in part by the Alpha P. Jamison Professorship endowment at Purdue University. The authors would like to also thank the reviewers for some constructive suggestions that have certainly improved the clarity of the presentation.

Author information

Correspondence to A. K. Bajaj.

Ethics declarations

Conflict of interest

The authors declare they have no ethical or financial conflict of interest.

Additional information

Publisher's Note

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

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Bilal, N., Tripathi, A. & Bajaj, A.K. On experiments in harmonically excited cantilever plates with 1:2 internal resonance. Nonlinear Dyn (2020). https://doi.org/10.1007/s11071-020-05517-6

Download citation

Keywords

  • Experimental nonlinear dynamics
  • Topology optimization
  • Internal resonances
  • Hyperelastic materials