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Near-grazing dynamics of base excited cantilevers with nonlinear tip interactions

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Abstract

In this article, nonsmooth dynamics of impacting cantilevers at different scales is explored through a combination of analytical, numerical, and experimental efforts. For off-resonance and harmonic base excitations, period-doubling events close to grazing impacts are experimentally studied in a macroscale system and a microscale system. The macroscale test apparatus consists of a base excited aluminum cantilever with attractive and repulsive tip interactions. The attractive force is generated through a combination of magnets, one located at the cantilever structure’s tip and another attached to a high-resolution translatory stage. The repulsive forces are generated through impacts of the cantilever tip with the compliant material that covers the magnet on the translatory stage. The microscale system is an atomic force microscope cantilever operated in tapping mode. In this mode, this microcantilever experiences a long-range attractive van der Waals force and a repulsive force as the cantilever tip comes close to the sample. The qualitative changes observed in the experiments are further explored through numerical studies, assuming that the system response is dominated by the fundamental cantilever vibratory mode. In both the microscale and macroscale cases, contact is modeled by using a quadratic repulsive force. A reduced-order model, which is developed on the basis of a single mode approximation, is employed to understand the period-doubling phenomenon experimentally observed close to grazing in both the macroscale and microscale systems. The associated near-grazing dynamics is examined by carrying out local analyses with Poincaré map constructions to show that the observed period-doubling events are possible for the considered nonlinear tip interactions. In the corresponding experiments, the stability of the observed grazing periodic orbits has been assessed by constructing the Jacobian matrix from the experimentally obtained Poincaré map. The present study also sheds light on the use of macroscale systems to understand near-grazing dynamics in microscale systems.

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References

  1. Moon, F.C., Shaw, S.W.: Chaotic vibrations of a beam with non-linear boundary conditions. Int. J. Non-Linear Mech. 18(6), 465–477 (1983)

    Article  MathSciNet  Google Scholar 

  2. Shaw, S.W., Holmes, P.J.: A periodically forced piecewise linear oscillator. J. Sound Vib. 90(1), 129–155 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  3. Shaw, S.W., Balachandran, B.: A Review of nonlinear dynamics of mechanical systems in year 2008. J. Syst. Des. Dyn. 2(3), 611–640 (2008)

    Google Scholar 

  4. Nordmark, A.B.: Non-periodic motion caused by grazing incidence in an impact oscillator. J. Sound Vib. 145(2), 279–297 (1991)

    Article  Google Scholar 

  5. Stensson, A., Nordmark, A.B.: Experimental investigation of some consequences of low velocity impacts in the chaotic dynamics of a mechanical system. Philos. Trans. R. Soc. Lond. A 347(1683), 439–448 (1994)

    Article  Google Scholar 

  6. Chin, W., Ott, E., Nusse, H.E., Grebogi, C.: Grazing bifurcations in impact oscillators. Phys. Rev. E 50(6), 4427–4444 (1994)

    Article  MathSciNet  Google Scholar 

  7. Hunt, J.P., Sarid, D.: Kinetics of lossy grazing impact oscillators. Appl. Phys. Lett. 72(23), 2969–2971 (1998)

    Article  Google Scholar 

  8. Molenaar, J., de Weger, J.G., van de Water, W.: Mappings of grazing impact oscillators. Nonlinearity 14, 301–321 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  9. de Weger, J.G., van de Water, W., Molenaar, J.: Grazing impact oscillations. Phys. Rev. E 62(2), 2030–2041 (2000)

    Article  Google Scholar 

  10. van de Water, W., Molenaar, J.: Dynamics of vibrating atomic force microscopy. Nanotechnology 11(6), 192–199 (2000)

    Article  Google Scholar 

  11. Dankowicz, H., Nordmark, A.B.: On the origin and bifurcations of stick-slip oscillations. Physica D 136, 280–302 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  12. di Bernardo, M., Budd, C.J., Champneys, A.R.: Normal form maps for grazing bifurcations in n-dimensional piecewise-smooth dynamical systems. Physica D 160, 222–254 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  13. di Bernardo, M., Budd, C.J., Champneys, A.R.: Corner collision implies border collision bifurcation. Physica D 154, 171–194 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  14. di Bernardo, M., Feigin, M.I., Hogan, S.J., Homer, M.E.: Local analysis of C-bifurcations in n-dimensional piecewise-smooth dynamical systems. Chaos Solitons Fractals 10(11), 1881–1908 (1991)

    Google Scholar 

  15. Ing, J., Pavlovskaia, E., Wiercigroch, M., Banerjee, S.: Experimental study of impact oscillator with one-sided elastic constraint. Philos. Trans. R. Soc. A 366, 679–704 (2008)

    Article  MATH  Google Scholar 

  16. Balachandran, B.: Dynamics of an elastic structure excited by harmonic and aharmonic impactor motions. J. Vib. Control 9(3–4), 265–279 (2003)

    Article  MATH  Google Scholar 

  17. Long, X.H., Lin, G., Balachandran, B.: Grazing bifurcations in elastic structures excited by harmonic impactor motions. Physica D 237(8), 1129–1138 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  18. Dankowicz, H., Zhao, X., Misra, S.: Near-grazing dynamics in tapping-mode atomic force microscopy. Int. J. Non-Linear Mech. 42, 697–709 (2007)

    Article  Google Scholar 

  19. Dick, A.J., Balachandran, B., Yabuno, H., Numatsu, K., Hayashi, K., Kuroda, M., Ashida, K.: Utilizing nonlinear phenomena to locate grazing in the constrained motion of a cantilever beam. Nonlinear Dyn. 57(3), 335–349 (2009)

    Article  MATH  Google Scholar 

  20. Lee, S.I., Howell, S.W., Raman, A., Reifenberger, R.: Nonlinear dynamics of microcantilevers in tapping mode atomic force microscopy: a comparison between theory and experiment. Phys. Rev. B 66(11), 115409 (2002)

    Article  Google Scholar 

  21. Yagasaki, K.: Nonlinear dynamics of vibrating microcantilevers in tapping-mode atomic force microscopy. Phys. Rev. B 70(24), 24541 (2004)

    Article  Google Scholar 

  22. Hashemi, N., Dankowicz, H., Paul, M.R.: The nonlinear dynamics of tapping mode atomic force microscopy with capillary force interactions. J. Appl. Phys. 103(9), 09351 (2008)

    Article  Google Scholar 

  23. Chakraborty, I., Balachandran, B.: Off-resonance cantilever dynamics in the presence of attractive and repulsive tip-interaction forces. Int. J. Struct. Stab. Dyn. 11(4), 603–620 (2011)

    Article  Google Scholar 

  24. Chakraborty, I., Balachandran, B.: Noise influenced elastic cantilever dynamics with nonlinear tip interaction forces. Nonlinear Dyn. 66(3), 427–439 (2011)

    Article  MathSciNet  Google Scholar 

  25. Nayfeh, A.H., Balachandran, B.: Applied Nonlinear Dynamics. Wiley, New York (1995)

    Book  MATH  Google Scholar 

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Acknowledgements

Support received through NSF grant No. CMMI-0800741 is gratefully acknowledged. The authors thank Mr. G. Chawla and Professor S. Solares of Mechanical Engineering, University of Maryland, for helping with the AFM experiments.

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  1. Department of Mechanical Engineering, University of Maryland, College Park, MD, 20742, USA

    Ishita Chakraborty & B. Balachandran

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  1. Ishita Chakraborty
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  2. B. Balachandran
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Correspondence to B. Balachandran.

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Chakraborty, I., Balachandran, B. Near-grazing dynamics of base excited cantilevers with nonlinear tip interactions. Nonlinear Dyn 70, 1297–1310 (2012). https://doi.org/10.1007/s11071-012-0534-8

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  • DOI: https://doi.org/10.1007/s11071-012-0534-8

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