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Parameter sensitivity of cantilever beam with tip mass to parametric excitation

  • Vamsi C. Meesala
  • Muhammad R. Hajj
Original Paper
  • 33 Downloads

Abstract

The sensitivity of the response of a parametrically excited cantilever beam with a tip mass to small variations in elasticity (stiffness) and the tip mass is performed. The governing equation of the first mode is derived, and method of multiple scales is used to determine the approximate solution based on the order of the expected variations. We demonstrate that the system can be designed so that small variations in either stiffness or tip mass can alter the type of bifurcation. Notably, we show that the response of a system designed for a supercritical bifurcation can change to yield a subcritical bifurcation with small variations in the parameters. Although such a trend is usually undesired, we argue that it can be used to detect small variations induced by fatigue or small mass depositions in sensing applications.

Keywords

Parametric excitation Cantilever beam-mass systems Sensitivity analysis Uncertainty quantification Mass/gas sensing Damage detection 

Notes

Acknowledgements

The authors would like to thank Dr. Mohammad Younis for sharing his expertise and providing valuable suggestions. The first author would also like to thank Dr. Ayoub Boroujeni for providing useful insights on carbon fiber/epoxy resin composites.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

References

  1. 1.
    Dwivedy, S., Kar, R.: Non-linear dynamics of a slender beam carrying a lumped mass under principal parametric resonance with three-mode interactions. Int. J. Non-Linear Mech. 36(6), 927–945 (2001)CrossRefzbMATHGoogle Scholar
  2. 2.
    Dwivedy, S., Kar, R.: Simultaneous combination and 1: 3: 5 internal resonances in a parametrically excited beam-mass system. Int. J. Non-Linear Mech. 38(4), 585–596 (2003)CrossRefzbMATHGoogle Scholar
  3. 3.
    To, C.: The response of mast antenna structures to transient disturbances. Ph.D. thesis, University of Southampton (1979)Google Scholar
  4. 4.
    To, C.: Vibration of a cantilever beam with a base excitation and tip mass. J. Sound Vib. 83(4), 445–460 (1982)CrossRefGoogle Scholar
  5. 5.
    Kim, K., Strganac, T.W.: Nonlinear responses of a cantilever wing with an external store. In: 44th AIAA Structures, Structural Dynamics, and Materials Conference, Norfolk, VA, AIAA Paper, no. 2003-1708 (2003)Google Scholar
  6. 6.
    Beran, P.S., Strganac, T.W., Kim, K., Nichkawde, C.: Studies of store-induced limit-cycle oscillations using a model with full system nonlinearities. Nonlinear Dyn. 37(4), 323–339 (2004)CrossRefzbMATHGoogle Scholar
  7. 7.
    Abbas, L., Chen, Q., Marzocca, P., Milanese, A.: Non-linear aeroelastic investigations of store (s)-induced limit cycle oscillations. Proc. Inst. Mech. Eng. Part G J. Aerosp. Eng. 222(1), 63–80 (2008)CrossRefGoogle Scholar
  8. 8.
    Nayfeh, A., Hammad, B., Hajj, M.: Discretization effects on flutter aspects and control of wing/store configurations. J. Vib. Control 18(7), 1043–1055 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Nayfeh, A.H., Ghommem, M., Hajj, M.R.: Normal form representation of the aeroelastic response of the goland wing. Nonlinear Dyn. 67(3), 1847–1861 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Na, S., Librescu, L.: Dynamic response of adaptive cantilevers carrying external stores and subjected to blast loading. J. Sound Vib. 231(4), 1039–1055 (2000)CrossRefGoogle Scholar
  11. 11.
    Shen, D., Ajitsaria, J., Choe, S.-Y., Kim, D.-J.: The optimal design and analysis of piezoelectric cantilever beams for power generation devices. MRS Online Proceedings Library Archive 888 (2005)Google Scholar
  12. 12.
    Daqaq, M.F., Stabler, C., Qaroush, Y., Seuaciuc-Osório, T.: Investigation of power harvesting via parametric excitations. J. Intell. Mater. Syst. Struct. 20(5), 545–557 (2009)CrossRefGoogle Scholar
  13. 13.
    Kim, M., Hoegen, M., Dugundji, J., Wardle, B.L.: Modeling and experimental verification of proof mass effects on vibration energy harvester performance. Smart Mater. Struct. 19(4), 045023 (2010)CrossRefGoogle Scholar
  14. 14.
    Erturk, A., Inman, D.J.: Piezoelectric Energy Harvesting. Wiley, New York (2011)CrossRefGoogle Scholar
  15. 15.
    Abdelkefi, A., Nayfeh, A., Hajj, M.: Effects of nonlinear piezoelectric coupling on energy harvesters under direct excitation. Nonlinear Dyn. 67(2), 1221–1232 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Abdelkefi, A., Yan, Z., Hajj, M.R.: Modeling and nonlinear analysis of piezoelectric energy harvesting from transverse galloping. Smart Mater. Struct. 22(2), 025016 (2013)CrossRefGoogle Scholar
  17. 17.
    Friswell, M.I., Ali, S.F., Bilgen, O., Adhikari, S., Lees, A.W., Litak, G.: Non-linear piezoelectric vibration energy harvesting from a vertical cantilever beam with tip mass. J. Intell. Mater. Syst. Struct. 23(13), 1505–1521 (2012)CrossRefGoogle Scholar
  18. 18.
    Li, X., Bao, M., Yang, H., Shen, S., Lu, D.: A micromachined piezoresistive angular rate sensor with a composite beam structure. Sens. Actuators A Phys. 72(3), 217–223 (1999)CrossRefGoogle Scholar
  19. 19.
    Yang, H., Bao, M., Yin, H., Shen, S.: A novel bulk micromachined gyroscope based on a rectangular beam-mass structure. Sens. Actuators A Phys. 96(2), 145–151 (2002)CrossRefGoogle Scholar
  20. 20.
    Bashir, R.: Biomems: state-of-the-art in detection, opportunities and prospects. Adv. Drug Deliv. Rev. 56(11), 1565–1586 (2004)CrossRefGoogle Scholar
  21. 21.
    Vashist, S.K.: A review of microcantilevers for sensing applications. J. Nanotechnol. 3, 1–18 (2007)Google Scholar
  22. 22.
    Johnson, B.N., Mutharasan, R.: Biosensing using dynamic-mode cantilever sensors: a review. Biosens. Bioelectron. 32(1), 1–18 (2012)CrossRefGoogle Scholar
  23. 23.
    Dai, M.D., Eom, K., Kim, C.-W.: Nanomechanical mass detection using nonlinear oscillations. Appl. Phys. Lett. 95(20), 203104 (2009)CrossRefGoogle Scholar
  24. 24.
    Kacem, N., Arcamone, J., Perez-Murano, F., Hentz, S.: Dynamic range enhancement of nonlinear nanomechanical resonant cantilevers for highly sensitive nems gas/mass sensor applications. J. Micromech. Microeng. 20(4), 045023 (2010)CrossRefGoogle Scholar
  25. 25.
    Zhang, W., Baskaran, R., Turner, K.L.: Effect of cubic nonlinearity on auto-parametrically amplified resonant mems mass sensor. Sens. Actuators A Phys. 102(1), 139–150 (2002)CrossRefGoogle Scholar
  26. 26.
    Zhang, W., Turner, K.L.: Application of parametric resonance amplification in a single-crystal silicon micro-oscillator based mass sensor. Sens. Actuators A Phys. 122(1), 23–30 (2005)CrossRefGoogle Scholar
  27. 27.
    Younis, M.I., Nayfeh, A.: A study of the nonlinear response of a resonant microbeam to an electric actuation. Nonlinear Dyn. 31(1), 91–117 (2003)CrossRefzbMATHGoogle Scholar
  28. 28.
    Meesala, V.C.: Modeling and analysis of a cantilever beam tip mass system, M.S. thesis, Virginia Tech. http://hdl.handle.net/10919/83378 (2018)
  29. 29.
    Nayfeh, A.H., Mook, D.T.: Nonlinear Oscillations. Wiley, New York (2008)zbMATHGoogle Scholar
  30. 30.
    Nayfeh, A.H.: Introduction to Perturbation Techniques. Wiley, New York (2011)zbMATHGoogle Scholar
  31. 31.
    Pratiher, B., Dwivedy, S.K.: Nonlinear response of a vertically moving viscoelastic beam subjected to a fluctuating contact load. Acta Mechanica 218(1–2), 65–85 (2011)CrossRefzbMATHGoogle Scholar
  32. 32.
    Pratiher, B.: Vibration control of a transversely excited cantilever beam with tip mass. Arch. Appl. Mech. 82(1), 31–42 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Rezaee, M., Fekrmandi, H.: A theoretical and experimental investigation on free vibration vehavior of a cantilever beam with a breathing crack. Shock Vib. 19(2), 175–186 (2012)CrossRefGoogle Scholar
  34. 34.
    Nayfeh, A.H.: Resolving controversies in the application of the method of multiple scales and the generalized method of averaging. Nonlinear Dyn. 40(1), 61–102 (2005)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of Biomedical Engineering and MechanicsVirginia TechBlacksburgUSA
  2. 2.Department of Civil, Environmental and Ocean EngineeringStevens Institute of TechnologyHobokenUSA

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