Dynamic transverse vibration characteristics of nonuniform nonlocal strain gradient beams using the generalized differential quadrature method

Regular Article
  • 31 Downloads

Abstract.

In this study, vibration response of small-scale tapered beams is investigated. Size-dependent effects are modeled in the framework of the nonlocal strain gradient theory and three different types of linearly cross-section variation are proposed by having width variation, thickness variation and a combination of them both. The small-scale beam is formulated using Euler-Bernoulli beam theory, nonlocal strain gradient theory and Hamilton’s principle. Equations of motion for all three types of nonuniformity are solved using the generalized differential quadrature method (GDQM). Results are presented and compared to those achieved for simplified models and effects of having different slenderness ratio are presented. Moreover, a comprehensive parametric study is proposed and the effects of varying nonuniformity terms, nonlocal and strain gradient parameters are precisely studied. Accordingly, with the vast application of tapered small-scale beams in many devices, especially scanning tunneling microscopes (STM) and atomic force microscopes (AFM), this study could be a step forward in understanding, predicting and controlling such kind of behaviors.

References

  1. 1.
    H. Zhang, C.M. Wang, E. Ruocco, N. Challamel, Eng. Struct. 126, 252 (2016)CrossRefGoogle Scholar
  2. 2.
    S.M. Abdelghany, K.M. Ewis, A.A. Mahmoud, M.M. Nassar, Beni-Suef Univ. J. Basic Appl. Sci. 4, 192 (2015)CrossRefGoogle Scholar
  3. 3.
    H. Malaeke, H. Moeenfard, J. Sound Vib. 366, 211 (2016)ADSCrossRefGoogle Scholar
  4. 4.
    M. Ahmadi, A. Nikkhoo, Appl. Math. Modell. 38, 2130 (2014)CrossRefGoogle Scholar
  5. 5.
    W. He, S.S. Ge, IEEE/ASME Trans. Mechatron. 20, 237 (2015)CrossRefGoogle Scholar
  6. 6.
    J.R. Banerjee, J. Phys.: Conf. Ser. 721, 012005 (2016)Google Scholar
  7. 7.
    Z. Qian, Y. Hui, F. Liu, S. Kar, M. Rinaldi, Single Transistor Oscillator Based on a Graphene-Aluminum Nitride Nano Plate Resonator, in Proceedings of the 2013 IEEE International Frequency Control Symposium (IFCS 2013), Prague, Czech Republic (IEEE, 2013) pp. 559--561Google Scholar
  8. 8.
    X. Tong, G.A. DiLabio, O.J. Clarkin, R.A. Wolkow, Nano Lett. 4, 357 (2004)ADSCrossRefGoogle Scholar
  9. 9.
    R. Reddy, B.R. Dorvel, J. Go, P.R. Nair, O.H. Elibol, G.M. Credo, J.S. Daniels, E.K.C. Chow, X. Su, M. Varma, M.A. Alam, R. Bashir, Biomed. Microdev. 13, 335 (2011)CrossRefGoogle Scholar
  10. 10.
    Y. Zhang, G. Chang, S. Liu, W. Lu, J. Tian, X. Sun, Biosensors Bioelectron. 28, 344 (2011)CrossRefGoogle Scholar
  11. 11.
    J. Ding, K. Zhang, G. Wei, Z. Su, RSC Adv. 5, 69745 (2015)CrossRefGoogle Scholar
  12. 12.
    Z. Jing, J. Zhan, Adv. Mater. 20, 4547 (2008)CrossRefGoogle Scholar
  13. 13.
    E. Detsri, Chin. Chem. Lett. 27, 1635 (2016)CrossRefGoogle Scholar
  14. 14.
    X. Tang, K.W.C. Lai, Quantitative Study of AFM-based Nanopatterning of Graphene Nanoplate, in 14th IEEE International Conference on Nanotechnology (IEEE, 2014) pp. 54--57Google Scholar
  15. 15.
    W. Jeong, M. Lee, H. Lee, H. Lee, B. Kim, J.Y. Park, Nanotechnology 27, 215601 (2016)ADSCrossRefGoogle Scholar
  16. 16.
    T. Nan, Y. Hui, M. Rinaldi, N.X. Sun, Sci. Rep. 3, 1985 (2013)ADSCrossRefGoogle Scholar
  17. 17.
    Y. Hui, J.S. Gomez-Diaz, Z. Qian, A. Alu, M. Rinaldi, Nat. Commun. 7, 11249 (2016)ADSCrossRefGoogle Scholar
  18. 18.
    E. Kröner, Int. J. Solids Struct. 3, 731 (1967)CrossRefGoogle Scholar
  19. 19.
    A.C. Eringen, J. Appl. Phys. 54, 4703 (1983)ADSCrossRefGoogle Scholar
  20. 20.
    A.C. Eringen, Nonlocal Continuum Field Theories (Springer-Verlag, New York, 2002) https://doi.org/10.1007/b97697
  21. 21.
    J.N. Reddy, Int. J. Eng. Sci. 45, 288 (2007)CrossRefGoogle Scholar
  22. 22.
    H.B. Khaniki, S. Hosseini-Hashemi, Int. J. Eng. Sci. 115, 51 (2017)CrossRefGoogle Scholar
  23. 23.
    C.M. Wang, Y.Y. Zhang, X.Q. He, Nanotechnology 18, 105401 (2007)ADSCrossRefGoogle Scholar
  24. 24.
    S.H. Hashemi, H.B. Khaniki, J. Mech. 33, 559 (2017)CrossRefGoogle Scholar
  25. 25.
    S.H. Hashemi, H.B. Khaniki, Alex. Eng. J. (2017) https://doi.org/10.1016/j.aej.2016.12.015
  26. 26.
    S.C. Pradhan, J.K. Phadikar, J. Sound Vib. 325, 206 (2009)ADSCrossRefGoogle Scholar
  27. 27.
    T. Murmu, S.C. Pradhan, Physica E 41, 1451 (2009)ADSCrossRefGoogle Scholar
  28. 28.
    J. Aranda-Ruiz, J. Loya, J. Fernández-Sáez, Compos. Struct. 94, 2990 (2012)CrossRefGoogle Scholar
  29. 29.
    S.H. Hashemi, H.B. Khaniki, Int. J. Eng. Trans. B: Appl. 29, 688 (2016)Google Scholar
  30. 30.
    S.H. Hashemi, H.B. Khaniki, Int. J. Nano Dimens. 8, 70 (2017)Google Scholar
  31. 31.
    J. Aranda-Ruiz, J. Loya, J. Fernández-Sáez, Compos. Struct. 94, 2990 (2012)CrossRefGoogle Scholar
  32. 32.
    G. Romano, R. Barretta, M. Diaco, F.M. de Sciarra, Int. J. Mech. Sci. 121, 151 (2017)CrossRefGoogle Scholar
  33. 33.
    G. Romano, R. Barretta, Int. J. Eng. Sci. 109, 240 (2016)CrossRefGoogle Scholar
  34. 34.
    G. Romano, R. Barretta, Int. J. Eng. Sci. 115, 14 (2017)CrossRefGoogle Scholar
  35. 35.
    G. Romano, R. Barretta, Compos. Part B: Eng. 114, 184 (2017)CrossRefGoogle Scholar
  36. 36.
    A. Apuzzo, R. Barretta, R. Luciano, F.M. de Sciarra, R. Penna, Compos. Part B: Eng. 123, 105 (2017)CrossRefGoogle Scholar
  37. 37.
    G. Romano, R. Barretta, M. Diaco, Int. J. Mech. Sci. 131, 490 (2017)CrossRefGoogle Scholar
  38. 38.
    R.M. Bergman, J. Appl. Math. Mech. 32, 1085 (1968)CrossRefGoogle Scholar
  39. 39.
    R.A. Toupin, Arch. Ration. Mech. Anal. 11, 385 (1962)MathSciNetCrossRefGoogle Scholar
  40. 40.
    R.D. Mindlin, H.F. Tiersten, Arch. Ration. Mech. Anal. 11, 415 (1962)CrossRefGoogle Scholar
  41. 41.
    N.A. Fleck, G.M. Muller, M.F. Ashby, J.W. Hutchinson, Acta Metallur. Mater. 42, 475 (1994)CrossRefGoogle Scholar
  42. 42.
    D.C.C. Lam, F. Yang, A.C.M. Chong, J. Wang, P. Tong, J. Mech. Phys. Solids 51, 1477 (2003)ADSCrossRefGoogle Scholar
  43. 43.
    H.M. Ma, X.L. Gao, J.N. Reddy, J. Mech. Phys. Solids 56, 3379 (2008)ADSMathSciNetCrossRefGoogle Scholar
  44. 44.
    H.B. Khaniki, S. Hosseini-Hashemi, Eur. Phys. J. Plus 132, 200 (2017)CrossRefGoogle Scholar
  45. 45.
    S.K. Park, X.L. Gao, J. Micromech. Microeng. 16, 2355 (2006)ADSCrossRefGoogle Scholar
  46. 46.
    B. Akgöz, Ö. Civalek, Int. J. Eng. Sci. 49, 1268 (2011)CrossRefGoogle Scholar
  47. 47.
    B. Akgöz, Ö. Civalek, Struct. Eng. Mech. 48, 195 (2013)CrossRefGoogle Scholar
  48. 48.
    B. Akgöz, Ö. Civalek, Compos. Struct. 98, 314 (2013)CrossRefGoogle Scholar
  49. 49.
    G. Romano, R. Barretta, M. Diaco, Continuum Mech. Thermodyn. 28, 1659 (2016)ADSMathSciNetCrossRefGoogle Scholar
  50. 50.
    C.W. Lim, G. Zhang, J.N. Reddy, J. Mech. Phys. Solids 78, 298 (2015)ADSMathSciNetCrossRefGoogle Scholar
  51. 51.
    F. Ebrahimi, M.R. Barati, J. Vib. Control (2016) https://doi.org/10.1177/1077546316678511
  52. 52.
    L. Li, Y. Hu, L. Ling, Physica E 75, 118 (2016)ADSCrossRefGoogle Scholar
  53. 53.
    H. Zeighampour, Y.T. Beni, I. Karimipour, Microfluid. Nanofluid. 21, 85 (2017)CrossRefGoogle Scholar
  54. 54.
    L. Lu, X. Guo, J. Zhao, Int. J. Eng. Sci. 116, 12 (2017)CrossRefGoogle Scholar
  55. 55.
    X.J. Xu, X.C. Wang, M.L. Zheng, Z. Ma, Compos. Struct. 160, 366 (2017)CrossRefGoogle Scholar
  56. 56.
    L. Li, Y. Hu, X. Li, Int. J. Mech. Sci. 115, 135 (2016)CrossRefGoogle Scholar
  57. 57.
    L. Li, Y. Hu, Int. J. Eng. Sci. 97, 84 (2015)CrossRefGoogle Scholar
  58. 58.
    T.Y. Wu, G.R. Liu, Comput. Mech. 24, 197 (1999)MathSciNetCrossRefGoogle Scholar
  59. 59.
    T.Y. Wu, G.R. Liu, Int. J. Numer. Methods Biomed. Eng. 16, 777 (2000)Google Scholar
  60. 60.
    L. Li, X. Li, Y. Hu, Int. J. Eng. Sci. 102, 77 (2016)CrossRefGoogle Scholar
  61. 61.
    C.M. Wang, Y.Y. Zhang, X.Q. He, Nanotechnology 18, 105401 (2007)ADSCrossRefGoogle Scholar

Copyright information

© Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Impact Research Laboratory, Department of Mechanical EngineeringIran University of Science and TechnologyTehranIran
  2. 2.Center of Excellence in Railway TransportationIran University of Science and TechnologyTehranIran

Personalised recommendations