Skip to main content
SpringerLink
Log in

Nonlinear vibration analysis of fluid-conveying microtubes

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

Non-classical beam theory is employed to study higher modes of free vibration of a microtube conveying fluid. The modified couple stress theory is utilized to model the size-dependent vibration of the microtube structure. It is proposed that the midplane stretching of the microtube needs to be taken into account, and subsequently, the nonlinear partial differential equation of motion in a non-dimensional form has been derived. Three modes of vibration are considered in this study, and the Galerkin procedure is utilized to obtain the nonlinear equations of motion. Analytical expressions for the nonlinear frequencies are developed, and the time responses of the structural model employing the variational iteration method are presented. Results obtained from the analytical procedure were compared with those computed by using numerical method, and close agreements are observed. A parametric sensitivity study is carried out to evaluate the performance and the accuracy of the proposed analytical method from an engineering application point of view.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

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

Similar content being viewed by others

References

  1. Wang, L.: Size-dependent vibration characteristics of fluid-conveying microtubes. J. Fluids Struct. 26, 675–684 (2010)

    Article  Google Scholar 

  2. Ke, L.-L., Wang, Y.-S., Yang, J., Kitipornchai, S.: Nonlinear free vibration of size-dependent functionally graded microbeams. Int. J. Eng. Sci. 50, 256–267 (2012)

    Article  MathSciNet  Google Scholar 

  3. Ke, L.-L., Wang, Y.-S.: Size effect on dynamic stability of functionally graded microbeams based on a modified couple stress theory. Compos. Struct. 93, 342–350 (2011)

    Article  Google Scholar 

  4. Alibert, J.-J., Seppecher, P., Dell’Isola, F.: Truss modular beams with deformation energy depending on higher displacement gradients. Math. Mech. Solids 8, 51–73 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  5. Şimşek, M.: Dynamic analysis of an embedded microbeam carrying a moving microparticle based on the modified couple stress theory. Int. J. Eng. Sci. 48, 1721–1732 (2010)

    Article  MATH  Google Scholar 

  6. Wang, Y.-G., Lin, W.-H., Liu, N.: Nonlinear free vibration of a microscale beam based on modified couple stress theory. Phys. E: Low-dimens. Syst. Nanostruct. 47, 80–85 (2013)

    Article  Google Scholar 

  7. Kong, S., Zhou, S., Nie, Z., Wang, K.: The size-dependent natural frequency of Bernoulli–Euler micro-beams. Int. J. Eng. Sci. 46, 427–437 (2008)

    Article  MATH  Google Scholar 

  8. Asghari, M., Kahrobaiyan, M., Ahmadian, M.: A nonlinear Timoshenko beam formulation based on the modified couple stress theory. Int. J. Eng. Sci. 48, 1749–1761 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chen, W.J., Li, X.P.: Size-dependent free vibration analysis of composite laminated Timoshenko beam based on new modified couple stress theory. Arch. Appl. Mech. 83, 431–444 (2013)

    Article  MATH  Google Scholar 

  10. Ke, L.-L., Wang, Y.-S.: Flow-induced vibration and instability of embedded double-walled carbon nanotubes based on a modified couple stress theory. Phys. E: Low-dimens. Syst. Nanostruct. 43, 1031–1039 (2011)

    Article  Google Scholar 

  11. Ma, H., Gao, X.-L., Reddy, J.: A microstructure-dependent Timoshenko beam model based on a modified couple stress theory. J. Mech. Phys. Solids 56, 3379–3391 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  12. Şimşek, M., Reddy, J.: Bending and vibration of functionally graded microbeams using a new higher order beam theory and the modified couple stress theory. Int. J. Eng. Sci. 64, 37–53 (2013)

    Article  MathSciNet  Google Scholar 

  13. Yang, T.-Z., Ji, S., Yang, X.-D., Fang, B.: Microfluid-induced nonlinear free vibration of microtubes. Int. J. Eng. Sci. 76, 47–55 (2014)

    Article  Google Scholar 

  14. Ahangar, S., Rezazadeh, G., Shabani, R., Ahmadi, G., Toloei, A.: On the stability of a microbeam conveying fluid considering modified couple stress theory. Int. J. Mech. Mater. Des. 7, 327–342 (2011)

    Article  Google Scholar 

  15. Eremeyev, V.A., Ivanova, E.A., Morozov, N.F.: On free oscillations of an elastic solids with ordered arrays of nano-sized objects. Continuum Mech. Thermodyn. 27, 583–607 (2015)

    Article  MathSciNet  Google Scholar 

  16. Seppecher, P., Alibert, J.-J., Isola, F.D.: Linear elastic trusses leading to continua with exotic mechanical interactions. J. Phys.: Conf. Ser. 319, 012018 (2011)

    Google Scholar 

  17. Wang, L., Xu, Y., Ni, Q.: Size-dependent vibration analysis of three-dimensional cylindrical microbeams based on modified couple stress theory: A unified treatment. Int. J. Eng. Sci. 68, 1–10 (2013)

    Article  MathSciNet  Google Scholar 

  18. Komijani, M., Reddy, J., Ferreira, A.: Nonlinear stability and vibration of pre/post-buckled microstructure-dependent FGPM actuators. Meccanica 49, 2729–2745 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  19. Akgöz, B., Civalek, Ö.: Free vibration analysis of axially functionally graded tapered Bernoulli–Euler microbeams based on the modified couple stress theory. Compos. Struct. 98, 314–322 (2013)

    Article  Google Scholar 

  20. Salamat-talab, M., Nateghi, A., Torabi, J.: Static and dynamic analysis of third-order shear deformation FG micro beam based on modified couple stress theory. Int. J. Mech. Sci. 57, 63–73 (2012)

    Article  Google Scholar 

  21. Xia, W., Wang, L., Yin, L.: Nonlinear non-classical microscale beams: static bending, postbuckling and free vibration. Int. J. Eng. Sci. 48, 2044–2053 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  22. Ma, H., Gao, X.-L., Reddy, J.: A nonclassical Reddy–Levinson beam model based on a modified couple stress theory. Int. J. Multiscale Comput. Eng. 8, 167–180 (2010)

  23. Kahrobaiyan, M., Asghari, M., Rahaeifard, M., Ahmadian, M.: A nonlinear strain gradient beam formulation. Int. J. Eng. Sci. 49, 1256–1267 (2011)

    Article  MathSciNet  Google Scholar 

  24. Salamat-talab, M., Shahabi, F., Assadi, A.: Size dependent analysis of functionally graded microbeams using strain gradient elasticity incorporated with surface energy. Appl. Math. Model. 37, 507–526 (2013)

    Article  MathSciNet  Google Scholar 

  25. Nateghi, A., Salamat-talab, M., Rezapour, J., Daneshian, B.: Size dependent buckling analysis of functionally graded micro beams based on modified couple stress theory. Appl. Math. Model. 36, 4971–4987 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  26. Nateghi, A., Salamat-talab, M.: Thermal effect on size dependent behavior of functionally graded microbeams based on modified couple stress theory. Compos. Struct. 96, 97–110 (2013)

    Article  Google Scholar 

  27. Ansari, R., Gholami, R., Shojaei, M.F., Mohammadi, V., Sahmani, S.: Size-dependent bending, buckling and free vibration of functionally graded Timoshenko microbeams based on the most general strain gradient theory. Compos. Struct. 100, 385–397 (2013)

    Article  Google Scholar 

  28. Ramezani, S.: A micro scale geometrically non-linear Timoshenko beam model based on strain gradient elasticity theory. Int. J. Non-Linear Mech. 47, 863–873 (2012)

    Article  Google Scholar 

  29. Asghari, M., Kahrobaiyan, M., Nikfar, M., Ahmadian, M.: A size-dependent nonlinear Timoshenko microbeam model based on the strain gradient theory. Acta Mech. 223, 1233–1249 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  30. Ansari, R., Gholami, R., Sahmani, S.: Study of small scale effects on the nonlinear vibration response of functionally graded Timoshenko microbeams based on the strain gradient theory. J. Comput. Nonlinear Dyn. 7, 031009 (2012)

    Article  Google Scholar 

  31. Zhao, J., Zhou, S., Wang, B., Wang, X.: Nonlinear microbeam model based on strain gradient theory. Appl. Math. Model. 36, 2674–2686 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  32. Ghayesh, M.H., Amabili, M., Farokhi, H.: Nonlinear forced vibrations of a microbeam based on the strain gradient elasticity theory. Int. J. Eng. Sci. 63, 52–60 (2013)

    Article  MathSciNet  Google Scholar 

  33. Younesian, D., Sadri, M., Esmailzadeh, E.: Primary and secondary resonance analyses of clamped–clamped micro-beams. Nonlinear Dyn. 76, 1867–1884 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  34. Panda, L., Kar, R.: Nonlinear dynamics of a pipe conveying pulsating fluid with parametric and internal resonances. Nonlinear Dyn. 49, 9–30 (2007)

    Article  MATH  Google Scholar 

  35. Zhang, Y.-L., Chen, L.-Q.: Internal resonance of pipes conveying fluid in the supercritical regime. Nonlinear Dyn. 67, 1505–1514 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  36. Ni, Q., Wang, Y., Tang, M., Luo, Y., Yan, H., Wang, L.: Nonlinear impacting oscillations of a fluid-conveying pipe subjected to distributed motion constraints. Nonlinear Dyn. 81, 893–906 (2015)

    Article  Google Scholar 

  37. Altenbach, H., Eremeyev, V.A.: On the shell theory on the nanoscale with surface stresses. Int. J. Eng. Sci. 49, 1294–1301 (2011)

    Article  MathSciNet  Google Scholar 

  38. Eremeyev, V.A., Altenbach, H.: Equilibrium of a second-gradient fluid and an elastic solid with surface stresses. Meccanica 49, 2635–2643 (2014)

  39. Kuang, Y., He, X., Chen, C., Li, G.: Analysis of nonlinear vibrations of double-walled carbon nanotubes conveying fluid. Comput. Mater. Sci. 45, 875–880 (2009)

    Article  Google Scholar 

  40. Rinaldi, S., Prabhakar, S., Vengallatore, S., Païdoussis, M.P.: Dynamics of microscale pipes containing internal fluid flow: damping, frequency shift, and stability. J. Sound Vib. 329, 1081–1088 (2010)

    Article  Google Scholar 

  41. Fu, Y., Zhang, J.: Modeling and analysis of microtubules based on a modified couple stress theory. Phys. E: Low-dimens. Syst. Nanostruct. 42, 1741–1745 (2010)

    Article  Google Scholar 

  42. He, J.-H.: Variational iteration method-a kind of non-linear analytical technique: some examples. Int. J. Non-Linear Mech. 34, 699–708 (1999)

    Article  MATH  Google Scholar 

  43. He, J.: A new approach to nonlinear partial differential equations. Commun. Nonlinear Sci. Numer. Simul. 2, 230–235 (1997)

    Article  Google Scholar 

  44. He, J.-H.: Variational iteration method for autonomous ordinary differential systems. Appl. Math. Comput. 114, 115–123 (2000)

    MathSciNet  MATH  Google Scholar 

  45. Askari, H., Saadatnia, Z., Esmailzadeh, E., Younesian, D.: Multi-frequency excitation of stiffened triangular plates for large amplitude oscillations. J. Sound Vib. 333, 5817–5835 (2014)

    Article  Google Scholar 

  46. Sadri, M., Younesian, D.: Nonlinear harmonic vibration analysis of a plate-cavity system. Nonlinear Dyn. 74, 1267–1279 (2013)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Science and Applied Science, University of Ontario Institute of Technology, Oshawa, ON, L1H 7K4, Canada

    Shamim Mashrouteh & Ebrahim Esmailzadeh

  2. Center of Excellence in Railway Transportation, School of Railway Engineering, Iran University of Science and Technology, Tehran, 16846-13114, Iran

    Mehran Sadri & Davood Younesian

  3. Faculty of Engineering and Applied Science, University of Ontario Institute of Technology, Oshawa, ON, L1H 7K4, Canada

    Davood Younesian

Authors
  1. Shamim Mashrouteh
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mehran Sadri
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Davood Younesian
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Ebrahim Esmailzadeh
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ebrahim Esmailzadeh.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mashrouteh, S., Sadri, M., Younesian, D. et al. Nonlinear vibration analysis of fluid-conveying microtubes. Nonlinear Dyn 85, 1007–1021 (2016). https://doi.org/10.1007/s11071-016-2739-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-016-2739-8

Keywords

Search

Navigation