Skip to main content
SpringerLink
Log in

Direct Multiscale Analysis of Stability of an Axially Moving Functionally Graded Beam with Time-Dependent Velocity

  • Published:
Acta Mechanica Solida Sinica Aims and scope Submit manuscript

Abstract

In this study, the transverse vibration of a traveling beam made of functionally graded material was analyzed. The material gradation was assumed to vary continuously along the thickness direction of the beam in the form of power law exponent. The effect of the longitudinally varying tension due to axial acceleration was highlighted, and the dependence of the tension on the finite support rigidity was also considered. A complex governing equation of the functionally graded beam was derived by the Hamilton principle, in which the geometric nonlinearity, material properties and axial load were incorporated. The direct multiscale method was applied to the analysis process of an axially moving functionally graded beam with time-dependent velocity, and the natural frequency and solvability conditions were obtained. Based on the conditions, the stability boundaries of subharmonic resonance and combination resonance were obtained. It was found that the dynamic behavior of axial moving beams could be tuned by using the distribution law of the functional gradient parameters.

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

Similar content being viewed by others

References

  1. Öz HR, Pakdemirli M. Vibrations of an Axially moving beam with time-dependent velocity. J Sound Vib. 1999;227(2):239–57.

    Article  Google Scholar 

  2. Zhang YW, Hou S, Xu KF, et al. Forced vibration control of an axially moving beam with an attached nonlinear energy sink. Acta Mech Solida Sin. 2017;30:674–82.

    Article  Google Scholar 

  3. Tang YQ, Zhang YX, Yang XD, et al. On Parametric Instability Boundaries of Axially Moving Beams with Internal Resonance. Acta Mech Solida Sin. 2018;31(4):470–83.

    Article  Google Scholar 

  4. Yang TZ, Fang B, Yang XD, et al. Closed-form approximate solution for natural frequency of axially moving beams. Int J Mech Sci. 2013;74:154–60.

    Article  Google Scholar 

  5. Wickert JA. Non-linear vibration of a traveling tensioned beam. Int J Non Linear Mech. 1992;27(3):503–17.

    Article  MATH  Google Scholar 

  6. Ding H, Huang LL, Mao XY, et al. Primary resonance of traveling viscoelastic beam under internal resonance. Appl Math Mech. 2017;38(1):1–14.

    Article  MathSciNet  MATH  Google Scholar 

  7. Thai HT, Vo TP. Bending and free vibration of functionally graded beams using various higher-order shear deformation beam theories. Int J Mech Sci. 2012;62(1):57–66.

    Article  Google Scholar 

  8. Wang X, Li S. Free vibration analysis of functionally graded material beams based on Levinson beam theory. Appl Math Mech (Engl Ed). 2016;37(7):861–78.

    Article  MathSciNet  MATH  Google Scholar 

  9. Pradhan KK, Chakraverty S. Effects of different shear deformation theories on free vibration of functionally graded beams. Int J Mech Sci. 2014;82(5):149–60.

    Article  Google Scholar 

  10. Jing LL, Ming PJ, Zhang WP, Fu LR, Cao YP. Static and free vibration analysis of functionally graded beams by combination Timoshenko theory and finite volume method. Compos Struct. 2016;138:192–213.

    Article  Google Scholar 

  11. Ke LL, Wang YS. Size effect on dynamic stability of functionally graded microbeams based on a modified couple stress theory. Compos Struct. 2011;93(2):342–50.

    Article  Google Scholar 

  12. Asghari M, Ahmadian MT, Kahrobaiyan MH, Rahaeifard M. On the size-dependent behavior of functionally graded micro-beams. Mater Des. 2010;31(5):2324–9.

    Article  Google Scholar 

  13. Ansari R, Gholami R, Sahmani S. Free vibration analysis of size-dependent functionally graded microbeams based on the strain gradient Timoshenko beam theory. Compos Struct. 2011;94(1):221–8.

    Article  Google Scholar 

  14. Tang Y, Yang T. Post-buckling behavior and nonlinear vibration analysis of a fluid-conveying pipe composed of functionally graded material. Compos Struct. 2018;185:393–400.

    Article  Google Scholar 

  15. Şimşek M, Kocatürk T. Free and forced vibration of a functionally graded beam subjected to a concentrated moving harmonic load. Compos Struct. 2009;90(4):465–73.

    Article  Google Scholar 

  16. Zhong Z, Yu T. Analytical solution of a cantilever functionally graded beam. Compos Sci Technol. 2007;67(3–4):481–8.

    Article  Google Scholar 

  17. Chakraborty A, Gopalakrishnan S. A spectrally formulated finite element for wave propagation analysis in functionally graded beams. Int J Solids Struct. 2003;40(10):2421–48.

    Article  MATH  Google Scholar 

  18. Sankar BV. An elasticity solution for functionally graded beams. Compos Sci Technol. 2001;61(5):689–96.

    Article  Google Scholar 

  19. Lv Z, Liu H. Nonlinear bending response of functionally graded nanobeams with material uncertainties. Int J Mech Sci. 2017;134:123–35.

    Article  Google Scholar 

  20. Fallah A, Aghdam MM. Nonlinear free vibration and post-buckling analysis of functionally graded beams on nonlinear elastic foundation. Eur J Mech. 2011;30(4):571–83.

    Article  MATH  Google Scholar 

  21. Sharabiani PA, Yazdi MR. Nonlinear free vibrations of functionally graded nanobeams with surface effects. Compos B Eng. 2013;45(1):581–6.

    Article  Google Scholar 

  22. Niknam H, Fallah A, Aghdam MM. Nonlinear bending of functionally graded tapered beams subjected to thermal and mechanical loading. Int J Non Linear Mech. 2014;65:141–7.

    Article  Google Scholar 

  23. Chen X, Zhang X, Lu Y, et al. Static and dynamic analysis of the postbuckling of bi-directional functionally graded material microbeams. Int J Mech Sci. 2019;151:424–43.

    Article  Google Scholar 

  24. Robinson MT, Adali S. Buckling of nonuniform and axially functionally graded nonlocal Timoshenko nanobeams on Winkler–Pasternak foundation. Compos Struct. 2018;206:95–103.

    Article  Google Scholar 

  25. Zhou X, Dai H, Wang L, et al. Dynamics of axially functionally graded cantilevered pipes conveying fluid. Compos Struct. 2018;190:112–8.

    Article  Google Scholar 

  26. An C, Su J. Dynamic behavior of axially functionally graded pipes conveying fluid. Math Probl Eng. 2017;6789634.

  27. Ke LL, Yang J, Kitipornchai S. An analytical study on the nonlinear vibration of functionally graded beams. Meccanica. 2010;45(6):743–52.

    Article  MathSciNet  MATH  Google Scholar 

  28. Rezaiee-Pajand M, Hozhabrossadati SM. Analytical and numerical method for free vibration of double-axially functionally graded beams. Compos Struct. 2016;152:488–98.

    Article  Google Scholar 

  29. Mote CD. A study of band saw vibrations. J Frankl Inst Eng Appl Math. 1965;279(6):430–44.

    Article  Google Scholar 

  30. Sahu S, Mitra A. Non-linear forced vibration of axially functionally graded tapered beam with different end conditions. In: International conference on computer modeling and simulation; 2014. p. 560–9.

  31. Chen LQ, Tang YQ. Combination and principal parametric resonances of axially accelerating viscoelastic beams: recognition of longitudinally varying tensions. J Sound Vib. 2011;330(23):5598–614.

    Article  Google Scholar 

  32. Mao XY, Ding H, Chen LQ. Forced vibration of axially moving beam with internal resonance in the supercritical regime. Int J Mech Sci. 2017;131–132:81–94.

    Article  Google Scholar 

  33. Wang L, Hu Z, Zhong Z. Non-linear dynamical analysis for an axially moving beam with finite deformation. Int J Non Linear Mech. 2013;54(3):5–21.

    Article  Google Scholar 

  34. Ghayesh MH. Nonlinear forced dynamics of an axially moving viscoelastic beam with an internal resonance. Int J Mech Sci. 2011;53(11):1022–37.

    Article  Google Scholar 

  35. Chen LQ, Yang XD, Cheng CJ. Dynamic stability of an axially accelerating viscoelastic beam. Eur J Mech. 2004;23(4):659–66.

    Article  MATH  Google Scholar 

  36. Chen SH, Huang JL, Sze KY. Multidimensional Lindstedt–Poincaré method for nonlinear vibration of axially moving beams. J Sound Vib. 2007;306(1–2):1–11.

    Article  Google Scholar 

  37. Mao XY, Ding H, Chen LQ. Internal resonance of a supercritically axially moving beam subjected to the pulsating speed. Nonlinear Dyn. 2019;95(1):631–51.

    Article  Google Scholar 

Download references

Acknowledgements

The authors acknowledge the support of National Natural Science Foundation of China (Nos. 11672187 and 11572182), Natural Science Foundation of Liaoning Province (201602573), the Key Research Projects of Shanghai Science and Technology Commission (No. 18010500100), Innovation Program of Shanghai Education Commission (No. 2017-01-07-00-09-E00019) and Beiyang Young Scholars of Tianjin University (2019XRX-0027).

Author information

Authors and Affiliations

  1. Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai, 200072, China

    Ting Yan & Liqun Chen

  2. Department of Mechanics, Shanghai University, Shanghai, 200444, China

    Ting Yan & Liqun Chen

  3. Department of Mechanics, Tianjin University, Tianjin, 300072, China

    Tianzhi Yang

  4. Tianjin Key Laboratory of Nonlinear Dynamics and Control, Tianjin, 300072, China

    Tianzhi Yang

Authors
  1. Ting Yan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tianzhi Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Liqun Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tianzhi Yang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yan, T., Yang, T. & Chen, L. Direct Multiscale Analysis of Stability of an Axially Moving Functionally Graded Beam with Time-Dependent Velocity. Acta Mech. Solida Sin. 33, 150–163 (2020). https://doi.org/10.1007/s10338-019-00140-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10338-019-00140-4

Keywords

Search

Navigation