Skip to main content
SpringerLink
Log in

Dynamics of axially functionally graded pipes conveying fluid

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

Abstract

Pipes conveying fluid near jet engines or rocket engines always subject to gradient temperature, which results in the gradient Young’s modulus. The influence of the Young’s modulus gradient on dynamics of pipes conveying fluid is studied for the first time. The pipe is treated as an axially functional gradient (AFG) Euler–Bernoulli beam. By using the generalized Hamilton’s principle, the nonlinear partial-differential-integral governing equation of the AFG pipe conveying fluid with simply supported boundaries is established. On the basis of it, the effects of gradient Young’s modulus on the natural characteristics and the non-trivial equilibrium configuration are analyzed. To simulate the pipe directly, the differential quadrature element method (DQEM) is introduced. The harmonic balance method is carried out to solve the response analytically. In the supercritical region, the non-trivial equilibrium configuration is superposed by modal shapes of a simply supported Euler–Bernoulli beam and verified by the DQEM. The results show that the gradually varied Young’s modulus along the axial direction leads to the asymmetric non-trivial equilibrium configuration. An increasing gradient of Young’s modulus can raise the critical fluid velocity of the buckled system and weaken the vibration. Unlike the pipe in the subcritical region, the pipe in the supercritical region will generate zero shift to the response. At the same time, the pipe changes the hard characteristic to the soft one, and the non-trivial equilibrium configuration introduces more resonance peaks to the system. The results also show that under the same external excitation, the increasing Young’s modulus gradient will strengthen the nonlinearity of the response and further enlarge the asymmetry of the vibration shape. This work further complements the theory of pipes conveying fluid.

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
Fig. 17

Similar content being viewed by others

Data Availability

The datasets generated during and/or analyzed during the current study are not publicly available due to the authors have no repository but are available from the corresponding author on reasonable request.

References

  1. Ibrahim, R.A.: Overview of mechanics of pipes conveying fluids—Part I: fundamental studies. J. Pressure Vessel Technol. 132, 034001 (2010)

    Google Scholar 

  2. Gao, P.-X., Yu, T., Zhang, Y.-L., Wang, J., Zhai, J.-Y.: Vibration analysis and control technologies of hydraulic pipeline system in aircraft: a review. Chin. J. Aeronaut. 34(4), 83–114 (2021)

    Google Scholar 

  3. Ali, H.H., Mustafa, A.W., Al-Bakri, F.F.: A new control design and robustness analysis of a variable speed hydrostatic transmission used to control the velocity of a hydraulic cylinder. Int. J. Dyn. Control. 9(3), 1078–1091 (2020)

    MathSciNet  Google Scholar 

  4. Païdoussis, M.P., Issid, N.T.: Dynamic stability of pipes conveying fluid. J. Sound Vib. 33(3), 267–294 (1974)

    Google Scholar 

  5. Plaut, R.H., Huseyin, K.: Instability of fluid conveying pipes under axial load. J. Appl. Mech. 42(4), 889–890 (1975)

    Google Scholar 

  6. Matsuzaki, Y., Fung, Y.-C.: Nonlinear stability analysis of a two-dimensional model of an elastic tube conveying a compressible flow. J. Appl. Mech. 46(1), 31–36 (1979)

    MATH  Google Scholar 

  7. Awrejcewicz, J., Kry’sko, V.A., Kry’sko, A.V.: Thermo-dynamics of plates and shells. Springer Science & Business Media, Germany (2007)

    Google Scholar 

  8. Yamashita, K., Kitaura, K., Nishiyama, N., Yabuno, H.: Non-planar motions due to nonlinear interactions between unstable oscillatory modes in a cantilevered pipe conveying fluid. Mech. Syst. Signal Pr. 178, 109183 (2022)

    Google Scholar 

  9. Łuczko, J., Czerwiński, A.: Three-dimensional dynamics of curved pipes conveying fluid. J. Fluids Struct. 91, 102704 (2019)

    Google Scholar 

  10. Zhou, J., Chang, X.-P., Xiong, Z.-J., Li, Y.-H.: Stability and nonlinear vibration analysis of fluid-conveying composite pipes with elastic boundary conditions. Thin-Walled Struct. 179, 109597 (2022)

    Google Scholar 

  11. Tan, X., Ding, H., Chen, L.-Q.: Nonlinear frequencies and forced responses of pipes conveying fluid via a coupled Timoshenko model. J. Sound Vib. 455, 241–255 (2019)

    Google Scholar 

  12. Ye, S.-Q., Ding, H., Wei, S., Ji, J.-C., Chen, L.-Q.: Non-trivial equilibriums and natural frequencies of a slightly curved pipe conveying supercritical fluid. Ocean Eng. 227, 108899 (2021)

    Google Scholar 

  13. Xie, W., Liang, Z., Jiang, Z., Zhu, L.: Dynamic responses of a flexible pipe conveying variable-density fluid and experiencing cross-flow and in-line coupled vortex-induced vibrations. Ocean Eng. 260, 111811 (2022)

    Google Scholar 

  14. Javadi, M., Noorian, M.A., Irani, S.: Nonlinear vibration analysis of cracked pipe conveying fluid under primary and superharmonic resonances. Int. J. Pres. Ves. Pip. 191, 104326 (2021)

    Google Scholar 

  15. Liang, F., Gao, A., Li, X.-F., Zhu, W.-D.: Nonlinear parametric vibration of spinning pipes conveying fluid with varying spinning speed and flow velocity. Appl. Math. Model. 95, 320–338 (2021)

    MathSciNet  MATH  Google Scholar 

  16. Ghadirian, H., Mohebpour, S., Malekzadeh, P., Daneshmand, F.: Nonlinear free vibrations and stability analysis of FG-CNTRC pipes conveying fluid based on Timoshenko model. Compos. Struct. 292, 115637 (2022)

    Google Scholar 

  17. Durmus, D., Balkaya, M., Kaya, M.O.: Comparison of the free vibration analysis of a fluid-conveying hybrid pipe resting on different two-parameter elastic soils. Int. J. Pres. Ves. Pip. 193, 104479 (2021)

    Google Scholar 

  18. Xu, W.-H., Jia, K., Ma, Y.-X., Wang, Y.-Y., Song, Z.-Y.: Multispan classification methods and interaction mechanism of submarine pipelines undergoing vortex-induced vibration. Appl. Ocean Res. 120, 103027 (2022)

    Google Scholar 

  19. Jensen, J.: Articulated pipes conveying fluid pulsating with high frequency. Nonlinear Dyn. 19(2), 171–191 (1999)

    MATH  Google Scholar 

  20. Ni, Q., Luo, Y.-Y., Li, M.-W., Yan, H.: Natural frequency and stability analysis of a pipe conveying fluid with axially moving supports immersed in fluid. J. Sound Vib. 403, 173–189 (2017)

    Google Scholar 

  21. Guo, Y., Zhu, B., Li, Y.: Nonlinear dynamics of fluid-conveying composite pipes subjected to time-varying axial tension in sub- and super-critical regimes. Appl. Math. Model. 101, 632–653 (2022)

    MathSciNet  MATH  Google Scholar 

  22. Nayfeh, A.H., Lacarbonara, W., Chin, C.M.: Nonlinear normal modes of buckled beams: three-to-one and one-to-one internal resonances. Nonlinear Dyn. 18(3), 253–273 (1999)

    MathSciNet  MATH  Google Scholar 

  23. Tan, X., Mao, X.-Y., Ding, H., Chen, L.-Q.: Vibration around non-trivial equilibrium of a supercritical Timoshenko pipe conveying fluid. J. Sound Vib. 428, 104–118 (2018)

    Google Scholar 

  24. Zhang, L., Chen, F.: Multi-pulse jumping orbits and chaotic dynamics of cantilevered pipes conveying time-varying fluid. Nonlinear Dyn. 97(2), 991–1009 (2019)

    MathSciNet  MATH  Google Scholar 

  25. Mao, X.-Y., Ding, H., Lim, C.W., Chen, L.-Q.: Super-harmonic resonance and multi-frequency responses of a super-critical translating beam. J. Sound Vib. 385, 267–283 (2016)

    Google Scholar 

  26. Mao, X.-Y., Ding, H., Chen, L.-Q.: Steady-state response of a fluid-conveying pipe with 3:1 internal resonance in supercritical regime. Nonlinear Dyn. 86(2), 795–809 (2016)

    Google Scholar 

  27. Zhou, S., Yu, T.-J., Yang, X.-D., Zhang, W.: Global dynamics of pipes conveying pulsating fluid in the supercritical regime. Int. J. Appl. Mech. 09(02), 1750029 (2017)

    Google Scholar 

  28. Duan, J., Chen, K., You, Y., Wang, R., Li, J.: Three-dimensional dynamics of vortex-induced vibration of a pipe with internal flow in the subcritical and supercritical regimes. Int. J. Nav. Arch. Ocean. 10(6), 692–710 (2018)

    Google Scholar 

  29. Li, Q., Liu, W., Lu, K., Yue, Z.: Three-dimensional parametric resonance of fluid-conveying pipes in the pre-buckling and post-buckling states. Int. J. Pres. Ves. Pip. 189, 104287 (2021)

    Google Scholar 

  30. Liang, F., Gao, A., Yang, X.-D.: Dynamical analysis of spinning functionally graded pipes conveying fluid with multiple spans. Appl. Math. Model. 83, 454–469 (2020)

    MathSciNet  MATH  Google Scholar 

  31. Chandrasekaran, S., Hari, S., Amirthalingam, M.: Functionally graded materials for marine risers by additive manufacturing for high-temperature applications: Experimental investigations. Structures. 35, 931–938 (2022)

    Google Scholar 

  32. Maknun, I.J., Natarajan, S., Katili, I.: Application of discrete shear quadrilateral element for static bending, free vibration and buckling analysis of functionally graded material plate. Compos. Struct. 284, 115130 (2022)

    Google Scholar 

  33. Chen, F.-J., Chen, J.-Y., Duan, R.-Q., Habibi, M., Khadimallah, M.A.: Investigation on dynamic stability and aeroelastic characteristics of composite curved pipes with any yawed angle. Compos. Struct. 284, 115195 (2022)

    Google Scholar 

  34. Gupta, V., Mittal, M.: QRS complex detection using STFT, chaos analysis, and PCA in standard and real-time ECG databases. J. Inst. Eng. India Ser. B. 100(5), 489–497 (2019)

    Google Scholar 

  35. Gupta, V., Mittal, M., Mittal, V.: R-peak detection based chaos analysis of ECG signal. Analog Integr. Circ. S. 102(3), 479–490 (2019)

    Google Scholar 

  36. Gupta, V., Mittal, M., Mittal, V.: R-peak detection using chaos analysis in standard and real time ECG databases. Irbm. 40(6), 341–354 (2019)

    Google Scholar 

  37. Gupta, V., Mittal, M., Mittal, V.: Chaos theory: an emerging tool for arrhythmia detection. Sens. Imaging. (2020). https://doi.org/10.1007/s11220-020-0272-9

    Article  Google Scholar 

  38. Gupta, V., Mittal, M., Mittal, V.: Chaos theory and ARTFA: emerging tools for interpreting ECG signals to diagnose cardiac arrhythmias. Wireless Pers. Commun. 118(4), 3615–3646 (2021)

    Google Scholar 

  39. Gupta, V., Mittal, M., Mittal, V., Saxena, N.K.: A critical review of feature extraction techniques for ECG signal analysis. J. Inst. Eng. India Ser. B. 102, 1049–1060 (2021)

    Google Scholar 

  40. Setoodeh, A.R., Afrahim, S.: Nonlinear dynamic analysis of FG micro-pipes conveying fluid based on strain gradient theory. Compos. Struct. 116, 128–135 (2014)

    Google Scholar 

  41. Bahaadini, R., Saidi, A.R., Hosseini, M.: Dynamic stability of fluid-conveying thin-walled rotating pipes reinforced with functionally graded carbon nanotubes. Acta Mech. 229(12), 5013–5029 (2018)

    MathSciNet  MATH  Google Scholar 

  42. Dehrouyeh-Semnani, A.M., Dehdashti, E., Yazdi, M.R.H., Nikkhah-Bahrami, M.: Nonlinear thermo-resonant behavior of fluid-conveying FG pipes. Int. J. Eng. Sci. 144, 103141 (2019)

    MathSciNet  MATH  Google Scholar 

  43. Deng, J.-Q., Liu, Y.-S., Zhang, Z.-J., Liu, W.: Size-dependent vibration and stability of multi-span viscoelastic functionally graded material nanopipes conveying fluid using a hybrid method. Compos. Struct. 179, 590–600 (2017)

    Google Scholar 

  44. Chen, A., Jian, S.: Dynamic behavior of axially functionally graded pipes conveying fluid. Math. Probl. Eng. 2017, 1–11 (2017)

    MathSciNet  MATH  Google Scholar 

  45. Tuo, Y.-H., Fu, G.-M., Sun, B.-J., Lou, M., Su, J.: Stability of axially functionally graded pipe conveying fluid: generalized integral transform solution. Appl. Ocean Res. 125, 103218 (2022)

    Google Scholar 

  46. Dai, J.-Y., Liu, Y.-S., Liu, H.-C., Miao, C.-X., Tong, G.-J.: A parametric study on thermo-mechanical vibration of axially functionally graded material pipe conveying fluid. Int. J. Mech. Mater. Des. 15(4), 715–726 (2019)

    Google Scholar 

  47. Zhao, Y.-Z., Hu, D., Wu, S., Long, X.-J., Liu, Y.-S.: Dynamics of axially functionally graded conical pipes conveying fluid. J. Mech. 37, 318–326 (2021)

    Google Scholar 

  48. Guo, Q., Liu, Y.-S., Chen, B.-Q., Zhao, Y.-Z.: An efficient stochastic natural frequency analysis method for axially varying functionally graded material pipe conveying fluid. Eur. J. Mech. A. Solids. 86, 104155 (2021)

    MathSciNet  MATH  Google Scholar 

  49. Liang, F., Yang, X.-D., Bao, R.-D., Zhang, W.: Frequency analysis of functionally graded curved pipes conveying fluid. Adv. Mater. Sci. Eng. 2016, 1–9 (2016)

    Google Scholar 

  50. Khodabakhsh, R., Saidi, A.R., Bahaadini, R.: An analytical solution for nonlinear vibration and post-buckling of functionally graded pipes conveying fluid considering the rotary inertia and shear deformation effects. Appl. Ocean Res. 101, 102277 (2020)

    Google Scholar 

  51. Zhen, Y.-X., Gong, Y.-F., Tang, Y.: Nonlinear vibration analysis of a supercritical fluid-conveying pipe made of functionally graded material with initial curvature. Compos. Struct. 268, 113980 (2021)

    Google Scholar 

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

    Google Scholar 

  53. Zhu, B., Xu, Q., Li, M., Li, Y.: Nonlinear free and forced vibrations of porous functionally graded pipes conveying fluid and resting on nonlinear elastic foundation. Compos. Struct. 252, 112672 (2020)

    Google Scholar 

  54. Liu, T., Li, Z.-M.: Nonlinear vibration analysis of functionally graded material tubes with conveying fluid resting on elastic foundation by a new tubular beam model. Int. J. Nonlin. Mech. 137, 103824 (2021)

    Google Scholar 

  55. Zhou, X.-W., Dai, H.-L., Wang, L.: Dynamics of axially functionally graded cantilevered pipes conveying fluid. Compos. Struct. 190, 112–118 (2018)

    Google Scholar 

  56. Shu, C., Chew, Y.T., Richards, B.E.: Generalized differential and integral quadrature and their application to solve boundary layer equations. Int. J. Numer. Meth. Fl. 21(9), 723–733 (1995)

    MathSciNet  MATH  Google Scholar 

  57. Wang, X., Wang, Y.: Free vibration analysis of multiple-stepped beams by the differential quadrature element method. Appl. Math. Comput. 219(11), 5802–5810 (2013)

    MathSciNet  MATH  Google Scholar 

  58. Wang, Y., Wang, X., Zhou, Y.: Static and free vibration analyses of rectangular plates by the new version of the differential quadrature element method. Int. J. Numer. Meth. Eng. 59(9), 1207–1226 (2004)

    MATH  Google Scholar 

Download references

Funding

The authors gratefully acknowledge the support of the National Natural Science Foundation of China (No. 12002195), the National Science Fund for Distinguished Young Scholars (No. 12025204), the Program of Shanghai Municipal Education Commission (Grant No. 2019-01-07-00-09-E00018), and the Pujiang Project of Shanghai Science and Technology Commission (Grant No. 20PJ1404000).

Author information

Authors and Affiliations

  1. Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai Institute of Applied Mathematics and Mechanics, School of Mechanics and Engineering Science, Shanghai University, 99 Shangda Road, Shanghai, 200444, China

    Xiao-Ye Mao, Jie Jing, Hu Ding & Li-Qun Chen

Authors
  1. Xiao-Ye Mao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jie Jing
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Hu Ding
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Li-Qun Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

All authors contributed to the study conception and design. Material preparation, data collection and analysis were performed by JJ and X-Y Mao. The first draft of the manuscript was written by X-Y Mao, and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

Corresponding author

Correspondence to Xiao-Ye Mao.

Ethics declarations

Conflict of interests

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mao, XY., Jing, J., Ding, H. et al. Dynamics of axially functionally graded pipes conveying fluid. Nonlinear Dyn 111, 11023–11044 (2023). https://doi.org/10.1007/s11071-023-08470-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-023-08470-2

Keywords

Search

Navigation