Skip to main content
Log in

Comparison of Natural Frequencies of Fluid-Conveying Functionally Graded Thin-Walled Pipes in Two Cases of Temperature Distributions

  • Published:
Mechanics of Composite Materials Aims and scope

The dynamic response of shear deformable functionally graded (FG) pipes conveying nd internal fluid is studied to determine the effects of temperature gradient, and to determine the effect of two cases of temperature distributions on their natural frequencies. Assuming that the material properties of the FG pipes obey a power-law distribution and are temperature-dependent, a differential governing equation is obtained using the extended Hamilton’s principle. A wavelet-based element of FG pipes considering shear deformations is developed and used to obtain ordinary differential equations for the system considered.

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

Access this article

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.

Similar content being viewed by others

References

  1. M. P. Paidoussis, Fluid–Structure Interactions. Slender Structures and Axial Flow, 1.., 2nd ed., Academic Press. London (2014).

    Google Scholar 

  2. M. H. Ghayesh and M. P. Paidoussis, “Three-dimensional dynamics of a cantilevered pipe conveying fluid, additionally supported by an intermediate spring array,” Int. J. Non-Linear Mech., 45, 507-524 (2010).

    Article  Google Scholar 

  3. B. H. Li, H. S. Gao, H. B. Zhai, Y. S. Liu, and Z. F. Yue, “Free vibration analysis of multi-span pipe conveying fluid with dynamic stiffness method,” Nuclear Eng. Des., 241, No. 3, 666-671 (2011).

    Article  CAS  Google Scholar 

  4. H. B. Zhai, Z. Y. Wu, Y. S. Liu, and Z. F. Yue, “In-plane dynamic response analysis of curved pipe conveying fluid subjected to random excitation,” Nuclear Eng. Des., 256 (March), 214-226. (2013).

    Article  CAS  Google Scholar 

  5. J. J. Gu, C. An, M. L. Duan, C. Levi, and J. Su, “Integral transform solutions of dynamic response of a clamped–clamped pipe conveying fluid,” Nuclear Eng. Des., 254, No. 1, 237-245 (2013).

    Article  CAS  Google Scholar 

  6. H. L. Dai, L. Wang, Q. Qian, and Q. Ni, “Vortex-induced vibrations of pipes conveying pulsating fluid,” Ocean Eng., 77 (February), 12-22 (2014).

    Article  Google Scholar 

  7. L. Wang, H. L. Dai, and Q. Qian, “Dynamics of simply supported fluid-conveying pipes with geometric imperfections,” J. Fluids and Struct., 29, 97-106 (2012).

    Article  CAS  Google Scholar 

  8. M. Hosseini and S. A. Fazelzaden, “Thermomechanical stability analysis of functionally graded thin-walled cantilever pipe with flowing fluid subjected to axial load,” Int. J. Structural Stability and Dynamics, 11, No. 3, 513-534 (2011).

    Article  Google Scholar 

  9. Zhong-Min Wang and Yan-Zhuang Liu, “Transverse vibration of pipe conveying fluid made of functionally graded materials using a symplectic method,” Nuclear Eng. Des., 298, 149-159 (2016).

    Article  CAS  Google Scholar 

  10. J. W. Xiang and M. Liang, “Multiple damage detection method for beams based on multi-scale elements using hermite cubic spline wavelet,” Computer Modeling in Engineering and Sci., 73, No. 3, 267-298 (2011).

    Google Scholar 

  11. A. Oke Wasiu and A. Khulief Yehia, “Vibration analysis of composite pipes using finite element method with B- Spline wavelets,” J. Mech. Sci. Technol., 30, No. 2, 623-635 (2016).

    Article  Google Scholar 

  12. Liviu Librescu, Sang Yong Oh, and Ohseop Song, “Spinning thin-walled beams made of functionally graded materials: modeling, vibration and instability,” Eur. J. Mech. - A/Solids, 23, Iss. 3, 499-515 (2004).

    Article  Google Scholar 

  13. J. C. Goswami, A. K. Chan, and C. K. Chui, “On solving first kind integral equations using wavelets on a bounded interval,” IEEE Transactions on Antennas and Propagation, 43, No. 6, 614-622 (1995).

    Article  Google Scholar 

  14. E. Quak and N. Weyrich, “Decomposition and reconstruction algorithms for spline wavelets on a bounded interval,”Appl. Computational Harmonic Analysis, 1, No. 3, 217-231 (1994).

    Article  Google Scholar 

Download references

Acknowledgements

The project was supported by the National Science Foundation of China (Grant No.51305350) and the Basic Research Foundation of NWPU (No.3102014JCQ01045)

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to J. H. Cao.

Additional information

Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 54, No. 5, pp. 925-940, September-October, 2018.

Appendices

Appendix A.

Mass, Damping, and Stiffness Matrices of Finite Element

$$ {\displaystyle \begin{array}{l}{m}^e={\int}_0^1\left[\begin{array}{cccc}\left({b}_1+{m}_f\right){N}_u^T{N}_u{l}_e& 0& 0& 0\\ {}0& \left({b}_5-2{\widehat{b}}_9+{b}_{15}\right)& 0& 0\\ {}0& 0& \left({b}_1+{m}_f\right){N}_{\upsilon}^T{N}_{\upsilon }{l}_e& 0\\ {}0& 0& 0& \left({b}_4-2{\widehat{b}}_8+{b}_{14}+{m}_f{r}_x^2\right){N}_{\theta}^T{N}_{\theta_x}{l}_e\end{array}\right] d\xi, \\ {}\kern4em {c}^e={\int}_0^1\left[\begin{array}{cccc}{m}_fU\left({N}_u^T{N}_u^{\hbox{'}}-|{N}_u^{\hbox{'}T}{N}_u\right)& 0& 0& 0\\ {}0& 0& 0& 0\\ {}0& 0& {m}_fU\left({N}_{\upsilon}^T{N}_{\upsilon}^{\prime }-{N}_{\upsilon}^{\prime T}{N}_{\upsilon}\right)& 0\\ {}0& 0& 0& 0\end{array}\right] d\xi \\ {}\kern5em +\left[\begin{array}{cccc}{\left.{m}_fU{N}_u^T\delta \left(z-L\right)\right|}_{\xi =1}& 0& 0& 0\\ {}0& 0& 0& 0\\ {}0& 0& {m}_fU{N}_{\upsilon}^T{N}_{\upsilon}\delta \left(z-L\right)& 0\\ {}0& 0& 0& 0\end{array}\right],\\ {}{k}^e=\left[\begin{array}{cccc}{a}_{44}-{m}_f{U}^2-P/{N}_u^{\prime T}{N}_u^{\prime }/{l}_e& {a}_{44}{N}_u^{\prime T}{N}_{\theta_y}& {a}_{45}{N}_u^{\prime T}{N}_{\upsilon}^{\prime }/{l}_e& {a}_{45}{N}_u^{\prime T}{N}_{\theta_x}\\ {}{a}_{44}{N}_{\theta_y}^T{N}_u^{\prime }& {a}_{44}{N}_{\theta_y}^T{N}_{\theta_y}& {a}_{45}{N}_{\theta_y}^T{N}_{\upsilon}^{\prime }& {a}_{44}{N}_{\theta_y}^T{N}_{\theta_x}\\ {}{a}_{54}{N}_{\upsilon}^{\prime T}{N}_u^{\prime }/{l}_e& {a}_{54}{N}_{\upsilon}^{\prime T}{N}_{\theta_y}& \left({a}_{55}-{m}_f{U}^2-P\right){N}_{\upsilon}^{\prime T}{N}_{\upsilon}^{\prime }/{l}_e& {a}_{55}{N}_{\upsilon}^{\prime T}{N}_{\theta_x}\\ {}{a}_{54}{N}_{\theta_x}^T{N}_u^{\prime }& {a}_{54}{N}_{\theta_x}^T{N}_{\theta_y}& {a}_{55}{N}_{\theta_x}^T{N}_{\upsilon}^{\prime }& {a}_{55}{N}_{\theta_x}^T{N}_{\theta_x}\end{array}\right] d\xi \\ {}\kern3em +\left[\begin{array}{cccc}{\left.{m}_f{U}^2{N}_u^{\prime T}{N}_u\delta \left(z-L\right)/{l}_e\right|}_{\xi =1}& 0& 0& 0\\ {}0& 0& 0& 0\\ {}0& 0& {\left.{m}_f{U}^2{N}_u^{\prime T}{N}_u\delta \left(z-L\right)/{l}_e\right|}_{\xi =1}& 0\\ {}0& 0& 0& 0\end{array}\right],\end{array}} $$

where ()· and ()′ denote ∂()/∂t and ∂()/∂ξ , respectively.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Cao, J.H., Liu, Y.S. Comparison of Natural Frequencies of Fluid-Conveying Functionally Graded Thin-Walled Pipes in Two Cases of Temperature Distributions. Mech Compos Mater 54, 635–646 (2018). https://doi.org/10.1007/s11029-018-9771-3

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11029-018-9771-3

Keywords

Navigation