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Stability and local bifurcation of parameter-excited vibration of pipes conveying pulsating fluid under thermal loading

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Abstract

The parametric excited vibration of a pipe under thermal loading may occur because the fluid is often transported heatedly. The effects of thermal loading on the pipe stability and local bifurcations have rarely been studied. The stability and the local bifurcations of the lateral parametric resonance of the pipe induced by the pulsating fluid velocity and the thermal loading are studied. A mathematical model for a simply supported pipe is developed according to the Hamilton principle. Two partial differential equations describing the lateral and longitudinal vibration are obtained. The singularity theory is utilized to analyze the stability and the bifurcation of the system solutions. The transition sets and the bifurcation diagrams are obtained both in the unfolding parameter space and the physical parameter space, which can reveal the relationship between the thermal field parameter and the dynamic behaviors of the pipe. The frequency response and the relationship between the critical thermal rate and the pulsating fluid velocity are obtained. The numerical results demonstrate the accuracy of the single-mode expansion of the solution and the stability and local bifurcation analyses. It also confirms the existence of the chaos. The presented work can provide valuable information for the design of the pipeline and the controllers to prevent the structural instability.

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References

  1. Semler, C., Li, G. X., and Paidoussis, M. P. The non-linear equations of motion of pipes conveying fluid. Journal of Sound and Vibration, 169(5), 577–599 (1994)

    Article  MATH  Google Scholar 

  2. Paidoussis, M. P. In Fluid-Structure Interactions: Slender Structures and Axial Flow, Academic Press Limited, London (1998)

    Google Scholar 

  3. Jin, J. D. and Song, Z. Y. Parametric resonances of supported pipes conveying pulsating fluid. Journal of Fluids and Structures, 20(6), 763–783 (2005)

    Article  Google Scholar 

  4. Panda, L. N. and Kar, R. C. Nonlinear dynamics of a pipe conveying pulsating fluid with parametric and internal resonances. Nonlinear Dynamics, 49, 9–30 (2007)

    Article  MATH  Google Scholar 

  5. Panda, L. N. and Kar, R. C. Nonlinear dynamics of a pipe conveying pulsating fluid with combination principle parametric and internal resonances. Journal of Sound and Vibration, 309, 375–406 (2008)

    Article  Google Scholar 

  6. Liang, F. and Wen, B. Forced vibrations with internal resonance of a pipe conveying fluid under external periodic excitation. Acta Mechanica Solids Sinica, 24, 477–483 (2011)

    Article  Google Scholar 

  7. Zhang, Y. L. and Chen, L. Q. External and internal resonances of the pipe conveying fluid in the supercritical regime. Journal of Sound and Vibration, 332, 2318–2337 (2013)

    Article  Google Scholar 

  8. Xu, J. and Yang, Q. B. Nonlinear dynamics of a pipe conveying pulsating fluid with combination principle parametric and internal resonances (I). Applied Mathematics and Mechanics (English Edition), 27(7), 943–951 (2006) DOI 10.1007/s 10483-006-0710-z

    Article  MathSciNet  MATH  Google Scholar 

  9. Xu, J. and Yang, Q. B. Nonlinear dynamics of a pipe conveying pulsating fluid with combination principle parametric and internal resonances (II) (in Chinese). Journal of Sound and Vibration, 27(7), 825–832 (2006)

    Google Scholar 

  10. Zhang, Y. L. and Chen, L. Q. Internal resonances of the pipe conveying fluid in the supercritical regime. Nonlinear Dynamics, 67, 1505–1514 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  11. Paidoussis, M. P. and Semler, C. Nonlinear and chaostic oscillations of a constrained cantilevered pipe conveying fluid a full nonlinear analysis. Nonlinear Dynamics, 4, 655–670 (1993)

    Article  Google Scholar 

  12. Jayaraman, K. and Narayanan, S. Chaostic oscillations in pipes conveying pulsating fluid. Nonlinear Dynamics, 10, 333–357 (1996)

    Article  Google Scholar 

  13. McDonald, R. J. and Namachchivaya, N. S. Pipes conveying pulsating fluid near a 0:1 resonance: local bifurcations. Journal of Fluids and Structures, 21, 629–664 (2005)

    Article  Google Scholar 

  14. McDonald, R. J. and Namachchivaya, N. S. Pipes conveying pulsating fluid near a 0:1 resonance: global bifurcation. Journal of Fluids and Structures, 21, 665–687 (2005)

    Article  Google Scholar 

  15. Xi, H. M., Zhang, W., and Yao, M. H. Periodic and chaotic oscillation of the fluid conveying pipes with pulse fluid (in Chinese). Journal of Dynamics and Control, 6(3), 243–246 (2008)

    Google Scholar 

  16. Wang, L. A further study on the non-linear dynamics of simply supported pipes conveying pulsating fluid. International Journal of Non-Linear Mechanics, 44, 115–121 (2009)

    Article  Google Scholar 

  17. Wang, L. and Ni, Q. Hopf bifurcation and chaostic motions of a tubular cantilever subject to cross flow and loose support. Nonlinear Dynamics, 59, 329–338 (2010)

    Article  MATH  Google Scholar 

  18. Modarres-Sadeghi, Y. and Paidoussis, M. P. Nonlinear dynamics of extensible fluid-conveying pipes supported at both ends. Journal of Fluids and Structures, 25, 535–543 (2009)

    Article  Google Scholar 

  19. Jin, J. D. and Zou, G. S. Bifurcations and chaotic motions in the autonomous system of a restrained pipe conveying fluid. Journal of Sound and Vibration, 260(5), 783–805 (2003)

    Article  Google Scholar 

  20. Nikolic, M. and Rajkovic, M. Bifurcations in nonlinear models of fluid-conveying pipes supported at both ends. Journal of Fluids and Structures, 22, 173–195 (2006)

    Article  Google Scholar 

  21. Drozyner, P. Determining the limits of piping vibration. Scientific Problems of Machines Operation and Maintenance, 1(165), 97–103 (2011)

    Google Scholar 

  22. Semler, C. and Paidoussis, M. P. Nonlinear analysis of parametric resonances of a planar fluidconveying cantilevered pipe. Journal of Fluids and Structures, 10, 787–825 (1996)

    Article  Google Scholar 

  23. Ganesan, N. and Kadoli, R. A study on the dynamic stability of a cylindrical shell conveying a pulsatile flow of hot fluid. Journal of Sound and Vibration, 274, 953–984 (2004)

    Article  Google Scholar 

  24. Sheng, G. G. and Wang, X. Dynamics characteristics of fluid-conveying functionally graded cylindrical shell under mechanical and thermal loads. Composite Structures, 93, 162–170 (2010)

    Article  Google Scholar 

  25. Qian, Q., Wang, L., and Ni, Q. Instability of simply supported pipes conveying fluid under thermal loads. Mechanics Research Communication, 36, 413–417 (2009)

    Article  MATH  Google Scholar 

  26. Takeuti, Y. Thermal Stress (in Chinese), Science Publishing Press, Beijing (1977)

    Google Scholar 

  27. Jekot, T. Nonlinear problems of thermal postbuckling of a beam. Journal of Thermal Stresses, 19, 356–367 (1996)

    Article  Google Scholar 

  28. Nayfeh, A. H. The Method of Normal Forms, Wiley, Singapore (2011)

    Book  MATH  Google Scholar 

  29. Golubisky, M. and Schaeffer, D. G. Singularities and Groups in Biurcation Theory, Springer-Verlag, New York (1984)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mechanics, College of Pipeline and Civil Engineering, China University of Petroleum, Qingdao, Shandong Province, 266580, China

    Demin Zhao & Jianlin Liu

  2. Department of Mechanical and Manufacturing Engineering, University of Manitoba, Winnipeg MB, R3T5V6, Canada

    C. Q. Wu

Authors
  1. Demin Zhao
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  2. Jianlin Liu
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  3. C. Q. Wu
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Corresponding author

Correspondence to Demin Zhao.

Additional information

Project supported by the National Natural Science Foundation of Shandong Province (No.ZR2013AL017), the National Natural Science Foundation of China (No. 11272357), and the Fundamental Research Funds for the Central Universities of China (No. 11CX04049A)

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Zhao, D., Liu, J. & Wu, C.Q. Stability and local bifurcation of parameter-excited vibration of pipes conveying pulsating fluid under thermal loading. Appl. Math. Mech.-Engl. Ed. 36, 1017–1032 (2015). https://doi.org/10.1007/s10483-015-1960-7

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  • DOI: https://doi.org/10.1007/s10483-015-1960-7

Keywords

Chinese Library Classification

2010 Mathematics Subject Classification

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