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Parametric vibration of a nonlinearly supported pipe conveying pulsating fluid

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Abstract

Changes of fluid speed may cause large vibrations in the pipe system. Generally, the fluid speed in a pipe system may not be constant all the time but has an oscillating property of the fluid, such as pulsating fluid. When a pipe conveys pulsating fluid, its dynamic response is significantly affected by the fluid. This paper investigates the vibration characteristics of a pipe conveying pulsating fluid with nonlinear supports at both ends. Based on the Hamilton principle, the governing equations and the boundary conditions for the pipes conveying pulsating fluid are determined. The multi-scale method combined with the modal revision method is introduced to obtain the approximate analytical results of the system steady-state response. Subsequently, the approximate analytical solution is verified numerically by the differential quadrature element method. The effects of the linear support stiffness, the nonlinear support stiffness, the fluid speed, and the viscoelastic coefficient on the stability boundary of parametric resonance are investigated. The effects of system parameters on steady-state responses of the pipe are also discussed in detail. The results reveal that the two methods are in good agreement. And it is shown that the linear support stiffness affects not only the parametric resonance amplitude but also the resonance region, while the nonlinear support stiffness only affects the parametric resonance amplitude, but not the resonance region.

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Data availability statement

The datasets generated or analyzed during the current study are available from the corresponding author on reasonable request.

References

  1. Andreikiv, O.E., Ivanyt’skyi, Y.L., Terlets’ka, Z.O., Kit, M.B.: Evaluation of the durability of a pipe of oil pipeline with surface crack under biaxial block loading. Mater. Sci. 40(3), 408–415 (2004)

    Google Scholar 

  2. Bulleri, B., Genoni, M., Gallone, G., Marchetti, A.: Multi-layer pipes for hydrocarbons conveyance. Macromol. Symp. 218(1), 363–371 (2004)

    Google Scholar 

  3. Lu, H., Huang, K., Wu, S.: Vibration and stress analyses of positive displacement pump pipeline systems in oil transportation stations. J. Pipel. Syst. Eng. Pract. 7(1), 05015002 (2016)

    Google Scholar 

  4. Zhao, Q., Sun, Z.: In-plane forced vibration of curved pipe conveying fluid by Green function method. Appl. Math. Mech.-Engl. Ed. 38(10), 1397–1414 (2017)

    MathSciNet  MATH  Google Scholar 

  5. Reddy, R.S., Panda, S., Natarajan, G.: Nonlinear dynamics of functionally graded pipes conveying hot fluid. Nonlinear Dyn. 99(3), 1989–2010 (2020)

    Google Scholar 

  6. Laithier, B.E., Païdoussis, M.P.: The equations of motion of initially stressed Timoshenko tubular beams conveying fluid. J. Sound Vibr. 79(2), 175–195 (1981)

    MATH  Google Scholar 

  7. PaÏdoussis, M.P., Li, G.X.: Pipes conveying fluid: a model dynamical problem. J. Fluids Struct. 7(2), 137–204 (1993)

    Google Scholar 

  8. Herrmann, G., Nemat-Nasser, S.: Instability modes of cantilevered bars induced by fluid flow through attached pipes. Int. J. Solids Struct. 3(1), 39–52 (1967)

    Google Scholar 

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

    Google Scholar 

  10. Sugiyama, Y., Tanaka, Y., Kishi, T., Kawagoe, H.: Effect of a spring support on the stability of pipes conveying fluid. J. Sound Vibr. 100(2), 257–270 (1985)

    Google Scholar 

  11. Ariaratnam, S.T., Sri Namachchivaya, N.: Dynamic stability of pipes conveying pulsating fluid. J. Sound Vibr. 107(2), 215–230 (1986)

    MATH  Google Scholar 

  12. Lottati, I., Kornecki, A.: The effect of an elastic foundation and of dissipative forces on the stability of fluid-conveying pipes. J. Sound Vibr. 109(2), 327–338 (1986)

    Google Scholar 

  13. Paidoussis, M.P., Sundararajan, C.: Parametric and combination resonances of a pipe conveying pulsating fluid. J. Appl. Mech. Trans. ASME 42(4), 780–784 (1975)

    MATH  Google Scholar 

  14. Panda, L.N., Kar, R.C.: Nonlinear dynamics of a pipe conveying pulsating fluid with combination, principal parametric and internal resonances. J. Sound Vibr. 309(3–5), 375–406 (2008)

    Google Scholar 

  15. Sri Namchchivaya, N.: Non-linear dynamics of supported pipe conveying pulsating fluid-I Subharmonic resonance. Int. J. Non-Linear Mech. 24(3), 185–196 (1989)

    MATH  Google Scholar 

  16. Sri Namchchivaya, N., Tien, W.M.: Non-linear dynamics of supported pipe conveying pulsating fluid-II Combination resonance. Int. J. Non-Linear Mech. 24(3), 197–208 (1989)

    MATH  Google Scholar 

  17. Stein, R.A., Tobriner, M.W.: Vibration of pipes containing flowing fluids. J. Appl. Mech. Trans. ASME 37(4), 906–916 (1970)

    MATH  Google Scholar 

  18. Misra, A.K., Païdoussis, M.P., Van, K.S.: On the dynamics of curved pipes transporting fluid. Part I: inextensible theory. J. Fluids Struct. 2(3), 221–244 (1988)

    MATH  Google Scholar 

  19. Païdoussis, M.P., Li, G.X., Moon, F.C.: Chaotic oscillations of the autonomous system of a constrained pipe conveying fluid. J. Sound Vibr. 135(1), 1–19 (1989)

    MathSciNet  MATH  Google Scholar 

  20. Ibrahim, R.A.: Overview of mechanics of pipes conveying fluids-part I: fundamental studies. J. Press. Vessel Technol. Trans. ASME 132(3), 034001 (2010)

    Google Scholar 

  21. Ibrahim, R.A.: Mechanics of pipes conveying fluids-part ii: applications and fluidelastic problems. J. Press. Vessel Technol. Trans. ASME 133(2), 024001 (2011)

    Google Scholar 

  22. Li, S.J., Karney, B.W., Liu, G.M.: FSI research in pipeline systems—A review of the literature. J. Fluids Struct. 57, 277–297 (2015)

    Google Scholar 

  23. Païdoussis, M.P.: Pipes conveying fluid: a fertile dynamics problem. J. Fluids Struct. 114, 103664 (2022)

    Google Scholar 

  24. Gorman, D.G., Reese, J.M., Zhang, Y.L.: Vibration of a flexible pipe conveying viscous pulsating fluid flow. J. Sound Vibr. 230(2), 379–392 (2000)

    Google Scholar 

  25. Zhang, Y.L., Chen, L.Q.: Internal resonance of pipes conveying fluid in the supercritical regime. Nonlinear Dyn. 67(2), 1505–1514 (2012)

    MathSciNet  MATH  Google Scholar 

  26. Adegoke, A.S., Adewumi, O., Fashanu, A., Oyediran, A.: Analysis of the nonlinear axial vibrations of a cantilevered pipe conveying pulsating two-phase flow. J. Comput. Appl. Mech. 51(2), 311–322 (2020)

    Google Scholar 

  27. Askarian, A.R., Abtahi, H., Firouz-Abadi, R.D., Haddadpour, H., Dowell, E.H.: Bending-torsional instability of a viscoelastic cantilevered pipe conveying pulsating fluid with an inclined terminal nozzle. J. Mech. Sci. Technol. 32(7), 2999–3008 (2018)

    Google Scholar 

  28. 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. Pressure Vessels Pip. 193, 104479 (2021)

    Google Scholar 

  29. Rahmati, M., Mirdamadi, H.R., Goli, S.: Divergence instability of pipes conveying fluid with uncertain flow velocity. Physica A 491, 650–665 (2018)

    MathSciNet  MATH  Google Scholar 

  30. Kheiri, M.: Nonlinear dynamics of imperfectly-supported pipes conveying fluid. J. Fluids Struct. 93, 102850 (2020)

    Google Scholar 

  31. Sri Namachchivaya, N., Tien, W.M.: Bifurcation behavior of nonlinear pipes conveying pulsating flow. J. Fluids Struct. 3(6), 609–629 (1989)

    MATH  Google Scholar 

  32. Yang, X., Yang, T., Jin, J.: Dynamic stability of a beam-model viscoelastic pipe for conveying pulsative fluid. Acta Mech. Solida Sin. 20(4), 350–356 (2007)

    Google Scholar 

  33. Mohammadi, J., Nikkhah-Bahrami, M.: Stability analyses of articulated rigid pipes conveying fluid with harmonic velocity using the method of multiple time scales. J. Mech. Sci. Technol. 34(3), 965–976 (2020)

    Google Scholar 

  34. Chen, W.H., Fan, C.N.: Stability analysis with lumped mass and friction effects in elastically supported pipes conveying fluid. J. Sound Vibr. 119(3), 429–442 (1987)

    Google Scholar 

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

    Google Scholar 

  36. Xu, J., Yang, Q.B.: Flow-induced internal resonances and mode exchange in horizontal cantilevered pipe conveying fluid (II). Appl. Math. Mech.Engl. Ed. 27(7), 943–951 (2006)

    MathSciNet  MATH  Google Scholar 

  37. Zhang, Y.F., Yao, M.H., Zhang, W., Wen, B.C.: Dynamical modeling and multi-pulse chaotic dynamics of cantilevered pipe conveying pulsating fluid in parametric resonance. Aerosp. Sci. Technol. 68, 441–453 (2017)

    Google Scholar 

  38. Zhang, Y.F., Liu, T., Zhang, W.: Nonlinear resonant responses, mode interactions, and multitime periodic and chaotic oscillations of a cantilevered pipe conveying pulsating fluid under external harmonic force. Complexity 2020 (2020)

  39. Zhou, K., Ni, Q., Guo, Z.L., Yan, H., Dai, H.L., Wang, L.: Nonlinear dynamic analysis of cantilevered pipe conveying fluid with local rigid segment. Nonlinear Dyn. 109(3), 1571–1589 (2022)

    Google Scholar 

  40. Wu, Q., Qi, G.: Global dynamics of a pipe conveying pulsating fluid in primary parametrical resonance: analytical and numerical results from the nonlinear wave equation. Phys. Lett. A 383(14), 1555–1562 (2019)

    MathSciNet  MATH  Google Scholar 

  41. 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(8), 1017–1032 (2015)

    MathSciNet  MATH  Google Scholar 

  42. Jayaraman, K., Narayanan, S.: Chaotic oscillations in pipes conveying pulsating fluid. Nonlinear Dyn. 10(4), 333–357 (1996)

    Google Scholar 

  43. Luczko, J., Czerwinski, A.: Parametric vibrations of flexible hoses excited by a pulsating fluid flow, part I: modelling, solution method and simulation. J. Fluids Struct. 55, 155–173 (2015)

    Google Scholar 

  44. Luczko, J., Czerwinski, A.: Experimental and numerical investigation of parametric resonance of flexible hose conveying non-harmonic fluid flow. J. Sound Vibr. 373, 236–250 (2016)

    Google Scholar 

  45. Seo, Y.S., Jeong, W.B., Jeong, S.H., Oh, J.S., Yoo, W.S.: Finite element analysis of forced vibration for a pipe conveying harmonically pulsating fluid. JSME Int. J. Ser. C-Mech. Syst. Mach. Elem. Manuf. 48(4), 688–694 (2006)

    Google Scholar 

  46. Liang, F., Yang, X., Zhang, W., Qian, Y., Melnik, R.V.N.: Parametric vibration analysis of pipes conveying fluid by nonlinear normal modes and a numerical iterative approach. Adv. Appl. Math. Mech. 11(1), 38–52 (2019)

    MathSciNet  MATH  Google Scholar 

  47. Li, Q., Liu, W., Lu, K., Yue, Z.F.: Nonlinear parametric vibration of a fluid-conveying pipe flexibly restrained at the ends. Acta Mech. Solid. Sin. 33(3), 327–346 (2020)

    Google Scholar 

  48. Wang, L.: A further study on the non-linear dynamics of simply supported pipes conveying pulsating fluid. Int. J. Non-Linear Mech. 44(1), 115–121 (2009)

    Google Scholar 

  49. Panda, L.N., Kar, R.C.: Nonlinear dynamics of a pipe conveying pulsating fluid with parametric and internal resonances. Nonlinear Dyn. 49(1–2), 9–30 (2007)

    MATH  Google Scholar 

  50. Li, Y.D., Yang, Y.R.: Nonlinear vibration of slightly curved pipe with conveying pulsating fluid. Nonlinear Dyn. 88(4), 2513–2529 (2017)

    Google Scholar 

  51. Peng, G., Xiong, Y., Gao, Y., Liu, L., Wang, M., Zhang, Z.: Non-linear dynamics of a simply supported fluid-conveying pipe subjected to motion-limiting constraints: two-dimensional analysis. J. Sound Vibr. 435, 192–204 (2018)

    Google Scholar 

  52. Balkaya, M., Orhan Kaya, M.: Analysis of the instability of pipes conveying fluid resting on two-parameter elastic soil under different boundary conditions. Ocean Eng. 241, 110003 (2021)

    Google Scholar 

  53. Yan, X., Wei, S., Mao, X.Y., Ding, H., Chen, L.Q.: Study on natural characteristics of fluid-conveying pipes with elastic supports at both ends. Chin. J. Theor. Appl. Mech. 54(5), 1341–1352 (2022)

    Google Scholar 

  54. Qian, Q., Wang, L., Ni, Q.: Nonlinear responses of a fluid-conveying pipe embedded in nonlinear elastic foundations. Acta Mech. Solida Sin. 21(2), 170–176 (2008)

    Google Scholar 

  55. 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 

  56. Wei, S., Yan, X., Fan, X., Mao, X., Ding, H., Chen, L.Q.: Vibration of fluid-conveying pipe with nonlinear supports at both ends. Appl. Math. Mech.Engl. Ed. 43(6), 845–862 (2022)

    MathSciNet  MATH  Google Scholar 

  57. Chen, L.Q., Ding, H.: Steady-state transverse response in coupled planar vibration of axially moving viscoelastic beams. J. Vib. Acoust. Trans. ASME 132(1), 011009 (2010)

    Google Scholar 

  58. Ding, H., Chen, L.Q.: On two transverse nonlinear models of axially moving beams. Sci. China Ser. E-Technol. Sci. 52(3), 743–751 (2009)

    MATH  Google Scholar 

  59. 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 

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

    MATH  Google Scholar 

  61. Malik, M., Bert, C.W.: Implementing multiple boundary conditions in the DQ solution of higher-order PDE’s: Application to free vibration of plates. Int. J. Numer. Methods Eng. 39(7), 1237–1258 (1996)

    MATH  Google Scholar 

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Funding

The research work was supported by the Project of the National Natural Science Foundation of China (Nos. 12072181, 12272211 and 12121002).

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Authors and Affiliations

  1. Shanghai Institute of Applied Mathematics and Mechanics, Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai Frontier Science Center of Mechanoinformatics, School of Mechanics and Engineering Science, Shanghai University, Shanghai, 200072, China

    Sha Wei, Xiong Yan, Xulong Li, Hu Ding & Li-Qun Chen

  2. Shanghai Institute of Aircraft Mechanics and Control, Zhangwu Road, Shanghai, 200092, China

    Sha Wei & Li-Qun Chen

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  1. Sha Wei
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  2. Xiong Yan
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  3. Xulong Li
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  4. Hu Ding
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Correspondence to Li-Qun Chen.

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Wei, S., Yan, X., Li, X. et al. Parametric vibration of a nonlinearly supported pipe conveying pulsating fluid. Nonlinear Dyn 111, 16643–16661 (2023). https://doi.org/10.1007/s11071-023-08761-8

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