GREEN’S FUNCTION METHOD IN THE INVESTIGATION OF DYNAMIC STABILITY OF A FLUID-CONVEYING PIPE

Lolov, Dimitar S.; Lilkova-Markova, Svetlana V.

doi:10.15593/perm.mech/2025.2.07

GREEN’S FUNCTION METHOD IN THE INVESTIGATION OF DYNAMIC STABILITY OF A FLUID-CONVEYING PIPE

Dimitar S. Lolov, Svetlana V. Lilkova-Markova

Author details

Dimitar S. Lolov
University of Architecture, Sofia, Bulgaria
Svetlana V. Lilkova-Markova
University of Architecture, Sofia, Bulgaria

https://doi.org/10.15593/perm.mech/2025.2.07

Fluid-conveying pipes represent a fundamental dynamic problem within the realm of fluid- structure interaction. They find extensive applications in various industries, including petroleum, nuclear engineering, aviation, aerospace, and nanostructures. This paper applies the Green’s function method to solve the stability problem of a fluid-conveying pipe, hinged at both ends and supported by intermediate linear-elastic supports. The objective is to examine the influence of the number and rigidity of these supports on the critical fluid velocity, which is the velocity at which the pipe loses stability. A numerical solution was performed for a straight pipe conveying fluid with specified geometric and physical characteristics, where the number and rigidity of the elastic sup- ports were considered as parameters. The numerical analysis presented herein includes graphs illustrating the dependence of the critical fluid velocity on the number of elastic supports for varying support rigidities. These results reveal that the elastic supports affect both the vibrational charac- teristics and the critical velocity of the conveyed fluid. The solution results are compared with those obtained using one of the most widely employed methods for analyzing the dynamic stability of pipe systems (Transfer Matrix Method – TMM). A good agreement between the results is observed. The paper aims at presenting a method for obtaining the exact solution to the differential equation governing the lateral displacements of a pipe system. This paper discusses the authors' perceived pros and cons of the Green's function method in comparison to the most popular methods for the dynamic investigation of fluid-conveying pipes.

Keywords

stabilityGreen’s functionfluid-conveying pipecritical velocity

Bibliography

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24.Liang F., Gao A., Yang X.D. Dynamical analysis of spinning functionally graded pipes conveying fluid with multiple spans. Applied Mathematical Modeling. 2020, Vol. 83, pp. 454-469. Scholar
25.Askarian A.R., Permoon M.R., Shakouri M. Vibration analysis of pipes conveying fluid resting on a fractional Kelvin-Voigt viscoelastic foundation with general boundary conditions. International Journal of Mechanical Sciences. 2020, Vol. 179, pp. 105702. Scholar
26.Cao J., Liu Y., Liu W. The effect of two cases of temperature distributions on vibration of fluid-conveying functionally graded thin-walled pipes. Journal of Strain Analysis. 2018, Vol. 53(5), pp. 324-331. Scholar
27.Ameen K.A., Al-Dulaimi M.J., Hatem A.A. Experimental study of vibration on pipe conveying fluid at different end conditions for different fluid temperatures. Engineering and Technology Journal. 2019, Vol. 37, pp. 512-515. Scholar
28.Li B., Wang Z., Jing L. Dynamic response of pipe conveying fluid with lateral moving supports. Shock and Vibration. 2018, Vol. 2018, pp. 1-17. Scholar
29.Al-Waily M., Al-Baghdadi M., Al-Khayat R. Flow Velocity and Crack Angle Effect on Vibration and Flow Characterization for Pipe Induced Vibration. International Journal of Mechanical & Mechatronics Engineering. 2017, Vol. 17(5), pp. 19-27. Scholar
30.Siba M., Wahmahmood W., Zakinauw M., Rasani R., Nassir M. Flow-induced vibrations in pipes: challenges and solutions - a review. Journal of Engineering Science and Technology. 2016, Vol. 11(3), pp. 362-382. Scholar
31.Faal R., Derakhshan D. Flow-Induced Vibration of Pipe-line on Elastic Support. Procedia Engineering. 2022, Vol. 14, pp. 2986-2993. Scholar
32.Shiwen L., Ronghui H., Zhao Y., Jiarui W., Shuang C., Jianyong L., Yi L. Numerical Simulation Research on Flow-Induced Vibration Characteristics of Fluid-Conveying Pipe Network. Nuclear Power Engineering. 2022, Vol. 43(1), pp. 187-191. Scholar
33.Jung D, Chung J. In-plane and out-of-plane motions of an extensible semi- circular pipe conveying fluid. Journal of Sound and Vibration, 2008, Vol.311(1), pp.408-420. Scholar
34.Liu G., Li S., Karney B.Y. Vibration analysis of curved pipes conveying fluid. Proceedings of the ASME 2014 Pressure Vessels & Piping Conference, Anaheim, California, USA. 2014, Vol.3. Scholar
35.Lolov D., Lilkova-Markova Sv. Free vibrations out-ofplane of curved planar pipe conveying fluid in two cases of supporting. Annual of UACEG, Mathematics. Mechanics. 2006, Vol. 72(2), pp.123-128. Scholar
36.Liang F., Yang X.-D., Bao R.-D., Zhang W. Frequency Analysis of Functionally Graded Curved Pipes Conveying Fluid. Advances in Materials Science and Engineering. 2016, Vol.2016, Article ID 7574216. Scholar
37.Zhao Q., Sun Z. Flow-induced vibration of curved pipe conveying fluid by a new transfer matrix method. Engineering applications of computational applied mechanics. 2018, Vol.12(1), pp. 780-790. Scholar
38.Zhao Q., Sun Z. In-plane forced vibration of curved pipe conveying fluid by Green function method. Applied Mathematics and Mechanics (English Edition). 2017, Vol.38(10), pp. 1397-1414. Scholar
39.Wu J., Shih P. The dynamic analysis of a multispam fluid-conveying pipe subjected to external load. Journal of sound and vibration. 2001, Vol.239(2), pp. 201-215. Scholar
40.Inspeger T., Horvath R. Pendulum with harmonic variation of the suspension point. Periodica Polytechnica. 2000, Vol. 44, pp. 39-46. Scholar
41.Akulenko L.D., Georgievskii D.V., Nesterov S.V. Spectrum of Transverse Vibrations of a Pipeline Element under Longitudinal Load. Doklady Physics. 2016, Vol. 61 (3), pp. 129-132. Scholar
42.Gay-Balmaz F., Georgievskii D., Putkaradze V. Stability of Helical Tubes Conveying Fluid. Journal of Fluids and Structures. 2018. Vol 78 (2), pp. 146-174. Scholar
43.Il'ichev A.T. Dynamics and spectral stability of soliton-like structures in fluid-filled membrane tubes. Russian Math. Surveys. 2020. Vol 75 (5), pp. 843-882 Scholar
44.Balkaya M., Kaya M. Analysis of the instability of pipes conveying fluid resting on two-parameter elastic soil under different boundary conditions. Ocean Engineering. 2021. Vol.241, pp.110003 Scholar
45.Zhang Yu, Pengzhou Li. Dynamical stability of pipe conveying fluid with various lateral distributed loads. Archive of Applied Mechanics. 2023. Vol.93, pp. 4093-4106 Scholar
46.Ding Hu, Ji J. Vibration control of fluid-conveying pipes: A state of the art review. Appl. Math. Mech.-Engl. 2023. Vol. 44, pp. 1423-1456 Scholar
47.Xueping C., Xiaoxiang H., Congjia Q., Yinghui L. Stability and nonlinear vibration of carbon nanotubes-reinforced composite pipes conveying fluid. Ocean Engineering. 2023. Vol.281, pp.114960 Scholar
48.Vassilev V., Valchev G. Dynamics and stability of DoubleWalled Carbon Nanotube cantilevers conveying fluid in an elastic medium. Dynamics. 2024. Vol. 4(2), pp. 222-232 Scholar

1.Paidoussis M. Fluid-structure interactions: Volume 1 Slender Structures and Axial flow. Academic Press, London, 2013. Scholar
2.Paidoussis M. Fluid-structure interactions: Volume 2 Slender Structures and Axial flow. Academic Press, London, 2016. Scholar
3.Gregory R., Paidoussis M. Unstable oscillation of tubular cantilevers conveying fluid I. Theory. Proceedings of the Royal Society of London. 1966, Vol. 293, No 1435, pp. 512-527. Scholar
4.Gregory R., Paidoussis M. Unstable oscillation of tubular cantilevers conveying fluid II. Experiments. Proceedings of the Royal Society of London. 1966, Vol. 293, No 1435, pp. 528-542. Scholar
5.Udotek E. Internal Fluid Flow Induced Vibration of Pipes. Journal of Mechanical Design and Vibration. 2018, Vol. 6, No. 1, pp. 1-8. Scholar
6.Faal R.T., Derakhshan D. Flow-Induced Vibration of Pipeline on Elastic Support. Procedia Engineering. 2011, Vol. 14, pp. 2986-2993. Scholar
7.Wheeler N. Dirac delta function identities, Reed College, Physics Department, USA, 1997. Scholar
8.Tembely M. Analytical solution of a forced vibrating tube transporting fluids. preprints.org, (in press). 2024. Scholar
9.Reddy, C.D., Lu C., Rajendran S., Liew K.M. Free vibration analysis of fluid conveying single-walled carbon nanotubes. Applied Physics Letters. 2007, Vol. 90, pp. 133122. Scholar
10.Yoon, J., Ru C., Mioduchowski A. Vibration and instability of carbon nanotubes conveying fluid. Composites Science and Technology. 2005, Vol.65 - pp. 1326 - 1336. Scholar
11.Lolov D.S., Lilkova-Markova Sv. V. Dynamic stability of double-walled carbon nanotubes. Journal of the Serbian Society for Computational Mechanics. 2018, Vol.12(1), pp.1-8. Scholar
12.Yoon J., Ru C.Q., Mioduchowski A, Flow-induced flutter instability of cantilever CNTs. Int. J. Solids Struct. 2006, Vol.43, pp. 3337-3349. Scholar
13.Yoon J., Ru C.Q., Mioduchowski A, Vibration and instability of CNTs conveying fluid. Compos. Sci. Technol. 2005, Vol. 65, pp. 1326-1336. Scholar
14.Wang X.Y., Wang X., Sheng G.G. The coupling vibration of fluid-filled carbon nanotubes. J. Phys. D: Appl. Phys. 2007, Vol.40, pp.2563-2572. Scholar
15.Dong K., Wang X. Wave propagation in carbon nanotubes under shear deformation. Nanotechnology. 2006, Vol. 17, pp. 2773-2782. Scholar
16.Dong K., Wang X., Sheng G.G. Wave dispersion characteristics in fluid-filled carbon nanotubes embedded in an elastic medium. Model. Simul. Mater. Sci. Eng. 2007, Vol.15, pp. 427-439. Scholar
17.Jeong W., Soo Y., Jeong S., Lee S., Yoo W. Stability Analysis of a pipe conveying periodically pulsating fluid using finite element method. JSME International Journal. 2006, Vol. 49, pp. 1116-1122. Scholar
18.Paidoussis M., Issid N. Dynamic stability of pipes conveying fluid. Journal of Sound and Vibration. 1974, Vol. 33(3), pp. 267-284. Scholar
19.Deng Q., Yang Z. Wave propagation in submerged pipe conveying fluid. Acta Mechanica Solida Sinica. 2019, Vol. 32(4), pp. 483-498. Scholar
20.Lolov D. S., Lilkova-Markova Sv. Dynamic stability of a fluid-immersed pipe conveying fluid and resting on a damped Winkler elastic foundation. Proceedings of XI International Conference Industrial Engineering and Environment Protection 2021 (IIZS 2021), October 7-8th, 2021, Zrenjanin, Serbia. pp.49-55. Scholar
21.Hellum A., Mukherjee R., Hull A. Flutter instability of a fluid-conveying fluid-immersed pipe affixed to a rigid body. Journal of Fluids and Structures. 2011, Vol.27, pp. 1086-1096. Scholar
22.Lin W., Qiao N. Vibration and stability of an axially moving beam immersed in fluid. International Journal of Solids and Structures. 2008, Vol. 45, pp. 1445-1457. Scholar
23.Huang Q., Lin T., Safarpour M. Flow-induced vibration attenuation of a viscoelastic pipe conveying fluid under sinusoidal flow, using a nonlinear absorber. Mechanics Based design of Structures and Machines. 2020, Vol. 50, pp. 1673-1703. Scholar
24.Liang F., Gao A., Yang X.D. Dynamical analysis of spinning functionally graded pipes conveying fluid with multiple spans. Applied Mathematical Modeling. 2020, Vol. 83, pp. 454-469. Scholar
25.Askarian A.R., Permoon M.R., Shakouri M. Vibration analysis of pipes conveying fluid resting on a fractional Kelvin-Voigt viscoelastic foundation with general boundary conditions. International Journal of Mechanical Sciences. 2020, Vol. 179, pp. 105702. Scholar
26.Cao J., Liu Y., Liu W. The effect of two cases of temperature distributions on vibration of fluid-conveying functionally graded thin-walled pipes. Journal of Strain Analysis. 2018, Vol. 53(5), pp. 324-331. Scholar
27.Ameen K.A., Al-Dulaimi M.J., Hatem A.A. Experimental study of vibration on pipe conveying fluid at different end conditions for different fluid temperatures. Engineering and Technology Journal. 2019, Vol. 37, pp. 512-515. Scholar
28.Li B., Wang Z., Jing L. Dynamic response of pipe conveying fluid with lateral moving supports. Shock and Vibration. 2018, Vol. 2018, pp. 1-17. Scholar
29.Al-Waily M., Al-Baghdadi M., Al-Khayat R. Flow Velocity and Crack Angle Effect on Vibration and Flow Characterization for Pipe Induced Vibration. International Journal of Mechanical & Mechatronics Engineering. 2017, Vol. 17(5), pp. 19-27. Scholar
30.Siba M., Wahmahmood W., Zakinauw M., Rasani R., Nassir M. Flow-induced vibrations in pipes: challenges and solutions - a review. Journal of Engineering Science and Technology. 2016, Vol. 11(3), pp. 362-382. Scholar
31.Faal R., Derakhshan D. Flow-Induced Vibration of Pipe-line on Elastic Support. Procedia Engineering. 2022, Vol. 14, pp. 2986-2993. Scholar
32.Shiwen L., Ronghui H., Zhao Y., Jiarui W., Shuang C., Jianyong L., Yi L. Numerical Simulation Research on Flow-Induced Vibration Characteristics of Fluid-Conveying Pipe Network. Nuclear Power Engineering. 2022, Vol. 43(1), pp. 187-191. Scholar
33.Jung D, Chung J. In-plane and out-of-plane motions of an extensible semi- circular pipe conveying fluid. Journal of Sound and Vibration, 2008, Vol.311(1), pp.408-420. Scholar
34.Liu G., Li S., Karney B.Y. Vibration analysis of curved pipes conveying fluid. Proceedings of the ASME 2014 Pressure Vessels & Piping Conference, Anaheim, California, USA. 2014, Vol.3. Scholar
35.Lolov D., Lilkova-Markova Sv. Free vibrations out-ofplane of curved planar pipe conveying fluid in two cases of supporting. Annual of UACEG, Mathematics. Mechanics. 2006, Vol. 72(2), pp.123-128. Scholar
36.Liang F., Yang X.-D., Bao R.-D., Zhang W. Frequency Analysis of Functionally Graded Curved Pipes Conveying Fluid. Advances in Materials Science and Engineering. 2016, Vol.2016, Article ID 7574216. Scholar
37.Zhao Q., Sun Z. Flow-induced vibration of curved pipe conveying fluid by a new transfer matrix method. Engineering applications of computational applied mechanics. 2018, Vol.12(1), pp. 780-790. Scholar
38.Zhao Q., Sun Z. In-plane forced vibration of curved pipe conveying fluid by Green function method. Applied Mathematics and Mechanics (English Edition). 2017, Vol.38(10), pp. 1397-1414. Scholar
39.Wu J., Shih P. The dynamic analysis of a multispam fluid-conveying pipe subjected to external load. Journal of sound and vibration. 2001, Vol.239(2), pp. 201-215. Scholar
40.Inspeger T., Horvath R. Pendulum with harmonic variation of the suspension point. Periodica Polytechnica. 2000, Vol. 44, pp. 39-46. Scholar
41.Akulenko L.D., Georgievskii D.V., Nesterov S.V. Spectrum of Transverse Vibrations of a Pipeline Element under Longitudinal Load. Doklady Physics. 2016, Vol. 61 (3), pp. 129-132. Scholar
42.Gay-Balmaz F., Georgievskii D., Putkaradze V. Stability of Helical Tubes Conveying Fluid. Journal of Fluids and Structures. 2018. Vol 78 (2), pp. 146-174. Scholar
43.Il'ichev A.T. Dynamics and spectral stability of soliton-like structures in fluid-filled membrane tubes. Russian Math. Surveys. 2020. Vol 75 (5), pp. 843-882 Scholar
44.Balkaya M., Kaya M. Analysis of the instability of pipes conveying fluid resting on two-parameter elastic soil under different boundary conditions. Ocean Engineering. 2021. Vol.241, pp.110003 Scholar
45.Zhang Yu, Pengzhou Li. Dynamical stability of pipe conveying fluid with various lateral distributed loads. Archive of Applied Mechanics. 2023. Vol.93, pp. 4093-4106 Scholar
46.Ding Hu, Ji J. Vibration control of fluid-conveying pipes: A state of the art review. Appl. Math. Mech.-Engl. 2023. Vol. 44, pp. 1423-1456 Scholar
47.Xueping C., Xiaoxiang H., Congjia Q., Yinghui L. Stability and nonlinear vibration of carbon nanotubes-reinforced composite pipes conveying fluid. Ocean Engineering. 2023. Vol.281, pp.114960 Scholar
48.Vassilev V., Valchev G. Dynamics and stability of DoubleWalled Carbon Nanotube cantilevers conveying fluid in an elastic medium. Dynamics. 2024. Vol. 4(2), pp. 222-232 Scholar

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Published: 30 June 2025
Issue date: 30 June 2025
Issue: No. 2 (2025)

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GREEN’S FUNCTION METHOD IN THE INVESTIGATION OF DYNAMIC STABILITY OF A FLUID-CONVEYING PIPE

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