Non-planar vibration characteristics and buckling behaviors of two fluid-conveying pipes coupled with an intermediate spring

doi:10.1007/s10483-025-3306-9

Abstract

Abstract:

This study investigates the dynamical behavior of two parallel fluid-conveying pipes by developing a non-planar dynamical model of the two pipes coupled with an intermediate spring. A systematic analysis is conducted to evaluate the effects of spring parameters on the non-planar vibration characteristics and buckling behaviors of the coupled system. The nonlinear governing equations are derived with Hamilton’s principle, subsequently discretized through Galerkin’s method, and finally numerically solved by the Runge-Kutta algorithm. Based on the linearized equations, an eigenvalue analysis is performed to obtain the coupled frequencies, modal shapes, and critical flow velocities for buckling instability. Quantitative assessments further elucidate the effects of the spring position and stiffness coefficient on the coupled frequencies and critical flow velocities. Nonlinear dynamic analyses reveal the evolution of buckling patterns and bifurcation behaviors between the lateral displacements of the two pipes and the flow velocity. Numerical results indicate that the intermediate spring increases the susceptibility to buckling instability in the out-of-plane direction compared with the in-plane direction. Furthermore, synchronized lateral displacements emerge in both pipes when the flow velocity of one pipe exceeds the critical threshold. This work is expected to provide a theoretical foundation for the stability assessment and vibration analysis in coupled fluid-conveying pipe systems.

Key words: coupled fluid-conveying pipe system, intermediate spring, non-planar vibration, buckling behavior

2010 MSC Number:

O326

Dali WANG, Tianli JIANG, Huliang DAI, Lin WANG. Non-planar vibration characteristics and buckling behaviors of two fluid-conveying pipes coupled with an intermediate spring. Applied Mathematics and Mechanics (English Edition), 2025, 46(10): 1829-1850.

TrendMD

Figures/Tables 18

Fig. 1

Fig. 2

Fig. 3

Fig. 4

Table 1

The first six coupled frequencies of the pipe system in the y- and z-directions with zero flow velocity at Ls=0.75L"

Mode number	Coupled frequency in the $y$ -direction	Coupled frequency in the $z$ -direction
1	9.869 6	9.869 6
2	18.333 8	13.135 8
3	39.478 4	39.478 4
4	49.740 5	41.662 0
5	88.826 4	88.826 4
6	91.588 1	89.323 3

Table 1

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

References 38

[1]	TANG, Y., ZHANG, H. J., CHEN, L. Q., DING, Q., GAO, Q. Y., and YANG, T. Z. Recent progress on dynamics and control of pipes conveying fluid. Nonlinear Dynamics, 113(7), 6253–6315 (2025)
[2]	CHEN, W., CAO, R., HU, J., DAI, H., and WANG, L. Research advances in large-deformation dynamics of slender fluid-conveying pipelines (in Chinese). Advances in Mechanics, 55(1), 113–174 (2025)
[3]	YAN, H., XIONG, F., JIANG, N., DAI, H. L., WANG, L., HUANG, Q., and NI, Q. Nonlinear vibration control of a simply-supported fluid-conveying pipe with nonlinear energy sink (in Chinese). Chinese Journal of Solid Mechanics, 40(2), 127–136 (2019)
[4]	PAÏDOUSSIS, M. P. and LI, G. X. Pipes conveying fluid: a model dynamical problem. Journal of Fluids and Structures, 7(2), 137–204 (1993)
[5]	BOURRIÈRES, F. and BÉNARD, H. Sur un Phénomène d’oscillation Auto-entretenue en Mécanique des Fluides réels, Blondel la Rougery, Gauthier-Villars (1939)
[6]	HOUSNER, G. W. Bending vibrations of a pipe line containing flowing fluid. Journal of Applied Mechanics, 19(2), 205–208 (1952)
[7]	NIORDSON, F. I. Vibrations of a Cylindrical Tube Containing Flowing Fluid, Elanders Boktr, Göteborg (1953)
[8]	FEODOS’EV, V. P. Vibrations and stability of a pipe when liquid flows through it. Inzhenernyi Sbornik, 1951(10), 169–170 (1951)
[9]	GREGORY, R. W. and PAÏDOUSSIS, M. P. Unstable oscillation of tubular cantilevers conveying fluid, I: theory. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, 293(1435), 512–527 (1966)
[10]	HOLMES, P. J. Pipes supported at both ends cannot flutter. Journal of Applied Mechanics, 45(3), 619–622 (1978)
[11]	CH’NG, E. and DOWELL, E. H. A Theoretical Analysis of Nonlinear Effects on the Flutter and Divergence of a Tube Conveying Fluid, AMS Report, Princeton University (1977)
[12]	LUNDGREN, T. S., SETHNA, P. R., and BAJAJ, A. K. Stability boundaries for flow induced motions of tubes with an inclined terminal nozzle. Journal of Sound and Vibration, 64(4), 553–571 (1979)
[13]	SEMLER, C., LI, G. X., and PAÏDOUSSIS, M. P. The non-linear equations of motion of pipes conveying fluid. Journal of Sound and Vibration, 169(5), 577–599 (1994)
[14]	MACIEL, V. S. F., KHEIRI, M., and FRANZINI, G. R. Passive suppression of flow-induced vibrations of a cantilevered pipe discharging fluid using non-linear vibration absorbers. International Journal of Non-Linear Mechanics, 144, 104053 (2022)
[15]	WANG, Y. K., LV, J. H., YANG, M., ZHANG, Y., QIN, T., and WANG, L. Stability analysis of an articulated flexible pipe conveying fluid with a rotational spring. Journal of Fluids and Structures, 133, 104277 (2025)
[16]	JAYAN, M. M. S., WANG, L. F., SELVAMANI, R., and RAMYA, N. Analysis of vibrational properties of horn-shaped magneto-elastic single-walled carbon nanotube mass sensor conveying pulsating viscous fluid using Haar wavelet technique. Acta Mechanica Solida Sinica, 37(5), 685–699 (2024)
[17]	TANG, Y., WANG, G., and DING, Q. Nonlinear fractional-order dynamic stability of fluid-conveying pipes constituted by the viscoelastic materials with time-dependent velocity. Acta Mechanica Solida Sinica, 35(5), 733–745 (2022)
[18]	CHEN, Y., YANG, X. D., and LIANG, F. Nonlocal thermal-mechanical vibration of spinning functionally graded nanotubes conveying fluid based on the Timoshenko model. Acta Mechanica Solida Sinica (2025) https://doi.org/10.1007/s10338-024-00574-5
[19]	HHAN, D. H., JIA, Q., GAO, Y. Y., JIN, Q. D., FANG, X., WEN, J. H., and YU, D. L. Local resonance metamaterial-based integrated design for suppressing longitudinal and transverse waves in fluid-conveying pipes. Applied Mathematics and Mechanics (English Edition), 45(10), 1821–1840 (2024) https://doi.org/10.1007/s10483-024-3166-8
[20]	JING, J., MAO, X. Y., DING, H., LI, H. G., and CHEN, L. Q. Modeling and mechanism of vibration reduction of pipes by visco-hyperelastic materials. Applied Mathematics and Mechanics (English Edition), 46(6), 1029–1048 (2025) https://doi.org/10.1007/s10483-025-3263-6
[21]	YU, T. C., LIANG, F., and YANG, H. L. Vibration energy harvesting of a three-directional functionally graded pipe conveying fluids. Applied Mathematics and Mechanics (English Edition), 46(5), 795–812 (2025) https://doi.org/10.1007/s10483-025-3249-8
[22]	HAO, M. Y., DING, H., MAO, X. Y., and CHEN, L. Q. Stability and nonlinear response analysis of parametric vibration for elastically constrained pipes conveying pulsating fluid. Acta Mechanica Solida Sinica, 36(2), 230–240 (2023)
[23]	WU, Y. F., CHEN, E. W., RUAN, Z. W., ZHANG, P. P., CHEN, P., and LU, Y. M. Nonlinear vibration analysis of a fluid-conveying pipe under harmonic excitation with elastic boundary constraints. International Journal of Non-Linear Mechanics, 170, 104981 (2025)
[24]	CHEN, W. J., QU, X. C., ZHANG, R. X., GUO, X. M., MA, H., and WEN, B. C. Nonlinear stress analysis of aero-engine pipeline based on semi-analytical method. Applied Mathematics and Mechanics (English Edition), 46(3), 521–538 (2025) https://doi.org/10.1007/s10483-025-3225-8
[25]	WEI, S., YAN, X., FAN, X., MAO, X. Y., DING, H., and CHEN, L. Q. Vibration of fluid-conveying pipe with nonlinear supports at both ends. Applied Mathematics and Mechanics (English Edition), 43(6), 845–862 (2022) https://doi.org/10.1007/s10483-022-2857-6
[26]	ALVIS, T., SAUNDERS, B. E., and ABDELKEFI, A. Dynamical responses of constrained pipe conveying fluids and its dependence on the modeling of the contact force. International Journal of Non-Linear Mechanics, 150, 104364 (2023)
[27]	FAN, C. Z., GUO, C. Q., XU, F., and WANG, T. L. Research on impact vibration response of hinged fluid-conveying pipe with bilateral gap constraints. International Journal of Non-Linear Mechanics, 162, 104726 (2024)
[28]	GHOLAMI, M. and EFTEKHARI, M. Nonlinear forced vibration in a subcritical regime of a porous functionally graded pipe conveying fluid with a retaining clip. Applied Mathematics and Mechanics (English Edition), 46(1), 101–122 (2025) https://doi.org/10.1007/s10483-025-3206-9
[29]	LI, H. F., SUN, Y. B., WEI, S., DING, H., and CHEN, L. Q. Random vibration of a pipe conveying fluid under combined harmonic and Gaussian white noise excitations. Acta Mechanica Solida Sinica (2025) https://doi.org/10.1007/s10338-025-00586-9
[30]	LI, M. W. and WANG, L. Parametric model reduction for a cantilevered pipe conveying fluid via parameter-dependent center and unstable manifolds. International Journal of Non-Linear Mechanics, 160, 104629 (2024)
[31]	LOUISE, E. R. Vibration and stability of vertical tubes conveying fluid subjected to planar excitation, Ph. D. dissertation, University of Wisconsin-Madison (1988)
[32]	NI, Q., ZHANG, Z. L., WANG, L., QIAN, Q., and TANG, M. Nonlinear dynamics and synchronization of two coupled pipes conveying pulsating fluid. Acta Mechanica Solida Sinica, 27(2), 162–171 (2014)
[33]	LÜ, L., HU, Y. J., WANG, X. L., LING, L., and LI, C. G. Dynamical bifurcation and synchronization of two nonlinearly coupled fluid-conveying pipes. Nonlinear Dynamics, 79(4), 2715–2734 (2015)
[34]	GAO, P. X., YU, T., ZHANG, Y. L., WANG, J., and ZHAI, J. Y. Vibration analysis and control technologies of hydraulic pipeline system in aircraft: a review. Chinese Journal of Aeronautics, 34(4), 83–114 (2021)
[35]	JIANG, T. L., ZHANG, L. B., GUO, Z. L., YAN, H., DAI, H. L., and WANG, L. Three-dimensional dynamics and synchronization of two coupled fluid-conveying pipes with intermediate springs. Communications in Nonlinear Science and Numerical Simulation, 115, 106777 (2022)
[36]	PAÏDOUSSIS, M. P. Fluid-Structure Interactions: Slender Structures and Axial Flow, Academic Press, London (1998)
[37]	HOLMES, P. J. Bifurcations to divergence and flutter in flow-induced oscillations: a finite dimensional analysis. Journal of Sound and Vibration, 53(4), 471–503 (1977)
[38]	WADHAM-GAGNON, M., PAÏDOUSSIS, M. P., and SEMLER, C. Dynamics of cantilevered pipes conveying fluid, part 1: nonlinear equations of three-dimensional motion. Journal of Fluids and Structures, 23(4), 545–567 (2007)