Stochastic stability analysis of fluid-conveying pipes under multiplicative Gaussian white noise excitations

doi:10.1007/s10483-026-3375-9

Abstract

Abstract:

This study investigates the stochastic stability of simply-supported fluid-conveying pipes under multiplicative Gaussian white noise excitations. The high-dimensional coupled stochastic differential equations for the fluid-conveying pipe are established through the generalized Hamilton principle and the Galerkin truncation method. The stochastic averaging method for quasi non-integrable Hamilton systems is used to decouple and reduce the dimension of the high-dimensional coupled pipe system with gyroscopic forces, yielding a one-dimensional Itô stochastic differential equation for the total energy of the pipe. According to the singular boundary classification theory of one-dimensional time-homogeneous diffusion processes, a stochastic stability criterion for the fluid-conveying pipe is proposed. The stability analyses of the time history responses of energy, displacement, and velocity for the pipe system are obtained via the Monte Carlo approach, thereby verifying the effectiveness of the proposed stability criterion in different parameter planes. Furthermore, the effects of system parameters, such as the fluid speed, multiplicative Gaussian white noise intensity, and pipe length, on the stochastic stability domain of the pipe system are discussed. The results indicate that as the fluid speed, the multiplicative Gaussian white noise intensity acting on the first-order mode, or pipe length increases, the stable domain of the system decreases. Conversely, the stable domain of the fluid-conveying pipe system increases as the multiplicative Gaussian white noise intensity acting on the second-order mode, pipe thickness, or Young’s modulus increases. It is worth noting that the appropriate increase in the multiplicative Gaussian white noise intensity acting on the second-order mode contributes to improving the stable domain of system. This method lays a theoretical foundation for the safe and stable operation of fluid-conveying pipes under random vibration.

Key words: fluid-conveying pipe, stochastic stability, multiplicative Gaussian white noise, stochastic averaging method, singular boundary classification theory

2010 MSC Number:

O324

Hufei LI, Sha WEI, Hu DING, Liqun CHEN. Stochastic stability analysis of fluid-conveying pipes under multiplicative Gaussian white noise excitations. Applied Mathematics and Mechanics (English Edition), 2026, 47(4): 741-766.

TrendMD

Figures/Tables 10

Fig. 1

Fig. 2

Fig. 3

Table 1

Physical parameters for the fluid-conveying pipe"

Item	Notation	Value
Outer diameter	$D out$ /m	0.02
Pipe thickness	h/m	0.004
Inner diameter	$D in$ /m	0.016
Young’s modulus	E/Pa	$7.2 × 1010$
Pipe length	L/m	1
Fluid density	$ρ f / (kg ⋅ m − 3)$	1 000
Pipe density	$ρ p / (kg ⋅ m − 3)$	2 700
Fluid speed	$Γ / (m ⋅ s − 1)$	10
Multiplicative Gaussian noise intensity acting on the first-order mode	D₁	0.02
Multiplicative Gaussian noise intensity acting on the second-order mode	D₂	0.03

Table 1

Fig. 4

Fig. 5

Fig. 6

Fig. 7

Fig. 8

Fig. 9

References 66

[1]	MA, S. C., NING, X., WANG, L., JIA, W. T., and XU, W. Complex response analysis of a non-smooth oscillator under harmonic and random excitations. Applied Mathematics and Mechanics (English Edition), 42(5), 641–648 (2021) https://doi.org/10.1007/s10483-021-2731-5
[2]	SUWEIS, S., FERRARO, F., GRILLETTA, C., AZAELE, S., and MARITAN, A. Generalized Lotka-Volterra systems with time correlated stochastic interactions. Physical Review Letters, 133(16), 167101 (2024)
[3]	GUO, C. X., GUO, B. L., and YANG, H. H²-regularity random attractors of stochastic non-Newtonian fluids with multiplicative noise. Applied Mathematics and Mechanics (English Edition), 35(1), 105–116 (2014) https://doi.org/10.1007/s10483-014-1776-7
[4]	ZHOU, X. Y., LI, H. X., and HUANG, Y. Stochastic averaging for estimating a ship roll in random longitudinal or oblique waves. Marine Structures, 75, 102814 (2021)
[5]	SAYED, M. and HAMED, Y. S. Stability and response of a nonlinear coupled pitch-roll ship model under parametric and harmonic excitations. Nonlinear Dynamics, 64(3), 207–220 (2011)
[6]	WEI, Z. C., LI, Y. X., KAPITANIAK, T., and ZHANG, W. Analysis of chaos and capsizing of a class of nonlinear ship rolling systems under excitation of random waves. Chaos: An Interdisciplinary Journal of Nonlinear Science, 34(4), 043106 (2024)
[7]	DENG, M. L., MU, G. J., and ZHU, W. Q. Random response of wake-oscillator excited by fluctuating wind. Journal of Applied Mechanics, 88(10), 101002 (2021)
[8]	ASWATHY, M. S. and SARKAR, S. Effect of stochastic parametric noise on vortex induced vibrations. International Journal of Mechanical Sciences, 153, 103–118 (2019)
[9]	LÜ, Q. F., LI, D. Y., ZHU, W. Q., and DENG, M. L. Reliability analysis of vortex-induced random vibration. ASCE-ASME Journal of Risk and Uncertainty in Engineering Systems, Part A: Civil Engineering, 11(3), 04025026 (2025)
[10]	ZHENG, Y. and HUANG, J. H. Stochastic stability of FitzHugh-Nagumo systems perturbed by Gaussian white noise. Applied Mathematics and Mechanics (English Edition), 32(1), 11–22 (2011) https://doi.org/10.1007/s10483-011-1389-7
[11]	LIU, W. Y., YIN, X. R., GUO, Z. J., and JIANG, S. Stochastic stabilization of quasi integrable and non-resonant Hamiltonian systems under combined Gaussian and Poisson white noise excitation. Probabilistic Engineering Mechanics, 79, 103733 (2025)
[12]	TUGNAIT, J. Stability of optimum linear estimators of stochastic signals in white multiplicative noise. IEEE Transactions on Automatic Control, 26(3), 757–761 (1981)
[13]	HUA, M. J. and WU, Y. Bifurcation in most probable phase portraits for a bistable kinetic model with coupling Gaussian and non-Gaussian noises. Applied Mathematics and Mechanics (English Edition), 42(12), 1759–1770 (2021) https://doi.org/10.1007/s10483-021-2804-8
[14]	XUE, J. R., ZHANG, Y. W., DING, H., and CHEN, L. Q. Vibration reduction evaluation of a linear system with a nonlinear energy sink under a harmonic and random excitation. Applied Mathematics and Mechanics (English Edition), 41(1), 1–14 (2020) https://doi.org/10.1007/s10483-020-2560-6
[15]	KUMAR, P., NARAYANAN, S., and GUPTA, S. Bifurcation analysis of a stochastically excited vibro-impact Duffing-van der Pol oscillator with bilateral rigid barriers. International Journal of Mechanical Sciences, 127, 103–117 (2017)
[16]	DEVARAJAN, K. and SANTHOSH, B. Nonlinear dynamics and performance analysis of modified snap-through vibration energy harvester with time-varying potential function. Applied Mathematics and Mechanics (English Edition), 43(2), 185–202 (2022) https://doi.org/10.1007/s10483-022-2812-8
[17]	CHEN, H. Y., GUO, Y. F., SONG, Q. Z., and YU, Q. Stochastic resonance of a fractional-order bistable system driven by corrected non-Gaussian noise and Gaussian noise. Journal of the Franklin Institute, 362(6), 107593 (2025)
[18]	MA, J. Z., XU, Y., LI, Y. G., TIAN, R. L., MA, S. J., and KURTHS, J. Quantifying the parameter dependent basin of the unsafe regime of asymmetric Lévy-noise-induced critical transitions. Applied Mathematics and Mechanics (English Edition), 42(1), 65–84 (2021) https://doi.org/10.1007/s10483-021-2672-8
[19]	RAMAKRISHNAN, S. and EDLUND, C. Stochastic stability of a piezoelectric vibration energy harvester under a parametric excitation and noise-induced stabilization. Mechanical Systems and Signal Processing, 140, 106566 (2020)
[20]	SHE, J., MA, S. J., and LI, H. F. Robust H∞ control of wind turbine drive system excited by multiplicative fractional Brownian noise. Journal of the Franklin Institute, 362(13), 107940 (2025)
[21]	LI, Y. J., WU, Z. Q., LAN, Q. X., CAI, Y. J., XU, H. F., and SUN, Y. T. Transition behaviors of system energy in a bi-stable van Ver Pol oscillator with fractional derivative element driven by multiplicative Gaussian white noise. Thermal Science, 26, 2727–2736 (2022)
[22]	GUO, F., WANG, X. Y., QIN, M. W., LUO, X. D., and WANG, J. W. Resonance phenomenon for a nonlinear system with fractional derivative subject to multiplicative and additive noise. Physica A: Statistical Mechanics and its Applications, 562, 125243 (2021)
[23]	CHEN, J. B., SUN, T. T., SPANOS, P. D., and LI, J. Efficient stochastic response analysis of high-dimensional nonlinear systems subject to multiplicative noise via the DR-PDEE. Journal of Computational Physics, 531, 113929 (2025)
[24]	ER, G. K., ZHU, H. T., IU, V. P., and KOU, K. P. PDF solution of nonlinear oscillators subject to multiplicative Poisson pulse excitation on displacement. Nonlinear Dynamics, 55(4), 337–348 (2009)
[25]	PAıDOUSSIS, M. P. Fluid-structure Interactions: Slender Structures and Axial Flow, Academic Press, California (1998)
[26]	FU, X. P., FU, S. X., ZHANG, M. M., REN, H. J., ZHAO, B., and XU, Y. W. Vortex-induced vibration of a flexible pipe under oscillatory sheared flow. Physical Review Fluids, 9(1), 014604 (2024)
[27]	BARKLEY, D. Theoretical perspective on the route to turbulence in a pipe. Journal of Fluid Mechanics, 803, P1 (2016)
[28]	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, 38, 843–856 (2025)
[29]	WEI, S., SUN, Y. B., DING, H., and CHEN, L. Q. Random vibration and reliability analysis of fluid-conveying pipe under white noise excitations. Applied Mathematical Modelling, 123, 259–273 (2023)
[30]	WEI, S., LI, H. F., SUN, Y. B., LIU, Y., DING, H., and CHEN, L. Q. Random vibration and first-passage of a pipe conveying fluid under colored noise excitations. Journal of Vibration and Control (2025) https://doi.org/10.1177/10775463251360963
[31]	SAZESH, S. and SHAMS, S. Vibration analysis of cantilever pipe conveying fluid under distributed random excitation. Journal of Fluids and Structures, 87, 84–101 (2019)
[32]	QU, W., ZHANG, H. L., SUN, W. Q., and LI, W. Stress response of the hydraulic composite pipe subjected to random vibration. Composite Structures, 255, 112958 (2021)
[33]	QU, W., JIANG, Z. Q., HE, Z. Z., DING, H., and YU, H. Nonlinear dynamic characteristic analysis of pre-stressed hydraulic composite pipe subjected to random vibration. Thin-Walled Structures, 213, 113249 (2025)
[34]	ZHU, H. T., GENG, G. Q., YU, Y., and XU, L. X. Probabilistic analysis on parametric random vibration of a marine riser excited by correlated Gaussian white noises. International Journal of Non-Linear Mechanics, 126, 103578 (2020)
[35]	GUO, X. M., GU, J. F., LI, H., SUN, K. H., WANG, X., ZHANG, B. J., ZHANG, R. W., GAO, D. W., LIN, J. Z., WANG, B., LUO, Z., SUN, W., and MA, H. Dynamic modeling and experimental verification of an L-shaped pipeline in aero-engine subjected to base harmonic and random excitations. Applied Mathematical Modelling, 126, 249–265 (2024)
[36]	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)
[37]	ALEXAKIS, A. and PÉTRÉLIS, F. Planar bifurcation subject to multiplicative noise: role of symmetry. Physical Review E. Statistical, Nonlinear, and Soft Matter Physics, 80(4), 041134 (2009)
[38]	ZHU, W. Q. Lyapunov exponent and stochastic stability of quasi-non-integrable Hamiltonian systems. International Journal of Non-Linear Mechanics, 39(4), 569–579 (2004)
[39]	DENG, Y. W., XU, B. B., CHEN, D. Y., and LIU, J. Stochastic global stability and bifurcation of a hydro-turbine generator. Communications in Nonlinear Science and Numerical Simulation, 72, 64–77 (2019)
[40]	SUN, J. J., LUO, Z. Q., and YAN, B. Stochastic stability of nonlinear mechanical metamaterial systems under combined Gaussian and Poisson white noises. Communications in Nonlinear Science and Numerical Simulation, 143, 108621 (2025)
[41]	WANG, X. Y., ZHANG, J. G., AN, X. L., HE, M. J., and WEI, L. X. Stochastic stability of an elastically constrained wheelset system under additive and multiplicative color noise excitations. Applied Mathematical Modelling, 145, 116120 (2025)
[42]	ZHANG, J. G., WANG, X. Y., HE, M. J., AN, X. L., and WEI, L. X. Stochastic bifurcation and nonlinear dynamics analysis of stochastic wheelset system with double time delays. Probabilistic Engineering Mechanics, 80, 103742 (2025)
[43]	WANG, P., YANG, S. P., LIU, Y. Q., LIU, P. F., ZHANG, X., and ZHAO, Y. W. Research on hunting stability and bifurcation characteristics of nonlinear stochastic wheelset system. Applied Mathematics and Mechanics (English Edition), 44(3), 431–446 (2023) https://doi.org/10.1007/s10483-023-2963-6
[44]	SUN, J. J., YING, Z. G., HUAN, R. H., and ZHU, W. Q. Asymptotic stability of controlled nonlinear stochastic systems considering the dynamics of sensors and actuators. International Journal of Structural Stability and Dynamics, 21(13), 2150180 (2021)
[45]	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
[46]	CHEN, L. Q., ZHANG, N. H., and ZU, J. W. The regular and chaotic vibrations of an axially moving viscoelastic string based on fourth order Galerkin truncaton. Journal of Sound and Vibration, 261(4), 764–773 (2003)
[47]	CHEN, L. Q., ZHANG, W., and ZU, J. W. Nonlinear dynamics for transverse motion of axially moving strings. Chaos, Solitons & Fractals, 40(1), 78–90 (2009)
[48]	BOLOTIN, V. V., GRISHKO, A. A., and PETROVSKY, A. V. Secondary bifurcations and global instability of an aeroelastic non-linear system in the divergence domain. Journal of Sound and Vibration, 191(3), 431–451 (1996)
[49]	WU, J. C. and LIU, X. B. Moment stability of viscoelastic system influenced by non-Gaussian colored noise. Journal of Sound and Vibration, 502, 116080 (2021)
[50]	ZHU, W. Q. and YANG, Y. Q. Stochastic averaging of quasi-nonintegrable-Hamiltonian systems. Journal of Applied Mechanics, 64(1), 157–164 (1997)
[51]	ITÔ K. On Stochastic Differential Equations, American Mathematical Soc., Providence, RI (1951)
[52]	ZHU, W. Q. Nonlinear stochastic dynamics and control in Hamiltonian formulation. Applied Mechanics Reviews, 59(4), 230–248 (2006)
[53]	KARNOPP, D. Computer simulation of stick-slip friction in mechanical dynamic systems. Journal of Dynamic Systems, Measurement, and Control, 107(1), 100–103 (1985)
[54]	HEO, B., BITTNER, H., SHUMWAY, M. L., and SHEN, I. Y. Identifying damping of a gyroscopic system through the half-power method and its applications to rotating disk/spindle systems. Journal of Vibration and Acoustics, 121(1), 70–77 (1999)
[55]	CHEN, C. Y., ZANETTE, D. H., CZAPLEWSKI, D. A., SHAW, S., and LÓPEZ, D. Direct observation of coherent energy transfer in nonlinear micromechanical oscillators. Nature Communications, 8(1), 15523 (2017)
[56]	SUN, J. J., HUAN, R. H., DENG, M. L., and ZHU, W. Q. A novel method for evaluating the averaged drift and diffusion coefficients of high DOF quasi-non-integrable Hamiltonian systems. Nonlinear Dynamics, 106(4), 2975–2989 (2021)
[57]	ABUNDO, M. On some properties of one-dimensional diffusion processes on an interval. Probability and Mathematical Statistics-Wroclaw University, 17, 277–310 (1997)
[58]	STROOCK, D. W. and VARADHAN, S. R. S. Diffusion processes with boundary conditions. Communications on Pure and Applied Mathematics, 24(2), 147–225 (1971)
[59]	FELLER, W. The parabolic differential equations and the associated semi-groups of transformations. Annals of Mathematics, 55(3), 468–519 (1952)
[60]	DENG, M. L. and ZHU, W. Q. Stochastic averaging of MDOF quasi integrable Hamiltonian systems under wide-band random excitation. Journal of Sound and Vibration, 305(4-5), 783–794 (2007)
[61]	LIU, W. Y., ZHU, W. Q., and XU, W. Stochastic stability of quasi non-integrable Hamiltonian systems under parametric excitations of Gaussian and Poisson white noises. Probabilistic Engineering Mechanics, 32, 39–47 (2013)
[62]	LIU, W. Y., ZHU, W. Q., and HUANG, Z. L. Effect of bounded noise on chaotic motion of Duffing oscillator under parametric excitation. Chaos, Solitons & Fractals, 12(3), 527–537 (2001)
[63]	LIN, Y. K. and CAI, G. Q. Probabilistic Structural Dynamics: Advanced Theory and Applications, McGraw-Hill, New York (1995)
[64]	DING, H., JI, J. C., and CHEN, L. Q. Nonlinear vibration isolation for fluid-conveying pipes using quasi-zero stiffness characteristics. Mechanical Systems and Signal Processing, 121, 675–688 (2019)
[65]	ZHU, W. Q., DENG M. L., and CAI, G. Q. Stochastic Averaging: Methods and Applications, Volume 2, Springer Nature, Beijing (2025)
[66]	ARIARATNAM, S. T. and NAMACHCHIVAYA, N. S. Dynamic stability of pipes conveying pulsating fluid. Journal of Sound and Vibration, 107(2), 215–230 (1986)