Abstract: Slender fluid-conveying pipes are widely used in critical engineering systems, including engine hydraulic circuits, aviation refueling equipment, nuclear heat exchangers, and offshore drilling platforms. At high flow velocities, these pipelines can experience flow-induced instabilities, such as buckling and flutter, which in severe cases may lead to system failure or catastrophic accidents. Flow-induced instabilities and the nonlinear vibration behavior of fluid-conveying pipelines represent typical fluid–structure interaction problems, the study of which is of great significance in structural dynamics and fluid–structure interaction mechanics. With ongoing research, focus has gradually shifted from linear analyses based on small-deformation assumptions to the study of large-deformation and large-displacement dynamics in flexible, slender pipelines subjected to high flow velocities or strong external loads. This paper provides a systematic review of recent progress in the study of nonlinear vibrations of slender fluid-conveying pipelines, with particular emphasis on the large-deformation bending dynamics.
First, the nonlinear characteristics of fluid-conveying pipeline systems and the validity of common modeling assumptions are discussed. Second, key modeling and solution approaches—including the Taylor expansion approximation, geometrically exact theory, absolute nodal coordinate formulation, and data-driven models—are presented. Next, the nonlinear dynamic characteristics and evolutionary behavior of two typical systems—end-supported and cantilevered pipelines—are reviewed, with a focus on recent findings concerning the transition of cantilevered pipelines from small-deformation assumptions to large-deformation responses. Finally, the current state of research on large-deformation dynamics in fluid-conveying pipelines is summarized, and several fundamental scientific questions that warrant further investigation are identified.
Fluid transport pipelines, also known as flow pipes, constitute a critical class of slender structures in engineering fields such as aviation, aerospace, marine engineering, nuclear power, hydropower, machinery, and chemical engineering (see Figures 1(a)–1(c)). These pipelines primarily serve to connect components and convey fluids, and their safety and reliability are crucial for the stable operation of the associated engineering systems. Numerous serious safety incidents caused by the vibration of flow pipes have been reported in aviation, aerospace, and marine equipment. For example, several aircraft refueling hoses have exhibited abnormal vibrations and subsequent failures during aerial refueling operations. Similarly, numerous offshore oil riser failures have been reported worldwide, most of which have been attributed to the complex flow-induced vibrations of pipelines subjected to ocean currents. However, the vibration of flow pipes has both detrimental and beneficial effects. Excessive vibration can, on the one hand, lead to fatigue damage or structural failure, but on the other hand, it can be harnessed for functional or bio-inspired applications. For example, inspired by the oscillatory motion of slender organisms such as snakes and eels, researchers have developed tubular-driven underwater biomimetic robots that exploit the flow-induced vibration mechanism of pipelines (see Figure 1(d)). Due to their broad range of engineering applications, the dynamics and control of fluid-conveying pipelines have attracted considerable attention from both academia and industry, spanning nearly a century of research (Paidoussis, 2022). Through theoretical modeling, computational analysis, and experimental investigation, researchers have extensively examined the stability and complex vibration mechanisms of fluid–structure coupled pipeline systems, with many findings successfully applied in engineering practice. In recent years, with the expansion of application scenarios, the design philosophy of fluid pipelines has shifted toward increased slenderness and flexibility. Consequently, the dynamics and control of flexible pipelines have emerged as a key focus of research. Owing to their low structural stiffness, slender flexible pipelines are susceptible to large-deformation (large-displacement or large-amplitude) vibration behaviors under internal flow excitation, presenting new challenges for dynamic modeling, computational analysis, and control design.
Research on fluid pipeline vibration dates back to the experimental work of Brillouin (1885), who first observed and documented the phenomenon of self-excited vibration in cantilevered fluid-conveying pipelines (Paidoussis, 2011, 2022). Since then, the dynamics of such pipelines have increasingly attracted attention from both academia and industry, with early investigations primarily focusing on the linear stability of pipeline systems. In the early 1950s, researchers including Feodos’ev (1951), Housner (1952), and Niordson (1953) derived the governing equations for the linear vibration of two-end-supported pipelines using diverse analytical approaches.
Their findings consistently revealed a fundamental conclusion: when the internal fluid velocity exceeds a certain critical threshold—referred to as the critical velocity—the pipeline system becomes dynamically unstable. Building on these pioneering studies, subsequent researchers refined and extended the theoretical framework of linear vibration for fluid-conveying pipelines. For instance, studies have investigated the instability characteristics of pipelines under various boundary conditions, including two-end-supported, cantilevered, and multi-supported configurations (Paidoussis, 1970; Lottati & Kornecki, 1986; Wu & Shih, 2001; Ding et al., 2019). Other studies examined the effects of steady and unsteady internal flows on system stability (Paidoussis & Issid, 1974; Zhang & Chen, 2013), while some researchers analyzed the linear dynamic behavior of articulated rigid fluid-conveying pipe systems (Benjamin, 1961a, 1961b). Through the collective efforts of researchers worldwide, understanding of the dynamic characteristics of fluid-conveying pipeline systems has steadily advanced (Paidoussis, 1987). The key findings can be summarized as follows:
Additionally, external axial forces, elastic constraints, damping, and gravity can all affect the critical flow velocity at which the system becomes unstable. It is important to note that, although linear theory and its governing equations can accurately determine system modes and predict critical velocities, they are insufficient for capturing the nonlinear dynamic behavior of pipelines after instability occurs.
To investigate the dynamics of fluid-conveying pipeline systems and elucidate their nonlinear vibration behavior beyond the onset of instability, researchers have conducted extensive nonlinear theoretical modeling and computational studies. The primary sources of nonlinearity in these systems include geometric nonlinearity, material nonlinearity, and external-force nonlinearity. Early studies on the nonlinear dynamics of fluid-conveying pipelines primarily focused on planar (two-dimensional) vibration characteristics (Semler et al., 1994; Semler & Paidoussis, 1995; Zhou et al., 2017; Tang et al., 2018).
These studies revealed a broad spectrum of nonlinear dynamic behaviors, including limit cycles, periodic, quasi-periodic, and chaotic motions. Among these studies, the work of Semler et al. (1994) is particularly significant. Using both the energy method and Newton’s method, they derived the two-dimensional nonlinear governing equations for the vibrations of cantilevered and end-supported fluid-conveying pipelines under the small-deformation assumption. The validity of these equations was confirmed by comparison with existing vibration formulations, and they continue to serve as a foundational reference in contemporary research. Building upon the theoretical framework established by Semler et al. (1994), numerous researchers have further examined the planar nonlinear dynamics of fluid-conveying pipelines by incorporating additional nonlinear effects, including magnetic field forces (Tang & Dowell, 1988; Dai & Wang, 2016), nonlinear inertial forces (Copeland & Moon, 1992; Paidoussis & Semler, 1998; Yi et al., 2020), nonlinear spring forces (Paidoussis & Semler, 1993b; Jin & Zou, 2003), and kinematic constraints (Paidoussis et al., 1991; Paidoussis & Semler, 1993a; Wang et al., 2017b; Paidoussis & Semler, 2016). These studies demonstrated that, under the combined influence of multiple nonlinear effects, fluid-conveying pipelines can exhibit complex dynamic responses, including quasi-periodic and chaotic motions. It should be noted that these studies of fluid–structure interactions in fluid-conveying pipelines primarily focused on the coupling between the pipe wall and the internal fluid flow, without considering interactions with external media.
As research progressed, scholars recognized that two-dimensional dynamic models could not fully capture the nonlinear vibration behavior of fluid-conveying pipelines, necessitating the development of three-dimensional dynamic models and associated computational analyses. The theoretical modeling by Lundgren et al. (1979) is considered one of the pioneering studies on the three-dimensional dynamics of cantilevered fluid-conveying pipelines. They analyzed a cantilevered pipeline system with an inclined outlet nozzle and derived the governing equations describing its non-planar vibration behavior. Subsequent studies by Bajaj and Sethna (1982) and Wadham-Gagnon et al. (1984) further explored the three-dimensional nonlinear dynamics of discontinuous articulated pipelines and continuous cantilevered pipelines. However, owing to the limitations of that period, the governing vibration equations in these studies varied considerably—some incorporated redundant parameters, while others omitted essential dynamic factors. Consequently, these early formulations were not widely adopted.
A major advancement was achieved by Wadham-Gagnon et al. (2007), who systematically derived the three-dimensional nonlinear vibration equations for cantilevered pipelines under the small-deformation assumption. Their model simultaneously captured the coupling of lateral vibrations in two orthogonal planes, making it one of the most comprehensive formulations for straight cantilevered pipelines at the time. Building upon this foundation, researchers have made significant contributions to the study of three-dimensional fluid-conveying pipeline dynamics. For example, several studies investigated the periodic and chaotic responses of pipelines under spring supports and gap constraints (Modarres-Sadeghi et al., 2007; Paidoussis et al., 2007; Wang et al., 2018), while others examined the forced vibration behavior of pipelines subjected to external excitations (Chen, 1970; Chang & Modarres-Sadeghi, 2014; Li & Yang, 2014). Additionally, several studies investigated flow-induced vibrations in pipelines interacting with external fluids, such as surrounding water or air (Hannoyer & Paidoussis, 1978; Dai et al., 2013; Abdelbaki et al., 2019; Païdoussis, 2021; Qu & Metrikine, 2021). In engineering practice, focus is placed not only on understanding the dynamic mechanisms of pipeline systems but also on controlling their vibrations. Vibration control in fluid-conveying pipelines is generally classified into passive and active strategies. Passive control typically employs three approaches: vibration absorption, resistance enhancement, and vibration isolation (Ding, 2022). Common techniques include spring–mass absorbers (Liu et al., 2019; Guo et al., 2021), tuned mass dampers (Rechenberger & Mair, 2017), quasi-zero-stiffness resonators (Cai et al., 2017; 2023), nonlinear energy sinks (Yang et al., 2014; Cao et al., 2022), and damping materials (Bakre et al., 2006). In contrast, active control modifies the system’s dynamic behavior through active or semi-active interventions, allowing real-time adjustment of system parameters according to vibration states to achieve precise regulation. Passive control offers the advantages of simplicity and low cost, while active control provides greater adaptability and responsiveness. Therefore, combining the strengths of both approaches through hybrid control strategies has emerged as a promising way to achieve cost-effective and high-performance vibration control in fluid-conveying pipeline systems.
The above comprehensive review demonstrates that both linear stability and nonlinear dynamic analyses of fluid-conveying pipelines have a long research history, and that researchers have developed a profound understanding of their dynamic behavior. Over the past several decades, numerous researchers have published comprehensive review papers on the dynamics of fluid-conveying pipelines (Paidoussis & Li, 1993b; Zhang et al., 2000; Xu & Yang, 2004; Paidoussis, 2022; Tang et al., 2013; et al., 2023). Most of these reviews have primarily focused on the linear dynamics or the nonlinear small-deformation dynamics of fluid-conveying pipeline systems. However, as pipeline designs evolve toward thinner and more flexible structures, the stiffness of the pipelines decreases, making them more susceptible to large-deformation dynamics under high internal flow velocities or strong external loads. When a pipeline undergoes large-deformation vibration, its lateral displacement may become comparable to its total length, leading to pronounced oscillations over a wide spatial range. Under these conditions, the conventional dynamic theory based on the small-deformation assumption becomes invalid. Encouragingly, significant progress has been achieved in recent years in understanding the large-deformation dynamics of fluid-conveying pipelines. In this context, the present paper reviews these emerging research trends to help readers and researchers quickly grasp the key scientific issues and latest developments in this field. It should be noted that although this paper primarily focuses on the large-deformation dynamics of fluid-conveying pipelines, representative studies on small-deformation dynamics are also briefly reviewed to provide a more comprehensive perspective on the topic. The deformation of slender pipelines can generally be decomposed into two components: axial tension–compression and bending. For the cantilevered pipelines considered in this study, bending deformation is far more pronounced than axial tension–compression. It should be noted that the term “large deformation” in this context primarily refers to large displacement amplitudes and bending deflections, rather than large midline strains. Since bending is the main contributor to large deformations in pipelines, these deformations usually do not produce significant strains, setting them apart from the notion of large deformation in solid mechanics.
