Nonlinearity in dynamic systems is a relative concept that depends on both the physical nature of the problem and the choice of coordinate system. For example, a circular arc appears linear in polar coordinates but nonlinear in Cartesian coordinates. Similarly, if curvature is used to quantify the large bending deformation of a slender pipe, the bending elastic force term appears linear; however, when displacement is used as the measure, the bending elastic force clearly exhibits complex nonlinear characteristics. It should be noted, however, that even though curvature provides a linear representation of the bending elastic force, the system itself is not necessarily linear, since the inertial forces under this formulation remain nonlinear. When analyzing the dynamics of slender pipeline systems, inertial forces play a crucial role. These forces are inherently variable but can be expressed in a linear form with respect to displacement. Consequently, in this article, all nonlinearities are described from the perspective of displacement, which is consistent with prevailing conventions in current academic research. Here, the axial coordinate along the pipeline centerline is denoted as s, the axial displacement as u, and the lateral displacement as v. For the dynamic problems considered, expressing geometric quantities in terms of displacement introduces nonlinear effects in the bending elastic force rather than in the inertial force. Using the energy method as an example, the derivation of the dynamic equations for a slender pipeline system generally involves three stages: expressing strain in terms of displacement (the geometric relationship), expressing stress in terms of strain (the constitutive relationship), and formulating the energy variation based on these two relationships.Nonlinear effects can arise at each of these stages, manifesting as geometric nonlinearity (Semler et al., 1994), material nonlinearity (Guo et al., 2023), and external-force nonlinearity (Paidoussis & Semler, 1993a). Geometric nonlinearity arises from the pipeline’s boundary conditions and has two main sources: one dominated by axial curvature and the other by centerline strain (E₀). In cantilevered pipelines, the ends are free from displacement constraints, allowing unrestricted deformation. Since bending strain develops more readily than tensile strain, even large displacements usually produce only small—and often negligible—centerline strain (Paidoussis & Semler, 1993a; Semler et al., 1994). In this case, the geometric nonlinearity originates primarily from variations in axial curvature. For pipelines supported at both ends (e.g., simply supported–simply supported, clamped–clamped, or simply supported–clamped), the displacement constraints at the boundaries induce noticeable centerline strain, thereby introducing nonlinear effects. When centerline strain is the dominant factor, the geometric nonlinearity is mainly governed by the von Kármán strain (Wang et al., 1994, 2009). Material nonlinearity arises when the pipeline material follows a nonlinear stress–strain relationship. For instance, pipelines made of soft materials such as rubber can experience large strains while still recovering their original shape, exhibiting hyperelastic behavior (Guo et al., 2023). External force nonlinearity is common in pipeline systems. Its sources include non-ideal boundary conditions—such as gap constraints (Paidoussis & Semler, 1993a), viscoelastic supports (Ghayesh et al., 2011), and contact friction (Guo et al., 2022b)—as well as additional vibration control components, including inertia vessels (Guo et al., 2021) and nonlinear energy sinks (Zhao et al., 2018).

Fluid-conveying pipeline systems can be categorized from various perspectives, including flow pattern, boundary condition, initial configuration, material composition, geometric scale, and motion dimensionality. This section provides a brief overview of these classifications and highlights the distinctive dynamic characteristics associated with each type. Based on flow patterns, fluid loading in pipelines can be classified into internal and external flows. Internal flows are further categorized as steady, transient, pulsating, or stochastic, and may involve single-phase, two-phase, or multiphase fluids. In the early stages of pipeline dynamics research, steady internal flow was the main focus and continues to be a central topic today. In steady-flow pipelines, both fluid density and velocity remain constant throughout the system. Instability in these systems arises from fluid-induced negative damping, resulting in self-excited oscillations. Transient internal flow occurs when fluid velocity changes abruptly over a short period, generating significant water hammer effects (Tijsseling). Common causes include valve closures, pump startups or shutdowns, pipeline damage, and seismic events. Water hammer can severely shorten pipeline lifespan, leading to fatigue damage or, in extreme cases, catastrophic failure. Theoretical studies of pipelines under transient flow have progressed from early water hammer models that ignored fluid–structure interaction, to coupled models including friction, Poisson effects, connections, and Bourdon effects, and finally to modern models fully incorporating fluid–structure interaction (Tijsseling, 1996). Pulsating internal flow, another common condition in engineering practice, has garnered significant attention. The fluid velocity in such flows is often expressed harmonically, for example, v=V（1+psinωt). Compared with steady flow, pulsating flow generates more complex dynamic behaviors in pipelines, such as parametric resonance and, in some cases, chaotic vibrations. Stochastic internal flow involves random variations in fluid velocity, and analyzing the dynamics of pipelines under such conditions typically requires the application of random vibration theory. Two-phase and multiphase flows involve mixtures of two or more fluid phases. In these cases, pipeline dynamics must consider fluid–structure interaction and include fluid-specific parameters that reflect the properties of the different phases. External flow in pipelines can be classified according to its direction relative to the pipe axis: transverse flow, which is perpendicular to the axis, and axial flow, which is parallel to it. High-velocity transverse flows can trigger vortex-induced vibration (VIV), a large-amplitude oscillation resulting from vortices shedding alternately on either side of the pipeline (Paidoussis et al., 2010). Clearly, the dynamic behavior of slender, flexible pipelines becomes increasingly complex when subjected simultaneously to both internal and external flows.

Based on boundary conditions, fluid-conveying pipelines can be broadly categorized as two-end-supported pipes and cantilevered pipes. These basic categories can be further refined by incorporating additional constraints along the pipe length, leading to elastically supported pipes, nonlinearly constrained pipes, and other specialized configurations. Two-end-supported pipelines can be further classified based on the nature of their end supports: fixed–fixed, fixed–hinged, hinged–hinged, and hinged–fixed. The end supports may permit axial sliding or be constrained against it. Regardless of the end-support type or whether axial sliding is allowed, these systems are conservative (neglecting material and external damping), and their instability under steady internal flow manifests as static buckling. In contrast, cantilevered pipes, fixed at one end and free at the other, are typical non-conservative systems. Under steady internal flow, they exhibit dynamic flutter instability when the flow velocity exceeds a critical threshold (Paidoussis, 1987). This self-excited vibration constitutes the most characteristic dynamic behavior of cantilevered pipeline systems. Pipelines with additional constraints—such as multiple supports, spring supports, or elastic foundations—constitute typical hyperstatic systems in pipeline dynamics. Multi-supported pipelines, featuring three or more support points, exhibit hyperstatic behavior, with the number and placement of supports significantly affecting both system stability and dynamic response. Supports such as springs or elastic foundations increase the overall stiffness of the pipeline, thereby improving system stability. Nonlinear constraints are also common in pipeline systems and typically arise from contact and non-contact interactions. Non-contact constraints include electromagnetic or electrostatic forces, whereas contact constraints often involve nonlinear spring forces or gap-collision effects.

The initial configuration of a pipeline is also a key criterion for classifying fluid-conveying systems. Pipelines are typically categorized as straight or curved and can further be distinguished by having either uniform or variable cross-sections. Straight pipes, characterized by an initially linear configuration, were the primary focus of early research on pipeline dynamics and can be regarded as a special case within more complex pipe geometries. Curved pipes, in contrast, possess an initially curved configuration and can be further classified into several subtypes: slightly curved, arc-shaped, straight–bend combination, and arbitrarily three-dimensional curved pipes. Slightly curved pipes usually result from construction or manufacturing imperfections rather than intentional design, and even minor deviations from straightness can affect the pipeline’s stability and dynamic response. Arc-shaped pipes—including quarter-circle, semicircle, and three-quarter-circle geometries—have fixed initial curvatures and represent a more specialized category of curved pipelines. Straight–bend combination pipes, composed of both straight and curved segments, are commonly used in nuclear industry heat exchangers. Their complex geometry leads to more intricate structural characteristics and typically results in richer dynamic behavior. Three-dimensional curved pipes can follow arbitrary spatial trajectories, such as spirals or hyperbolic curves, and are mainly employed in heat exchangers and condenser systems. Additionally, articulated pipes, consisting of two or more segments connected by hinges, form a distinct geometric category. Research indicates that as the number of articulated segments increases, the fundamental dynamic behavior of the segmented pipe gradually approaches that of a continuous pipe, consistent with empirical observations (Bajaj & Sethna, 1982, 1984). Fluid-conveying pipelines can also be categorized based on material composition and distribution, distinguishing between uniform and non-uniform materials. Non-uniform pipelines comprise heterogeneous materials, including functionally graded materials (FGMs) and periodically varying materials. FGMs have garnered significant attention in pipeline dynamics research and are usually classified according to axial or radial property gradients. Functionally graded materials (FGMs) have been used to customize the mechanical properties of fluid-conveying pipelines; for instance, a pipeline with a continuously varying elastic modulus along a specific direction can effectively reduce or eliminate stress concentrations (Paidoussis, 2022). Based on geometric scale, pipelines can be classified as macroscale, microscale, or nanoscale, with microscale and nanoscale pipelines having transverse dimensions on the order of micrometers and nanometers, respectively. At these scales, classical continuum theory alone is generally inadequate, requiring consideration of scale effects from both the solid material and the fluid. These effects can be investigated using molecular dynamics simulations or approximated via modified continuum theories. Due to their higher computational efficiency, modified continuum theories are widely employed in the analysis of microscale and nanoscale pipelines. Commonly used constitutive theories in this context include nonlocal elasticity theory, modified couple stress theory, strain gradient theory, and surface elasticity theory (Paidoussis, 2022).

The dimension of motion can also serve as a classification criterion for fluid pipelines, dividing systems into planar (two-dimensional) and three-dimensional (spatial) dynamic systems. Although a straight fluid pipeline can be regarded as a one-dimensional structure—meaning its displacement vector varies only along a single coordinate, s—the displacement may occur in either a planar or a spatial direction. This distinction leads to two categories of dynamic problems: two-dimensional and three-dimensional. Early studies, limited by modeling techniques and computational resources, focused primarily on two-dimensional dynamic modeling and analysis of fluid-conveying pipelines. Indeed, when system parameters lie within certain specific ranges, pipelines often deform and vibrate predominantly within a single plane, making a two-dimensional model adequate for basic dynamic analysis. However, as system complexity increases, fluid pipelines inevitably experience three-dimensional vibrations under multi-source load excitation, making nonplanar modeling and analysis essential for accurately predicting their dynamic behavior. Fluid pipeline systems can also be classified as linear or nonlinear. Early linear analyses effectively identified the modes of instability and the system’s critical flow velocity. However, linear analysis alone cannot capture the nonlinear dynamic behavior that may emerge after instability. Consequently, it is typically used as a foundation for nonlinear investigations, making nonlinear dynamic modeling and computation a central focus in modern pipeline dynamics research. It is worth noting that the curvature-based third-order Taylor expansion model proposed by Wadham-Gagnon et al. (2007) has inherent limitations when applied to the large-deformation dynamics of fluid-conveying pipelines, which is the primary focus of this paper. To address this limitation, recent studies have developed a variety of theoretical and computational models specifically targeting large-deformation pipeline dynamics, thereby advancing the theoretical framework for the nonlinear behavior of flexible fluid-conveying pipelines.