## 1 Introduction  
Hydraulic pipeline systems offer significant advantages, including high power density and exceptional stiffness under load. They are extensively utilized in the fields of aviation, aerospace, and the ship industry for the efficient transfer of power flow, energy flow, and mass flow [1; 2; 3]. In the ship industry, the design of hydraulic pipeline systems has historically prioritized function, often neglecting the critical aspects of vibration and noise. In reality, hydraulic pipeline systems have emerged as significant sources of vibrational radiation within ship environments. This is primarily due to factors such as pump-induced flow pulsations, environmental vibration, and fluid-structure interaction characteristics [4]. As underwater vessels, including submarine and unmanned underwater vehicles, increasingly demand high levels of acoustic stealth performance, it has become imperative to investigate the vibration and noise characteristics of hydraulic pipeline systems, as well as their control methods. Fluid-structure interaction (FSI) is a critical aspect that must not be overlooked when examining the vibration of hydraulic pipeline systems. In the past decades, many scholars have studied pipeline FSI regarding many aspects, such as mechanism explanation [5; 6], model construction [7] and engineering applications [8]. Pipeline FSI belongs to one of the general FSI. For hydraulic pipeline systems, the noise excitation source is usually the pulsating pressure of the fluid, so the nonlinear effect is usually ignored in the analysis process while the fluid-solid interaction between the pipeline and the fluid medium in the linear domain is emphasized. There are three types of interaction between fluid and the pipeline: Poisson coupling, friction coupling, and junction coupling. Poisson coupling and friction coupling exert their effects on the entire pipeline and can be described using first-order partial differential equations. In contrast, junction coupling primarily occurs at points of discontinuity in the pipeline or where there is a change in the direction of fluid flow; it must be expressed in conjunction with the pipeline equation and relevant boundary conditions. The pipeline FSI model has evolved from a classical water hammer model, which solely accounts for the characteristics of fluid pressure waves, to a comprehensive 14-equation model based on the Timoshenko beam theory. This advanced model incorporates not only the axial and lateral motions but also the torsional dynamics of the pipeline [9; 10]. In terms of computational methods, it primarily encompasses the method of characteristics (MOC), the finite element method (FEM), and the transfer matrix method (TMM). The MOC is a widely utilized method for addressing hyperbolic partial differential equations. In the domain of pipeline FSI, it is frequently employed for the time-domain transient response analysis of pressure waves within the pipeline. [PERSON] et al. [7] applied the MOC to investigate the pressure and stress responses in a spatially filled pipe featuring an elbow joint when the valve closes abruptly based on a 14-equation model. [PERSON] et al. [11] integrated an eight-equation model with the MOC to analyze the impact response of a planar liquid-filled elbow, providing an in-depth discussion on various boundary conditions. [PERSON] and [PERSON] [12] introduced an MOC based on the modified compatibility equation, and this method improved algorithm accuracy by circumventing simplifications associated with this equation. Nevertheless, the MOC is subject to multiple interpolation errors related to characteristic lines and presents challenges in managing complex boundary conditions within pipeline systems, which constrains its broader engineering applications [13]. The FEM is a powerful tool for analyzing the modal and dynamic response of structures, with numerous applications in the field of pipeline FSI.