## 1 Introduction
Based on the finite element theory, [PERSON] and [PERSON] [14] employed spline functions as shape functions to develop a motion model for a three-dimensional liquid-filled pipeline. Based on the 14-equation beam model, [PERSON] et al. [15] developed a finite element expression for pipelines that incorporated both Poisson coupling and junction coupling. The study focused on the water hammer effect as the target scenario. The findings indicate that neglecting FSI can lead to an overestimation of the calculated structural velocity. Furthermore, through recognizing the advantages of the MOC in addressing fluid dynamic characteristics alongside the efficacy of the FEM in handling structural vibration, many researchers have utilized the MOC-FEM to investigate pipeline FSI characteristics [16; 17]. However, it is important to note that the FEM represents both pipe movement and fluid behavior through nodes and elements; thus, its accuracy and efficiency are contingent upon the number of discrete elements utilized. For complex long-span pipelines, the FEM incurs substantial computational costs, which remains a primary factor limiting its broader engineering application. For ship hydraulic pipeline systems, frequency domain response characteristics are of paramount importance. The TMM serves as an effective tool for analyzing the frequency domain vibration characteristics of chain-linked systems. This method discretizes the system into a series of sets comprising domain transfer matrices and point transfer matrices. By sequentially connecting these matrices, a global transfer matrix is formed, which incorporates boundary conditions to facilitate the realization of system modes and responses. This approach offers significant advantages in terms of both simplicity and efficiency. [PERSON] [10] examines the impact of frequency-domain-related friction terms, categorizing the state variables in the 14-equation model into a pair of conjugate groups. He derives the transfer matrix for pipes based on the general solution form of the wave equation. [PERSON] and [PERSON] [18] transformed the axial, transverse, and torsional vibration models of straight pipes from multivariate first-order equations to one-dimensional high-order equations, thereby obtaining analytical solutions for each state variable and subsequently deriving the transfer matrix; however, this approach is limited to straight pipes. [PERSON] et al. [19] utilized the decoupling method of the MOC and introduced the L-MOC to derive the transfer matrix for the axial four equations governing straight pipe sections. This methodology has since become a classic technique for constructing transfer matrices in liquid-filled pipes. Building upon [PERSON] et al. 's work, [PERSON] et al. [20] developed a solution addressing pipeline frequency-domain characteristics that incorporates complex constraints and extended the L-MOC method to encompass the 14-equation model. In fact, although the methods for constructing transfer matrices in the aforementioned literature differ, they all address partial differential equations to derive the transfer relationships of state variables at both ends of the pipeline and are fundamentally interconnected. Several studies have indicated that numerical instability arises when calculating the frequency domain characteristics of long-span liquid-filled pipelines using the TMM based on the 14-equation model. Research conducted by [PERSON] [21] and [PERSON] [22] highlighted that this instability primarily occurs during calculations involving lateral vibrations of pipelines. Furthermore, investigations by [PERSON] et al. [8] demonstrated that in systems containing curved pipes, axial excitation can also induce instability due to the coupling between axial and lateral vibrations within these curved structures. In order to address the high-frequency instability issues associated with the TMM, various enhanced methods have been proposed sequentially. [PERSON] et al.