## 1 Introduction
[23] introduced the concept of branching within the original system, which enhances TMM stability by integrating a side branch system into the boundary of the main path system, thereby shortening the transfer distance between variables. [PERSON] [24] elaborated on this method's calculation process in detail based on [PERSON]'s theory; however, it necessitates constructing a complex intermediate matrix and is not universally applicable. The Riccati Transfer Matrix Method (RTMM) offers a means to reduce the dimensions of the transfer matrix while simultaneously improving stability by transforming a two-point boundary value problem into a one-point initial value problem. Nonetheless, this approach is limited to simple boundary conditions and proves unsuitable for intricate hydraulic pipelines [13; 25]. [PERSON] et al. [26] and [PERSON] et al. [5; 6] proposed corresponding enhancement strategies for applying the TMM to tackle instability challenges in structural dynamics and the pipeline fluid-structure interaction (FSI). Although their computational processes differ, they all fundamentally segment the original system into distinct sub-elements to minimize the \"characteristic length\". This treatment methodology is uniformly referred to as the Piecewise Transfer Matrix Method (PTMM) herein. [PERSON] and [PERSON] proposed a foundational framework for dividing subunits in the context of liquid-filled piping problems using the PTMM [21; 27]. Subsequently, [PERSON] et al. [8] applied the PTMM to analyze the frequency domain characteristics of aviation hydraulic piping systems. Similar to the FEM, the accuracy and stability of the PTMM are contingent upon the number of sub-elements. As pipeline system complexity increases, the dimensions of the global transfer matrix derived from the PTMM also expand, leading to decreased efficiency and thereby limiting its applicability. In addressing elastic wave propagation in layered media, [PERSON] et al. [28] introduced a method known as the mixed variable transfer matrix method. The central premise involves transforming the traditional transfer matrix into a mixed variable transfer matrix. Here, \"mixed variable\" refers to utilizing potential variables at one end of the system alongside flow variables at the opposite end as fundamental parameters. [PERSON] et al. [29] were pioneers in applying this method within pipeline FSI contexts and demonstrated its effectiveness; however, their investigations were confined to simple straight pipes under ideal boundary conditions, leaving further exploration regarding this method's broader applicability unaddressed. Ship hydraulic pipelines typically span long distances in space. Given this context, it is essential to develop an efficient, accurate, and stable method for analyzing the FSI frequency domain characteristics of long-span hydraulic pipelines. This paper constructs the frequency-domain transfer matrix for a typical hydraulic pipeline based on the FSI 14-equation model and employs the concept of mixed variable matrices. Furthermore, we derived a recursive algorithm for the mixed variable transfer matrix, enabling the effective prediction of the frequency domain vibration response of the hydraulic pipeline by incorporating the matrix representation of complex boundary constraints. This approach is also referred to as the mixed variable transfer matrix method (MVTMM) in this paper. The proposed method was validated using FSI test data published by Dubee University and through an FSI module developed within commercial finite element software. Subsequently, we conducted experiments on a section of a U-shaped pipeline utilized in a ship's steering device. It is important to note that previous experiments concerning FSI in pipelines predominantly involved free boundary conditions based on force hammer impact methods [30; 31], which differ significantly from the operational environment encountered in actual hydraulic pipeline systems.