## 1 Introduction  
Currently, internal combustion engines remain the primary power source and are widely used in various sectors, including transportation, industrial, and agricultural production [1]. They power not only vehicles, ships, and aircraft but also engineering machinery, agricultural equipment, and mobile power station facilities. However, a significant drawback of internal combustion engines is their high exhaust noise. The most direct and effective method to reduce this noise is by installing exhaust mufflers [2]. The reactive muffler is the most commonly used traditional exhaust muffler for internal combustion engines. To further enhance the acoustic attenuation performance of exhaust mufflers, a split-stream rushing exhaust muffler designed to improve overall performance by reducing the internal airflow velocity was proposed in this study [3]. Moreover, theoretical calculations, numerical simulations, and experimental validations of this new muffler have been conducted in previous studies [4] and demonstrated its feasibility and superiority. [PERSON] et al. [5] analyzed the acoustic performance of the muffler using acoustic wave theory and derived the transmission matrix of the split-stream rushing muffler unit based on one-dimensional acoustic theory, which enabled the calculation of transmission loss. The reliability of this analysis was validated through numerical simulations and experimental results. [PERSON] et al. [6] proposed a new coupling exhaust muffler for the diesel engine based on this new split-stream muffler unit. They developed a mathematical model to calculate the transmission loss without considering airflow through the one-dimensional transmission matrix method. Subsequently, they conducted experimental verification of the airflow-free transmission loss for the new muffler and performed an airflow-free transmission test on the original muffler in the engine. A comparison of the results showed that the new muffler demonstrated superior acoustic attenuation and aerodynamic performance compared with the original muffler. In practical applications, acoustic waves transmitted through the exhaust muffler can induce structural vibrations in the muffler shell, which in turn affect the original acoustic field. This interaction between acoustic waves and structural vibrations significantly influences the overall performance of the muffler. Consequently, addressing acoustic-structure coupling problems becomes essential. Common approaches to studying acoustic-structure coupling include analytical methods and the finite element method (FEM). Compared with analytical methods, FEM is more widely used by researchers for solving problems related to acoustic-structure coupling owing to its versatility and accuracy. [PERSON] and [PERSON] [7; 8] established the energy equation for acoustic solids. They were the first to treat the sound field as a continuous elastic medium and derived the theoretical expressions for sheet vibration and sound field coupling using the complementary energy principle and Hamiltonian variation principle. This work laid the theoretical foundation for addressing acoustic-structure coupling problems through the FEM. [PERSON] et al. [9] further advanced this field by developing a finite element model that coupled a vehicle body structure with its interior sound cavity. They derived the FEM formulas for structural vibrations and sound pressure fluctuations under external excitations. [PERSON] et al. [10; 11] developed an acoustic-structure coupling model for car cabins using finite element software and validated its accuracy through acoustic modal testing. Similarly, [PERSON] et al. [12] investigated the acoustic-structure coupling effect of a diesel generator set on its transmission loss through finite element analysis. Their findings revealed that these influences mainly occurred at various structural modal orders and acoustic modal frequencies. [PERSON] et al.