# Modal Analysis and Optimization of Tractor Exhaust System
## 1 Introduction  
The agricultural sector plays a vital role in meeting the basic needs of human life. The use of modern production methods in agriculture is a crucial requirement to increase production efficiency. Tractor powertrain systems are essential for performing fundamental agricultural operations such as seeding, fertilizing, spraying, and harvesting. As technology advances, new methods have been employed in tractor production, and research in this field has deepened. Engines, the primary source of vibration in internal combustion engines, subject exhaust systems to intense vibrations [1]. Vibration can be defined as the periodic or random occurrence of mechanical oscillations around an equilibrium point [2]. These oscillations are usually caused by rapid linear or rotational movements of components in a system. Factors such as speed, position, load conditions, roughness of moving surfaces, and operating environment of components are important parameters that affect the vibration level [3, 4, 5]. Since rotating machinery and powertrains are considered to be the primary sources of vibration, these components as well as assembled and thin sheet parts are highly affected and damaged by vibration [6, 7, 8]. Therefore, the exhaust system is a critical subsystem that filters harmful gasses produced by the engine and reduces noise levels before they are released into the environment. Understanding the vibration characteristics of the exhaust system and taking precautions against these vibrations is essential [9]. The dynamic characterization of vehicle structures can be examined in detail throughsimulations performed in a virtual environment before vehicle prototypes are produced. In this process, numerical methods such as finite element analysis can be used to analyze important characteristics such as vehicle dynamics, durability, and NVH (Noise, Vibration, and Harshness). In this way, potential problems can be controlled, and vehicle performance, life, and passenger comfort can be improved [10]. In addition, severe vibrations generated during vehicle operation can seriously threaten the health of operators, leading to various health problems [11; 12]. Exhaust systems consist of multiple components that trap harmful gasses and particulates to protect the environment and reduce vehicle noise [13]. This study focuses on the design and analysis methods of exhaust systems. Studies in the literature form the reference points of the current study. [PERSON] et al. [14] analyzed the exhaust system by modeling it in one dimension. They showed that the finite element model gives accurate results in predicting the low-frequency rigid body motion of the automobile exhaust system. [PERSON] et al. [15] investigated the vertical vibration characteristics of the exhaust system and developed a simplified model of the system. This study was validated by simulation and experimental comparisons. [PERSON] [16] observed the dynamics of an automobile exhaust system and developed a finite element model. In another study, [PERSON] [17] evaluated the vertical vibrations of a car exhaust system. This study stated that simplified models can be used to quickly predict system vibrations in the vertical direction. [PERSON] et al. [18] presented the dynamic behavior of an exhaust system containing pipes, mufflers, and a catalyst with a nonlinear flexible connection. The study demonstrated the validity of the modeling by showing a strong agreement between theoretical and experimental results. [PERSON] et al. [19] analyzed the NVH characteristics of a tractor exhaust system using the finite element method. The natural frequencies and mode shapes of the exhaust system were determined, and the spectral density of random vibrations occurring under different operating conditions of the tractor was examined. [PERSON] et al. [20] created three-dimensional models using ANSYS Workbench for the design of exhaust systems for diesel engines. Modal analysis methods of vibrating structures include test modal analysis and finite element modal analysis [21; 22].