## 3 Dynamic Modeling
### Tailgate Model with Free-Free Boundary Conditions  
When constructing the tailgate model, our selection of the major components considers their structural characteristics. As introduced in Section 2, the tailgate is composed of a trim, inner panel, outer panel, and glass from inside the vehicle. It includes structures in which components such as spoilers, garnishes, and taillight covers are attached to the outer panel. However, the essential elements embodying the structural characteristics of the tailgate are the inner and outer panels and the glass. The other parts have a relatively low rigidity compared to steel or glass, performing auxiliary functions and making their exclusion from the modeling process feasible. However, spoilers and taillight covers both possess a significant mass and are attached to specific areas outside the tailgate. Excluding these from the model could oversimplify the model and potentially compromise its accuracy. Therefore, these components were included in our model. As a result, the free-free-state tailgate model in this paper consists of the inner and outer panels, glass, spoiler, and taillight cover. Before establishing the model, simplifying the 3D model is essential for efficient finite-element modeling, instead of using the completed tailgate shape directly. The core structure of the tailgate, including the inner and outer panels, typically has a complex shape to satisfy various requirements in the final design stage. However, these elements have a minimal impact on the dynamic characteristics of the tailgate. Moreover, many elements can be created unnecessarily during the mesh generation process for analysis, thus disrupting the efficient modeling process. Additionally, to address the issue of booming noise effectively, it is crucial to build a simple model rather than a full model. To avoid excessively compromising the accuracy of the model, the simplification of the tailgate panels focuses on the elements that play a significant role in the overall dynamic characteristics. Unnecessary holes were filled and features such as fillets were removed, resulting in a simplified model for analysis, as shown in Figure 10. During this process, the CATIA 3D modeling software (V5-6R2020) was utilized. The selection approach for the appropriate elements corresponding to the characteristics of the main modeling targets is as follows. Shell elements are applied to the simplified tailgate inner and outer panels, reinforcements, and glass, which have thin sheet structures  
Figure 10: Simplification of the tailgate 3D CAD model: (**a**) outer panel; and (**b**) inner panel. and play a crucial role in the bending deformation acting on the tailgate surface. Shell elements are suitable modeling elements that adequately reflect the effects of bending deformation while using a minimal number of elements. Typically, when solid elements are used for thin sheets, a large number of elements may be required to accurately capture the bending stiffness, leading to unnecessarily long analysis times or inaccurate results. Therefore, selecting the shell elements and inputting appropriate cross-sectional information is desirable for addressing this issue. On the other hand, the spoiler and lamp, chosen as the main modeling elements of the tailgate, are attached to specific areas of the outer panel of the tailgate. A distributed mass is applied to model these components and they are integrated into the outer panel. The tailgate's inner and outer panels, as well as the reinforcements, are all made of steel material, with the same Young's modulus of 200 GPa and a density of 7850 kg/m\({}^{3}\). The actual thicknesses of the panels and reinforcements were applied for the required thickness of the shell elements. For the tailgate glass, the material properties of glass, including [PERSON]'s modulus of 76 GPa and a density of 1853 kg/m\({}^{3}\), were used, and the actual thickness was input in order to model it using shell elements.