## 3 Dynamic Modeling
### Tailgate Model Mounted on the Vehicle
Based on the introduced structure of the guide bumper, it not only plays a role in providing stiffness for deformation in the \(z\)-direction, where the guide bumper surface contacts the tailgate surface, but also has structural features that can resist rotation in the \(y\)-direction when the two surfaces are in contact. Therefore, the guide bumper is simplified with equivalent stiffness for the translational stiffness in the \(z\)-direction and rotational stiffness in the \(y\)-direction, denoted as \(k_{z}\) and \(\kappa_{g}\), respectively. To define the equivalent stiffness caused by the guide bumper, a finite-element analysis was conducted following these steps: To obtain the stiffness for translational movement in the \(z\)-direction, as depicted in Figure 15a,b, deformation occurring in the \(z\)-direction when the guide bumper was attached to the tailgate was simulated; the force generated on the opposite side, where it connects to the vehicle body, was then extracted. The stiffness for translational movement in the \(z\)-direction was determined based on the relationship between force and displacement. Furthermore, as shown in Figure 15c,d, when the guide bumper experienced a rotational displacement in the \(y\)-direction upon contact with the tailgate, the equivalent rotational stiffness was obtained from the counteracting moment generated on the opposite side of the guide bumper connected to the vehicle body. The acquired stiffness values were then applied to the free-free model to construct a final tailgate model. Figure 14: Components of the guide bumper: (**a**) steel part; (**b**) rubber part; (**c**) plastic part; and (**d**) guide bumper assembly. To validate the final tailgate model with the boundary conditions applied to the previously validated free-free state of the tailgate, the measured mode shapes of the tailgate in the state connected to the actual vehicle, analyzed in Section 2, were compared with the modal analysis results of the final model. As shown in Table 3, the analysis results of the constructed tailgate model reveal first, second, and third natural frequencies of 21.5, 27.8, and 34.7 Hz, respectively. Rigid body modes in the fore-aft direction, along with the deformation modes of the tailgate, appear at the first mode (21.5 Hz) and second mode (27.8 Hz). Notably, at the second natural frequency of 27.8 Hz, a significant deformation occurred at the central bottom part of the tailgate. This characteristic aligns with the tailgate modes obtained in the experimental setup described in Section 2. Additionally, the torsional mode observed at 34.7 Hz shows opposite phase displacements at all four corners, resembling the results obtained in the actual experiments and confirming the validity of the finalized model. Considering the mode shapes, the mode at 34.7 Hz exhibits localized deformation in some areas, potentially contributing to noise generation. However, because the dominant behavior involves the overall twisting of the tailgate, its contribution to noise generation is expected to be minimal. Therefore, the next section explores strategies for enhancing the first and second natural modes.