# Idle Vibration Reduction of a Diesel Sport Utility Vehicle
## 3 Dynamic Modeling  
On the basis of the experimental analysis reported in the previous section, a dynamic model for the diesel SUV has been established. To analyze the behavior of the low-frequency region below 30 Hz, the dynamic model presented in this study possesses 13 degrees of freedom, as displayed in Figure 9. Three degrees of freedom are assigned to the powertrain, two to the radiator, one to the intercooler, and seven to the body floor. First, as shown in Figure 7a-c, the vibration modes dominating the behavior of the powertrain in the low-frequency region are the bounce, pitch, and roll modes. Accordingly, three degrees of freedom are accorded to the rigid-body motion of the powertrain, and the generalized coordinates for the bounce, pitch, and roll motions are expressed as \(z_{\text{P}}\), \(\theta_{\text{Py}}\), and \(\theta_{\text{Py}}\), respectively. As depicted in Figure 7d,e, the radiator exhibits bounce and roll modes in the low-frequency region; therefore, the generalized coordinates \(z_{\text{R}}\) and \(\theta_{\text{R}x}\) for the bounce and roll motions are respectively assigned to the radiator. Similarly, the intercooler possesses a bounce mode solely in the low-frequency region (refer to Figure 7f); thus, the generalized coordinate \(z_{1}\) for the bounce motion is assigned to the intercooler. Figure 9: Dynamic model of a diesel SUV with 13 degrees of freedom. Conversely, the body floor features rigid-body modes such as bounce and pitch modes (refer to Figure 8a,b) as well as flexible modes owing to the bending and torsional deformations (refer to Figure 8c,d) in the low-frequency region. Therefore, both the rigid-body and flexible modes should be considered for the dynamic modeling. Considering these two modes, this study proposes a body floor model comprising three parts: the front, middle, and rear floors, as represented in Figure 9. The motion of the front floor is described with the generalized coordinates for the bounce, pitch, and roll motions (respectively denoted by \(z_{1}\), \(\theta_{1y}\), and \(\theta_{1x}\)), the motion of the middle floor is described with the coordinates for the pitch and roll motions (respectively denoted by \(\theta_{2y}\) and \(\theta_{2x}\)), and the motion of the rear floor is described with the coordinates for pitch and roll motions (respectively denoted by \(\theta_{3y}\) and \(\theta_{3x}\)). Degrees of freedom were not imposed on the bounce motions of the middle and rear floors because the midpoint of the interface between the front and middle floors and the midpoint of the interface between the middle and rear floors were assumed to be constrained by revolute joints. The inertia properties, such as masses and mass moments of inertia, are required to construct the dynamic model of the SUV. The masses of relatively small components such as the radiator, intercooler, and powertrain were measured using a scale. However, masses and mass moments of inertia, which are difficult to measure, were analytically extracted using three-dimensional solid computer-aided design models. On the basis of the nodal lines of bending and torsional deformations obtained from the ODS analysis, the body floor was categorized into three segments: the front, middle, and rear floors. The inertia properties used in the dynamic model are summarized in Table 1.  
In addition, to establish the dynamic model, the stiffness and damping properties related to the radiator, intercooler, powertrain, and floors should be evaluated. As discussed earlier, the radiator is connected to the front floor via four rubber bushings, and the intercooler is connected to the front floor via three rubber bushings. The radiator and the intercooler are each connected to the powertrain through two rubber hoses. Moreover, the powertrain is supported at three positions by the engine, transmission, and roll mounts.