# Idle Vibration Reduction of a Diesel Sport Utility Vehicle
## 3 Dynamic Modeling
By substituting these values into Equation (3), the equivalent stiffness and damping are respectively calculated as \(k=53.3\) N/mm and \(c=57.5\) kg/s. Similarly, the stiffnesses and damping coefficients of bushings and hoses obtained through experiments are summarized in Table 2.  
Figure 10: Equivalent stiffnesses and damping coefficients between the powertrain, radiator, intercooler, and front floor. After independently obtaining the tire stiffness and suspension stiffness, these stiffnesses, which were assumed to be connected in series, were used to derive the equivalent stiffness of the tire and suspension combination. The tire stiffness was measured as per the static load-deflection test proposed by [PERSON] et al. [22]. The static load -deflection test determines the vertical load of a tire by measuring the vertical deflection under a vertical load from a regression curve that represents the load as a function of displacement, after which the function is differentiated with the displacement. In the static load-deflection test, the vertical deflection of a tire is measured by varying the vertical load. After expressing the load as a function of deflection, the tire stiffness is obtained by differentiating this function by the deflection at a specific deflection value. Moreover, using the suspension kinematics and compliance test system [23], the vertical stiffness of the vehicle suspension was deter  
Figure 11: Measurement setup of the stiffnesses and damping coefficients for mounts, rubber bushings, and hoses: (**a**) the photo of actual device and (**b**) the corresponding physical model. Figure 12: Dynamic stiffness for the rubber bushing of the radiator: (**a**) the real part and (**b**) the imaginary part. mined by measuring the vertical deflection under the applied vertical load. The equivalent vertical stiffnesses of the front and rear suspension/tire combinations obtained from these methods are 28.9 and 31.6 N/mm, respectively. The equivalent bending and torsional stiffnesses between the front, middle, and rear body floors were obtained using a commercial package for finite element analysis. The process of obtaining equivalent stiffness between the front and middle body floors are described herein. As shown in Figure 13, the front body floor was discretized into finite elements with the commercial package, and the middle and rear floors were modeled as stationary rigid bodies. The boundary between the front and middle body floors was considered to be fixed. The load conditions for obtaining the equivalent bending and torsional stiffnesses between the front and middle floors are displayed in Figure 13a,b, respectively. To determine the bending stiffness, as shown in Figure 13a, uniform vertical forces of \(F=10\) N were applied to both ends of the front-end module in the same direction. In addition, to determine the torsional stiffness, as illustrated in Figure 13b, vertical forces in opposite directions were applied at both ends. The deflections at the front corner of the front-end module for the load conditions, which are depicted in Figure 13a,b, obtained from the finite element analysis are 5.7 and 0.85 mm, respectively. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Component** & **Part** & **Stiffness (N/mm)** & **Damping Coefficient (kg/s)** \\ \hline \multirow{3}{*}{Radiator} & Rubber bushing & 53.3 & 57.5 \\  & Upper hose & 5.0 & 157.7 \\  & Lower hose & 12.8 & 88.9 \\ \hline \multirow{3}{*}{Intercooler} & Rubber bushing & 55.2 & 54.4 \\  & Upper hose & 10.2 & 150.3 \\  & Lower hose & 13.6 & 189.8 \\ \hline \multirow{3}{*}{Powertrain} & Engine mount & 500.2 & 1223.8 \\  & Transmission mount & 497.5 & 375.1 \\ \cline{1-1}  & Roll mount & 2911.6 & 2553.4 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Equivalent stiffnesses and damping coefficients of the bushings, hoses, and mounts at the idle frequency of 27.7 Hz.