# Idle Vibration Reduction of a Diesel Sport Utility Vehicle
## 3 Dynamic Modeling
The equivalent springs and dampers of these mounts, bushings and hoses are displayed in Figure 10. The equivalent stiffness and damping coefficient of the radiator rubber bushing are respectively denoted by \(k_{\text{R}}\) and \(c_{\text{R}}\), and the stiffness and damping coefficient of the intercooler rubber bushing are respectively represented by \(k_{\text{I}}\) and \(c_{\text{I}}\). The upper radiator hose exhibits a stiffness of \(k_{\text{H1}}\) and damping coefficient of \(c_{\text{H1}}\), whereas the lower radiator hose possesses a stiffness of \(k_{\text{H2}}\) and damping coefficient of \(c_{\text{H2}}\). Similarly, the upper intercooler hose demonstrates a stiffness of \(k_{\text{H3}}\) and damping coefficient \(c_{\text{H3}}\), whereas the lower intercooler hose exhibits a stiffness of \(k_{\text{H4}}\) and damping coefficient of \(c_{\text{H4}}\). Finally, the stiffness and damping coefficient are respectively denoted by \(k_{\text{P1}}\) and \(c_{\text{P1}}\) for the engine mount, by \(k_{\text{P2}}\) and \(c_{\text{P2}}\) for the transmission mount, and by \(k_{\text{P3}}\) and \(c_{\text{P3}}\) for the roll mount. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Component**} & \multicolumn{2}{c}{**Mass (kg)**} & \multicolumn{5}{c}{**Mass Moment of Inertia (kg\(\cdot\)m\({}^{2}\))**} \\ \cline{2-9}  & **m** & \(\mathbf{I_{x}}\) & \(\mathbf{I_{y}}\) & \(\mathbf{I_{z}}\) & \(\mathbf{I_{xy}}\) & \(\mathbf{I_{yz}}\) & \(\mathbf{I_{zx}}\) \\ \hline Powertrain & 301.4 & 11.55 & 24.01 & 23.14 & 1.49 & 0.04 & \(-\)4.45 \\ Radiator & 11.5 & 0.57 & 0.33 & 0.52 & 0 & 0 & 0 \\ Intercooler & 5.0 & 0.13 & 0.12 & 0.02 & 0 & 0 & 0 \\ Front floor & 523.5 & 119.45 & 113.37 & 171.94 & \(-\)0.59 & 1.21 & \(-\)9.43 \\ Middle floor & 426.1 & 148.26 & 81.02 & 131.76 & \(-\)0.39 & 0.10 & \(-\)0.96 \\ Rear floor & 474.3 & 115.50 & 120.27 & 134.17 & 0.58 & \(-\)0.45 & \(-\)1.50 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Inertia properties used to construct the dynamic model. The equivalent stiffnesses and damping coefficients of the mounts, bushings, and hoses were obtained from the experiments by using the measurement setup illustrated in Figure 11a. Under the given condition that the mount, bushing, or hose, which is the test object, was placed between the mass block and the rigid wall, the mass was excited with a shaker at the idling frequency. As shown in this figure, the excitation force and displacement of the mass were measured with a force transducer and a gap sensor, respectively. The equation of motion for the physical model of the measurement setup, which is illustrated in Figure 11b, can be expressed as:  
\[m\ddot{x}+c\dot{x}+kx=F\mathrm{e}^{\mathrm{j}\omega t} \tag{1}\]  
where the superposed dot represents differentiation with respect to time; \(x\) denotes the displacement of the mass; \(m\) represents the mass of the block (0.5 kg); \(c\) and \(k\), respectively, symbolize the damping coefficient and stiffness of the specimen; and \(F\) and \(\omega\) denote the amplitude and frequency of the excitation force, respectively. Upon assuming \(x=X\mathrm{e}^{\mathrm{j}\omega t}\), the dynamic stiffness, defined as the ratio of the excitation force and the displacement, is expressed as follows:  
\[F/X=k-m\omega^{2}+\mathrm{j}\omega c \tag{2}\]  
Considering the real and imaginary parts of the experimentally obtained dynamic stiffness, which are denoted by \(k_{r}\) and \(k_{i}\), respectively, the stiffness and damping coefficient for the specimen can be obtained as follows:  
\[k=k_{r}+m\omega^{2},\ c=k_{i}/\omega \tag{3}\]  
The excitation frequency, \(\omega\), was selected as the idling frequency of 27.7 Hz. The dynamic stiffness for the rubber bushing of the radiator is presented in Figure 12, wherein Figure 12a,b correspond to the real and imaginary parts of the dynamic stiffness, respectively. In this figure, the real and imaginary parts at the idling frequency of 27.7 Hz are respectively denoted by \(k_{r}=38.2\) N/mm and \(k_{i}=10.0\) N/mm.