# Idle Vibration Reduction of a Diesel Sport Utility Vehicle
## 4 Derivation of the Equations of Motion
\begin{bmatrix}\mathbf{K}&\mathbf{A}_{L}^{\mathrm{T}}\\ \mathbf{A}_{L}&\mathbf{0}\end{bmatrix} \begin{Bmatrix}\mathbf{Q}\\ \boldsymbol{\Lambda}\end{bmatrix}=\omega^{2} \begin{bmatrix}\mathbf{M}& \mathbf{0}\\ \boldsymbol{0}&\mathbf{0}\end{bmatrix} \begin{Bmatrix}\mathbf{Q}\\ \boldsymbol{\Lambda}\end{bmatrix} \tag{26}\]  
Equation (26) equation yields the characteristic equation expressed as  
\[\det\biggl{[}\mathbf{A}_{L}\Bigl{(}\mathbf{K}-\omega^{2}\mathbf{M}\Bigr{)}^{ -1}\mathbf{A}_{L}^{\mathrm{T}}\biggr{]}=0 \tag{27}\]  
In Table 3, the natural frequencies, which are computed from Equation (25) using the properties detailed in Section 3, are compared with the natural frequencies obtained from the modal tests, as reported in Section 2. According to the information listed in this table, the analytically calculated natural frequencies are in agreement with the experimentally measured frequencies. Subsequently, the mass effects of the radiator and intercooler on the vibration level during engine idling were investigated through a simulation and an experiment, and the proposed dynamic model was verified again by demonstrating that the mass effects exhibit identical tendencies in the simulation as well as the experiment. Prior to verification, the simulation process was considered to obtain the frequency response function for the constrained system, which is defined in Equations (24) and (25). Let the displacement and Lagrange multiplier vectors be \(\mathbf{q}=\mathbf{Q}e^{\mathrm{i}\omega t}\) and \(\boldsymbol{\lambda}=\boldsymbol{\Lambda}e^{\mathrm{i}\omega t}\), and let the applied load be \(\mathbf{f}=\mathbf{F}e^{\mathrm{i}\omega t}\); thus, Equations (24) and (25) can be rewritten as follows:  
\[\mathbf{H}(\omega)^{-1}\mathbf{Q}+\mathbf{A}_{L}^{\mathrm{T}} \mathbf{A}=\mathbf{F} \tag{28}\] \[\mathbf{A}_{L}\mathbf{Q}=\mathbf{0} \tag{29}\]  
where \(\mathbf{H}(\omega)=\bigl{(}\mathbf{K}-\omega^{2}\mathbf{M}+\mathrm{i}\omega \mathbf{C}\bigr{)}^{-1}\). From Equations (28) and (29), the displacement response vector \(\mathbf{Q}\) can be expressed as \(\mathbf{Q}=\hat{\mathbf{H}}(\omega)\mathbf{F}\), where \(\hat{\mathbf{H}}(\omega)\) is the frequency response function for constrained system expressed as follows. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Component**} & \multirow{2}{*}{**Mode**} & \multicolumn{4}{c}{**Natural Frequency (Hz)**} \\ \cline{3-5}  & & **Simulation** & **Experiment** & **Difference** \\ \hline \multirow{3}{*}{Powertrain} & Bounce mode & 6.7 & 4.8 & \(-1.9\) \\  & Pitch mode & 15.3 & 13.0 & \(-2.3\) \\  & Roll mode & 21.1 & 18.5 & \(-2.6\) \\ \hline \multirow{3}{*}{Radiator} & Bounce mode & 19.4 & 20.0 & 0.6 \\  & Roll mode & 30.3 & 30.3 & 0.0 \\ \hline Intercooler & Bounce mode & 31.0 & 29.3 & \(-1.7\) \\ \hline \multirow{3}{*}{Body floor} & Bounce mode & 1.2 & 3.2 & 2 \\  & Pitch mode & 1.7 & 3.7 & 2 \\ \cline{1-1}  & Bending mode & 25.2 & 23.8 & \(-1.4\) \\ \cline{1-1}  & Torsional mode & 28.0 & 28.4 & 0.4 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparison of the natural frequencies obtained by simulation and experiment. \[\hat{\mathbf{H}}(\omega)=\mathbf{H}(\omega)\bigg{\{}\mathbf{I}-\mathbf{A}_{L}^{ \mathrm{T}}\Big{[}\mathbf{A}_{L}\mathbf{H}(\omega)\mathbf{A}_{L}^{\mathrm{T}} \Big{]}^{-1}\mathbf{A}_{L}\mathbf{H}(\omega)\bigg{\}} \tag{30}\]  
The vibration levels were experimentally measured in the idle engine state by adding mass blocks of various sizes to the radiator or intercooler as illustrated in Figure 17. The added masses were 1, 2, 3, and 4 kg for the radiator and 0.5, 1.0, 1.5, and 2.0 kg for the intercooler. The vibration levels were measured at the driver's seat position on the body floor at an engine speed of 830 rpm, i.e., an idling frequency of 27.7 Hz. In addition, using the added blocks in the experiments for the radiator and intercooler, simulations were performed with the dynamic model established in this study.