# Idle Vibration Reduction of a Diesel Sport Utility Vehicle
## 3 Dynamic Modeling
Figure 13: Finite element models to obtain equivalent stiffnesses between the front and middle body floors: (**a**) the load condition for the bending stiffness and (**b**) the load condition for the torsional stiffness. To determine the equivalent stiffnesses, the rigid front body floor illustrated in Figure 9 has been schematically redrawn in Figure 14, wherein Figure 14a features the side view of the SUV, and its front view is displayed in Figure 14b. As depicted in this figure, the force, \(F\), is the same force of 10 N that was imposed in the finite element analysis, and \(l_{1}\) and \(w_{1}\) denote the length of the front floor (907 mm) and the half-width of the vehicle (428.5 mm), respectively. In Figure 14, \(k_{b1}\) and \(k_{t1}\) respectively symbolize the bending and torsional stiffnesses between the front and middle floors, which are identical to those represented in Figure 9, and \(\delta_{1b}\) and \(\delta_{1t}\) respectively denote the bending and torsional deflections. From the free-body diagrams in Figure 14, the bending and torsional stiffnesses are obtained as  
\[k_{b1}=2 Fl_{1}^{2}/\delta_{1b},\;k_{t1}=2 Fw_{1}^{2}/\delta_{1t} \tag{4}\]  
Substituting the bending and torsion deflections of \(\delta_{1b}=5.7\) mm and \(\delta_{1t}=0.85\) mm, which were obtained from the finite element analysis, into Equation (4), the bending and torsional stiffnesses between the front and middle floors were calculated as \(2.886\times 10^{6}\) and \(4.320\times 10^{6}\) N/mm, respectively. Similarly, the bending and torsional stiffnesses between the middle and rear body floors were determined as \(3.584\times 10^{6}\) and \(2.917\times 10^{6}\) N/mm. Subsequently, the excitation forces and moments generated from the engine in the powertrain were analyzed. These excitations are induced by the gas pressure resulting from the explosion inside the cylinder as well as the unbalanced inertia force owing to the piston, connecting rod, and crank. Generally, the primary and secondary inertia forces of the engine are negated by the crank and the balance shaft. Furthermore, the excitation forces and the pitch excitation moment are relatively low compared to the roll excitation moment. Therefore, in this study, the roll excitation moment generated by the gas pressure was considered as the main excitation force in the idle state. Considering the explosion mechanism in a cylinder, the piston, connecting rod and crank are illustrated in Figure 15, wherein \(P\) denotes the gas pressure, \(l\) represents the length of the connecting rod, \(r\) depicts the length of the crank arm, and \(\omega\) symbolizes the angular velocity of the crankshaft. The gas torque, \(M\), owing to the gas pressure at time \(t\) is expressed as follows [24]:  
\[M=PAr\Big{(}\sin\omega t+\frac{r}{2l}\sin 2\omega t\Big{)} \tag{5}\]  
Figure 14: Rigid model of the front body floor to obtain equivalent stiffnesses between the front and middle body floors: (**a**) the bending stiffness and (**b**) the torsional stiffness. where \(A\) denotes the piston area. In case of the four-cylinder diesel engine considered for this study, because the explosions occur with phase differences of 180 degrees in the two pairs of cylinders, the total gas torque, \(M_{t}\), generated by the four cylinders is expressed as follows:  
\[M_{t}=\frac{2 PAr^{2}}{l}\sin 2\omega t \tag{6}\]  
The diesel SUV employed in this study possessed a gas pressure of \(P=9.45\) N/mm\({}^{2}\), connecting rod length of \(l=145\) mm, crank arm length of \(r=48\) mm, and piston area of \(A=5728\) mm\({}^{2}\). By substituting these values into Equation (6), the magnitude of the total moment was computed as \(1.72\times 10^{6}\) Nmm from Equation (6). For the idling frequency of 27.7 Hz, the angular frequency, \(2\omega\), is calculated as 348.09 rad/s. However, as reported in a study, only 1-5% of the total gas torque generally contributes to the vibration excitation torque [25].