# Idle Vibration Reduction of a Diesel Sport Utility Vehicle
## 4 Derivation of the Equations of Motion  
As detailed in this section, the equations of motion for the dynamic model, which were proposed in the previous section, were derived and verified. The equations of motion were derived using Lagrange equations with constraints, which are expressed as follows:  
\[\frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{\partial T}{\partial\dot{q}_{k}} \right)+\frac{\partial F_{d}}{\partial\dot{q}_{k}}-\frac{\partial T}{\partial q _{k}}+\frac{\partial V}{\partial q_{k}}+\sum_{l=1}^{6}\lambda_{l}\frac{ \partial\xi_{l}}{\partial q_{k}}=f_{k} \tag{7}\]  
where \(q_{k}\) (\(k=1,\,2,\,\cdots,\,19\)) represent 19 coordinates describing the motions of the dynamic model: \(z_{P}\), \(\theta_{Px}\), \(\theta_{Py}\), \(z_{R}\), \(\theta_{Rx}\), \(z_{L}\), \(z_{1}\), \(\theta_{1x}\), \(x_{2}\), \(y_{2}\), \(z_{2}\), \(\theta_{2x}\), \(\theta_{2y}\), \(x_{3}\), \(y_{3}\), \(z_{3}\), \(\theta_{3x}\), and \(\theta_{3y}\). In the aforementioned equation, \(T\), \(V\), and \(F_{d}\) denote the kinetic energy, potential energy, and Rayleigh's dissipation function, respectively, \(f_{k}\) symbolizes the generalized nonconservative forces, \(g_{l}\) represent the constraints, and \(\lambda_{l}\) depict the Lagrange multipliers. Notably, the coordinates presented herein include 6 coordinates of \(x_{2}\), \(y_{2}\), \(z_{2}\), \(x_{3}\), \(y_{3}\), and \(z_{3}\) in addition to the 13 independent generalized coordinates mentioned in Section 3. To express the kinetic and potential energies more effectively in terms of the coordinates, the additional 6 coordinates were introduced. The constraints related to the additional coordinates are discussed herein. First, consider the kinetic energy of the dynamic model for the diesel SUV, which can be expressed as  
\[T=\frac{1}{2}\sum m_{i}\mathbf{v}_{i}^{\mathrm{T}}\mathbf{v}_{i}+\frac{1}{2} \sum\mathbf{\omega}_{i}^{\mathrm{T}}\mathbf{I}_{i}\mathbf{\omega}_{i} \tag{8}\]  
Figure 15: Schematic diagram of an engine crank mechanism. where \(i\) represents the six parts of the SUV: the powertrain, radiator, intercooler, front floor, middle floor, and rear floor, \(m_{i}\) indicate the masses of these parts, \(\mathbf{v}_{i}\) represents the velocity vectors of the mass centers, \(\mathbf{\omega}_{i}\) denotes the angular velocity vectors, and \(\mathbf{I}_{i}\) symbolizes the inertia matrices about the mass centers. The first term in the right-hand side of the equation represents the translational kinetic energy, whereas the second term is for the rotational kinetic energy. The velocity vector of mass center and angular velocity vectors for each part were defined in the global coordinate system. The kinetic energy was determined by substituting the velocity and angular velocity vectors and the inertia properties of Table 1 into Equation (8). The potential energy and Rayleigh's dissipation function owing to the stiffness and damping properties of the mounts, bushings, hoses, and body floor connections are expressed as  
\[V=\frac{1}{2}\sum k_{j}\big{(}\Delta l_{j}\big{)}^{2}+\frac{1}{2}\sum k_{r}( \Delta\theta_{r})^{2} \tag{9}\]  
\[F_{d}=\frac{1}{2}\sum c_{s}\big{(}\Delta v_{s}\big{)}^{2} \tag{10}\]  
where subscript \(j\), \(r\) represent the equivalent translational and rotational springs, respectively, \(k_{j}\) and \(k\), are the equivalent translational and rotational stiffnesses, respectively. Subscript \(s\) represent the equivalent translational dampers, \(c_{s}\) are the equivalent damping coefficients. \(\Delta l_{j}\) are the stretches of the equivalent translational springs, \(\Delta\theta\), are the angular changes of the equivalent torsional springs, and \(\Delta v_{s}\) are the velocity changes in the equivalent dampers.