# Ride Comfort Prediction on Urban Road Using Discrete
## 3 Numerical Simulation  
### Vehicle Model  
In the past decades, numerical simulation has been proven to be an effective approach for investigating the dynamic response of vehicles on an uneven road profile; many vehicle models were suggested, including the quarter-car model, the half-car model, and the full-car model. By using proper validated mechanical parameters, these multi-DOF (degree of freedom) models allow us to solve dynamic vehicle responses in both the time domain [23] and frequency domain [24].  
In this study, we focus mainly on the effect of pavement roughness on the whole-body vibration. Therefore, we adopted an eight DOF vehicle model, which was an extension of the quarter-car model, which allows us to extract the accelerations on different seats. The model parameters are referenced from lecture [19]. As shown in Figure 3, this model took the effects of both of the two uneven road profiles of the left and right wheel paths into consideration, which could better simulate the rolling and pitching motions of the vehicle.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Weighted Acceleration** & **Comfort Levels** \\ \hline Less than 0.315 m/s\({}^{2}\) & Not uncomfortable \\
0.315 m/s\({}^{2}\) to 0.63 m/s\({}^{2}\) & A little uncomfortable \\
0.5 m/s\({}^{2}\) to 1.0 m/s\({}^{2}\) & Fairly uncomfortable \\
0.8 m/s\({}^{2}\) to 1.6 m/s\({}^{2}\) & Uncomfortable \\
1.25 m/s\({}^{2}\) to 2.5 m/s\({}^{2}\) & Very uncomfortable \\ Greater than 2.0 m/s\({}^{2}\) & Extremely uncomfortable \\ \hline \hline \end{tabular}
\end{table}
Table 1: Weighted acceleration and comfort levels.  
A degree of freedom for seat was defined in this model to solve the accelerations on the seat. The position of the seat can be adequately adjusted to match the position of drivers or passengers. Moreover, we assumed that the vehicle did not accelerate, brake, or steer during the simulation, and only the vertical responses of the seat were considered.  
According to [PERSON]'s law, the vibration of the full-car model is described in Equation (13).  
\[[\mathbf{M}]\Big{\{}\ddot{\mathbf{Z}}\Big{\}}+[\mathbf{C}]\Big{\{}\dot{\mathbf{ Z}}\Big{\}}+[\mathbf{K}]\{\mathbf{Z}\}=\{\mathbf{F}(t)\} \tag{13}\]  
where \(\ddot{\mathbf{Z}}\), \(\dot{\mathbf{Z}}\), and \(\mathbf{Z}\) are the acceleration vector, the velocity vector, and the displacement vector, M, C, and K are the mass matrix, the damping matrix, and the spring matrix, and \(\mathbf{F}(t)\) represents the external force of the vibration system. Assuming that the road profiles of the right and left wheel paths are \(z_{1}(t)\) and \(z_{2}\), respectively, \(\mathbf{F}(t)\) can be written as Equation (14).  
\[F(t)=\begin{bmatrix}-k_{i}z_{1}(t)\\ -k_{i}z_{2}(t)\\ -k_{i}z_{1}\Big{(}t-\frac{l}{v}\Big{)}\\ -k_{i}z_{2}\Big{(}t-\frac{l}{v}\Big{)}\\ \mathbf{0}_{4\times 1}\end{bmatrix} \tag{14}\]  
Also, we obtained the state space representation of Equation (12), as expressed in Equations (15) and (16).  
\[\begin{split}\begin{cases}\ddot{\mathbf{Z}}\\ \dot{\mathbf{Z}}\end{cases}=\begin{bmatrix}-\mathbf{M}^{-1}\mathbf{C}&- \mathbf{M}^{-1}\mathbf{K}\\ \mathbf{0}&\mathbf{I}\end{bmatrix}\begin{cases}\dot{\mathbf{Z}}\\ \mathbf{Z}\end{cases}+\begin{cases}\mathbf{M}^{-1}\\ \mathbf{0}\end{cases}\mathbf{F}(t)\end{split} \tag{15}\]  
\[\ddot{\mathbf{Z}}=\begin{bmatrix}-\mathbf{M}^{-1}\mathbf{C}&-\mathbf{M}^{-1} \mathbf{K}\end{cases}\begin{cases}\dot{\mathbf{Z}}\\ \mathbf{Z}\end{cases}+\begin{bmatrix}\mathbf{M}^{-1}\end{bmatrix}\mathbf{F}(t) \tag{16}\]  
Assuming that the seat denotes the nth DOF, we can extract the vertical acceleration by the output of the nth element of the vector. The simulations are implemented on Matlab 2020b, with a time interval of 0.001 s.