## 3 Modal Simulation Analysis
### Eleven-DOFs Dump Truck Model  
The dump truck is simplified to an 11-DOFs model comprising three DOFs associated with the vehicle body (\(Z_{b}\), \(\theta\), \(\Phi\)), six DOFs associated with the bounce and rolling motions of three integral axles (\(Z_{A}\), \(Z_{B}\), \(Z_{C}\), \(Z_{D}\), \(Z_{E}\), \(Z_{F}\)), and two DOFs describe the pitch motions of the balanced suspension (\(\theta_{C}\), \(\theta_{D}\)), as seen in Figure 6.  
Figure 4: Identification stabilization diagrams of ACS-SSI and SSI. PSD: Power Spectral Density  
Figure 5: Schematic diagram of simulation analysis. To simplify the modeling of the balanced suspension, the leaf spring is divided into two separate stiffnesses (\(k_{E}\) and \(k_{F}\)) and damping (\(c_{E}\) and \(c_{F}\)). Both ends of the leaf springs are separately linked with the front and rear axle through the rigid stabilizer rod. The front and rear axle vibrations are coupled together with the stabilizer rod [33]. The equivalent balance suspension model is shown in Figure 7.  
The dynamic model of the dump truck, considering the integral axle and the balanced suspension, can be written in matrix form as:  
\[\mathbf{M}\ddot{\mathbf{Z}}+\mathbf{C}\dot{\mathbf{Z}}+\mathbf{K}Z=\mathbf{K_{t }Q_{g}}+\mathbf{C_{t}\dot{Q}_{g}}, \tag{3}\]  
where \(\mathbf{M}\) is the matrix representing vehicle body mass, \(\mathbf{Z}\) is the vector of vehicle dynamic responses, \(\mathbf{C}\) is the matrix of vehicle damping, \(\mathbf{K}\) is the stiffness matrix of the vehicle system, \(\mathbf{C_{t}}\) is the matrix of tire damping, \(\mathbf{K_{t}}\) is the stiffness matrix of the tire and \(\mathbf{Q_{g}}\) is the vector of displacement inputs acting on the tire. Figure 6: Eleven-DOFs dynamic model of a dump truck. Figure 7: Dynamic model of balanced suspension. The differential equations of Equation (3) can be transformed into the state space equation (Equation (4)) to facilitate the simulation, reducing the need for solution time and computing resources under the premise of ensuring system integrity and accuracy. \[\left\{ \begin{array}{l}\dot{x}=\mathbf{A}x+\mathbf{B}u\\ Y=\mathbf{C}x+\mathbf{D}u\end{array} \right., \tag{4}\]  
The conventional method for equation transformation is achieved by selecting an appropriate intermediate variables \(x\), regardless of the derivative term in the input variable \(u\). The road roughness input \(\mathbf{Q_{g}}\) was directly converted into the force input acting on the vehicle \(K_{t}Q_{g}+C_{t}\dot{Q}_{g}=f(t)\) through numerical differential operation. Then:  
\[\dot{x}=\left[ \begin{array}{cc}\mathbf{0}&\mathbf{I}\\ -\mathbf{M^{-1}K}&-\mathbf{M^{-1}C}\end{array} \right]x+\left\{ \begin{array}[] {c}\widehat{\mathbf{0}}\\ \mathbf{M^{-1}}\end{array} \right\}f(t), \tag{5}\]  
where \(\mathbf{I}\) and \(\mathbf{0}\) represent identity and null matrices with similar dimensions as the property matrices, respectively; \(\widehat{0}\) is a null column vector whose length is the same as that of forcing vector \(f\), and \(x\) is a vector containing the states (displacement and velocity vectors). The state variables \(x\) were selected as Equation (6):  
\[ \begin{array}{ll}x&=[Z,\ \dot{Z}]\\ &=[Z_{b},\theta,\phi,Z_{f},\theta_{f},Z_{m},\theta_{m},Z_{r},\theta_{r},\theta _{C},\theta_{D},\dot{Z}_{b},\dot{\theta},\dot{\phi},\dot{Z}_{f},\dot{\theta}_ {f},\dot{Z}_{m},\dot{\theta}_{m},\dot{Z}_{r},\dot{\theta}_{r},\dot{\theta}_{C},\dot{\theta}_{D}]\end{array} , \tag{6}\]  
Although input order reduction has been realized in Equation (5) via the conversion from \(K_{t}Q_{g}+C_{t}\dot{Q}_{g}\) to \(f(t)\), the essence of this order reduction is an approximate computational process of a numerical differentiation algorithm based on the Runge-Kutta method. Order reduction itself could cause truncation error input and reduce the reliability and accuracy of the system output. Therefore, to oversimplify the input vector, it is essential to keep the input derivative for obtaining reliability and accuracy.