# Vibration Response Law of Aircraft Taxing under Random Roughness Excitation
## 2 Random Vibration Models
### Aircraft Nonlinear Ground Dynamics Model
\[m_{\mathrm{f}}\ddot{\mathrm{z}}_{\mathrm{f}}-C_{\mathrm{f}}\Big{(}\dot{ \mathrm{Z}}_{\mathrm{s}}+d\dot{\Phi}-\dot{\mathrm{z}}_{\mathrm{f}}\Big{)}-K_{ \mathrm{f}}(Z_{\mathrm{s}}+d\Phi-z_{\mathrm{f}})+c_{\mathrm{f}}\dot{\mathrm{z }}_{\mathrm{f}}+k_{\mathrm{f}}z_{\mathrm{f}}-c_{\mathrm{f}}\dot{\mathrm{q}}_{ \mathrm{f}}-k_{\mathrm{f}}\dot{\mathrm{q}}_{\mathrm{f}}=0 \tag{4}\]  
\[ \begin{split} m_{\mathrm{r}}\ddot{\mathrm{z}}_{\mathrm{r}}-C_{ \mathrm{r}}\Big{(}\dot{\mathrm{Z}}_{\mathrm{s}}-e\dot{\Phi}+b\dot{\Psi}-\dot{ \mathrm{z}}_{\mathrm{r}}\Big{)}-K_{\mathrm{r}}(Z_{\mathrm{s}}-e\Phi+b\dot{ \Psi}-z_{\mathrm{r}})+c_{\mathrm{r}}\dot{\mathrm{z}}_{\mathrm{r}}+k_{\mathrm{ r}}z_{\mathrm{r}}-c_{\mathrm{r}}\dot{\mathrm{q}}_{\mathrm{r}}\\ -k_{\mathrm{r}}q_{\mathrm{r}}=0\end{split}  \tag{5}\]  
\[ \begin{split} m_{\mathrm{l}}\ddot{\mathrm{z}}_{\mathrm{l}}-C_{ \mathrm{l}}\Big{(}\dot{\mathrm{Z}}_{\mathrm{s}}-e\dot{\Phi}-a\dot{\Psi}-\dot{ \mathrm{z}}_{\mathrm{l}}\Big{)}-K_{\mathrm{l}}(Z_{\mathrm{s}}-e\Phi-a\Psi-z_{ \mathrm{l}})+c_{\mathrm{l}}\dot{\mathrm{z}}_{\mathrm{l}}+k_{\mathrm{l}}z_{ \mathrm{l}}-c_{\mathrm{l}}\dot{\mathrm{q}}_{\mathrm{l}}\\ -k_{\mathrm{l}}q_{\mathrm{l}}=0\end{split}  \tag{6}\]  
Combining Equations (1)-(6), the matrix equation for the spatial motion of the 6-DOF aircraft ground dynamics model is established as shown in Equation (7). \[[M_{\mathrm{a}}][\ddot{X}]+[C_{\mathrm{a}}][\dot{X}]+[K_{\mathrm{a}}][X]=[K_ {\mathrm{f}}][q]+[C_{\mathrm{t}}][\dot{q}] \tag{7}\]  
where \(M_{\mathrm{s}}\) denotes the aircraft sprung mass; \(m_{\mathrm{f}}\), \(m_{\mathrm{r}}\), and \(m_{\mathrm{l}}\) denote the unsprung mass of the front, right rear, and left rear landing gear, respectively; \(C_{\mathrm{f}}\), \(C_{\mathrm{r}}\), and \(C_{\mathrm{l}}\) denote the damping coefficients of the front, right rear, and left rear landing gear buffer, respectively; \(K_{\mathrm{f}}\), \(K_{\mathrm{r}}\), and \(K_{\mathrm{l}}\) denote the stiffness coefficients of the front, right rear, and left rear landing gear buffer, respectively; \(c_{\mathrm{f}}\), \(c_{\mathrm{r}}\), and \(c_{\mathrm{l}}\) denote the damping coefficients of the front, right rear, and left rear landing gear tire, respectively; \(k_{\mathrm{f}}\), \(k_{\mathrm{r}}\), and \(k_{\mathrm{l}}\) denote the stiffness coefficients of the front, right rear, and left rear landing gear tire, respectively; \(e\) and \(d\) denote the horizontal distances from the rear and front landing gear to the \(x\)-axis of the aircraft, respectively; \(b\) and \(a\) denote the horizontal distances from the right rear and left rear landing gear to the \(y\)-axis of the aircraft, respectively; \(I_{\mathrm{x}}\) and \(I_{\mathrm{y}}\) denote the rotational inertia of the aircraft around the \(x\)-axis and \(y\)-axis, respectively; \(Z_{\mathrm{s}}\), \(\Phi\), and \(\Psi\) denote the vertical displacement, pitch rotation displacement, and lateral rotation displacement of the sprung mass, respectively; \(z_{\mathrm{f}}\), \(z_{\mathrm{r}}\), and \(z_{\mathrm{l}}\) denote the vertical displacements of the front, right rear, and left rear landing gear unsprung mass, respectively; \(q_{\mathrm{f}}\), \(q_{\mathrm{r}}\), and \(q_{\mathrm{l}}\) denote the roughness excitation of the front, right rear, and left rear landing gear tire, respectively; \(F_{\mathrm{a}}\) is the lift force acting on the aircraft; \([M_{\mathrm{a}}]\) is the mass matrix of the aircraft model, which is a matrix of 6 rows and 6 columns. \([M_{\mathrm{a}}]\) includes the mass of the six degrees of freedom of the aircraft, that is, \(M_{\mathrm{s}}\), \(I_{\mathrm{x}}\), \(I_{\mathrm{y}}\), \(m_{\mathrm{f}}\), \(m_{\mathrm{r}}\), and \(m_{\mathrm{l}}\). \([C_{\mathrm{a}}]\) is the damping matrix of the aircraft model, which is a matrix of 6 rows and 6 columns. \([C_{\mathrm{a}}]\) includes the damping coefficient of the aircraft, that is, \(C_{\mathrm{f}}\), \(C_{\mathrm{r}}\), \(C_{\mathrm{l}}\), \(c_{\mathrm{f}}\), \(c_{\mathrm{r}}\), and \(c_{\mathrm{l}}\).