# Vibration Response Law of Aircraft Taxing under Random Roughness Excitation
## 2 Random Vibration Models  
### Aircraft Nonlinear Ground Dynamics Model  
Throughout the taxiing process, pavement roughness excitation is transmitted through the tire to the landing gear, which absorbs some of the energy and transfers the vibration to the fuselage. The aircraft is a typical nonlinear model due to the strong nonlinearity of both the tire and the landing gear. In frequency domain dynamic response analysis, nonlinear elements are typically represented by equivalent linear elements. Therefore, in this section, the nonlinear elements are initially represented by equivalent linear letters (with the equivalent linearization process described in the next section), and the nonlinear aircraft model is established. Initially, based on the theory of multi-rigid-body system dynamics, the aircraft is simplified into two parts: the sprung mass and unsprung mass. Additionally, the mechanical properties of the landing gear buffer and the tire are simulated as a spring-plus-damping model, and the lift, \(F_{\text{s}}\), is applied to the center of gravity of the aircraft during takeoff and landing on the runway in a straight line. This yields the ground dynamics model of the 6-DOF aircraft [25], as illustrated in Figure 1.  
At any given moment, the aircraft is in a state of dynamic equilibrium. Therefore, the motion balance equations for each degree of freedom of the aircraft ground dynamics model under pavement roughness excitation can be derived using the D'Alembert principle. Figure 1: Ground dynamics model of a 6-DOF aircraft: (**a**) The overall diagram of the ground dynamics model of a 6-DOF aircraft; (**b**) local diagram of the ground dynamics model of a 6-DOF aircraft. The vertical vibration balance equation for the sprung mass is shown as Equation (1). \[ \begin{split} M_{\mathrm{s}}\ddot{\ddot{\mathrm{Z}}}_{\mathrm{s}}+C_{ \mathrm{f}}\Big{(}\dot{\mathrm{Z}}_{\mathrm{s}}+d\dot{\Phi}-\dot{\mathrm{z}}_{ \mathrm{f}}\Big{)}+K_{\mathrm{f}}(Z_{\mathrm{s}}+d\Phi-z_{\mathrm{f}})+C_{ \mathrm{r}}\Big{(}\dot{\mathrm{Z}}_{\mathrm{s}}-e\dot{\Phi}+b\dot{\Psi}-\dot{ \mathrm{z}}_{\mathrm{r}}\Big{)}+\,K_{\mathrm{r}}(Z_{\mathrm{s}}\\ -e\Phi+b\Psi-z_{\mathrm{r}})+C_{\mathrm{l}}\Big{(}\dot{\mathrm{Z} }_{\mathrm{s}}-e\dot{\Phi}-a\dot{\Psi}-\dot{\mathrm{z}}_{\mathrm{l}}\Big{)}+K_{ \mathrm{l}}(Z_{\mathrm{s}}-e\Phi-a\Psi-z_{\mathrm{l}})+F_{\mathrm{a}}=0\end{split}  \tag{1}\]  
The pitch rotation balance equation for the sprung mass is shown as Equation (2). \[ \begin{split} I_{\mathrm{x}}\ddot{\Phi}+C_{\mathrm{f}}\Big{(} \dot{\mathrm{Z}}_{\mathrm{s}}+d\dot{\Phi}-\dot{\mathrm{z}}_{\mathrm{f}}\Big{)} \!d+K_{\mathrm{f}}(Z_{\mathrm{s}}+d\Phi-z_{\mathrm{f}})\!d-C_{\mathrm{r}}\Big{(} \dot{\mathrm{Z}}_{\mathrm{s}}-e\dot{\Phi}+b\dot{\Psi}-\dot{\mathrm{z}}_{ \mathrm{r}}\Big{)}-K_{\mathrm{r}}(Z_{\mathrm{s}}\\ -e\Phi+b\Psi-z_{\mathrm{r}})e-C_{\mathrm{l}}\Big{(}\dot{\mathrm{Z }}_{\mathrm{s}}-e\dot{\Phi}-a\dot{\Psi}-\dot{\mathrm{z}}_{\mathrm{l}}\Big{)}e- K_{\mathrm{l}}(Z_{\mathrm{s}}-e\Phi-a\Psi-z_{\mathrm{l}})e=0\end{split}  \tag{2}\]  
The lateral rotation balance equation for the sprung mass is shown as Equation (3). \[ \begin{split} I_{\mathrm{y}}\ddot{\Psi}+C_{\mathrm{r}}\Big{(} \dot{\mathrm{Z}}_{\mathrm{s}}-e\dot{\Phi}+b\dot{\Psi}-\dot{\mathrm{z}}_{ \mathrm{r}}\Big{)}\!b+K_{\mathrm{r}}(Z_{\mathrm{s}}-e\Phi+b\Psi-z_{\mathrm{r} })\!b-\\ C_{\mathrm{l}}\Big{(}\dot{\mathrm{Z}}_{\mathrm{s}}-e\dot{\Phi}-a \dot{\Psi}-\dot{\mathrm{z}}_{\mathrm{l}}\Big{)}a-K_{\mathrm{l}}(Z_{\mathrm{s }}-e\Phi-a\Psi-z_{\mathrm{l}})a=0\end{split}  \tag{3}\]  
The vertical vibration balance equations for the unsprung mass of the front, right rear, and left rear landing gear are shown as Equations (4)-(6), respectively.