# Vibration Response Law of Aircraft Taxing under Random Roughness Excitation
## 2 Random Vibration Models
### Statistical Linearization of the Nonlinear System
#### 2.2.2 Calculation of Equivalent Stiffness and Equivalent Damping by Statistical Linearization  
In a nonlinear vibration system with n degrees of freedom, the mass of each inertial element is \(m_{1},\ m_{2},\ldots,m_{n}\), respectively, and the excitation force \(\{F(t)\}=\{F_{1}(t),F_{2}(t),\ldots,F_{n}(t)\}^{\rm T}\) acting on the system is an n-dimensional joint Gaussian process vector with zero mean. Then, in the inertial coordinate system \(y_{1},y_{2},\ldots,y_{n}\), the differential equation of motion for this system is shown in Equation (10).  
\[[M]\{\ddot{Y}\}+\{G(Y,\dot{Y})\}=\{F(t)\} \tag{10}\]  
where \([M]=diag[m_{i}]\) is the inertial matrix of the system in the coordinate system \(y_{1},y_{2},\ldots,y_{n}\); \(G(Y,\dot{Y})\) is the sum of elastic force and damping force acting on \(m_{i}\); and \(F(t)\) is the exciting force applied to \(m_{i}\).  
Suppose the transformation between coordinates \(y_{1},y_{2},\ldots,y_{n}\) and coordinates \(z_{1},z_{2},\ldots,z_{n}\) can be expressed as Equation (11).  
\[\{Y\}=[A]\{Z\} \tag{11}\]  
Taking Equation (11) into Equation (10), the equation of motion of the system represented by the relative coordinates \(z_{1},z_{2},\ldots,z_{n}\) is obtained, as shown in Equation (12).  
\[\left[M^{\prime}\right]\{\ddot{Z}\}+\left\{G^{\prime}\Big{(}Z,\dot{Z}\Big{)} \right\}=\{F(t)\} \tag{12}\]  
where \([M^{\prime}]=[M][A]\); and \(\{G^{\prime}(Z,\dot{Z})\}=\left\{G^{\prime}(Y,\dot{Y})\right\}=\left[g_{1}( z,\dot{z}),g_{2}(z,\dot{z}),\ldots g_{n}(z,\dot{z})\right]^{\rm T}\).  
Assume that the statistically equivalent linear system of the initial nonlinear system satisfies the differential equation shown in Equation (13).  
\[\left[M^{\prime}\right]\{\ddot{Z}\}+[C_{e}]\{\dot{Z}\}+[K_{e}]\{Z\}=\{F(t)\} \tag{13}\]  
where \([K_{e}]\) and \([C_{e}]\) are the equivalent linear stiffness matrix and the equivalent damping matrix, respectively.  
\[K_{eij}=E\left[\frac{\partial G_{i}\Big{(}Z,\dot{Z}\Big{)}}{\partial Z_{j}} \right],\ \ i,j=1,2,\ldots,n \tag{14}\]  
\[C_{eij}=E\left[\frac{\partial G_{i}\Big{(}Z,\dot{Z}\Big{)}}{\partial\dot{Z}_{j }}\right],\ \ i,j=1,2,\ldots,n \tag{15}\]  
Figure 3: Variation curve of landing gear damping force with moving speed.  
Since \(f_{i}(Z_{i})\) is the elastic force of elastic element \(i\), then \(df_{i}(Z_{i})/dZ_{i}\) is the stiffness of elastic element \(i\). Therefore, \(E[df_{i}(Z_{i})/dZ_{i}]\) should be the statistical equivalent stiffness coefficient \(K_{ei}\) of elastic element \(i\), which can be expressed as Equation (15a).  
\[K_{ei}=E\left[\frac{df_{i}(Z_{i})}{dZ_{i}}\right],\ \ \ i=1,2,\ldots,n\] (15a) Similarly, the statistically equivalent damping coefficient, \[C_{ei}\], of each damping element can be derived as shown in Equation ( 15b ). \[C_{ei}=E\left[\frac{df_{i}\left(\dot{Z}_{i}\right)}{d\dot{Z}_{i}}\right],\ \ \ i=1,2,\ldots,n \tag{15b}\]  
Consequently, the equivalent stiffness coefficient and the equivalent damping coefficient can be treated as the expected values of Equations (8) and (9) for varying displacement and velocity conditions. In reality, the probability distribution of the vertical vibration displacement of the landing gear and tire during aircraft taxiing becomes relevant. After estimating the probability distribution, Equations (15a) and (15b) can be employed to obtain the statistically equivalent linear coefficient for each element.