# Vibration Response Law of Aircraft Taxing under Random Roughness Excitation
## 2 Random Vibration Models
### Pavement Roughness Models
#### 2.3.3. Pavement Roughness Excitation Input  
By utilizing the relationship between temporal frequency \(f\), time angular frequency \(\omega\), and spatial frequency \(n\) (as described in Equations (18a) and (18b)), the pavement roughness model in the spatial domain can be transformed into pavement roughness excitation input in the time domain.  
\[f=nv \tag{18a}\] \[\omega=2\pi f \tag{18b}\]  
where \(v\) is the aircraft taxiing speed.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**No.**} & **International Air Transport** & \multirow{2}{*}{**Runway Type**} & \multirow{2}{*}{**Runway Length/m**} \\  & **Association** & & \\  & **(IATA) Code** & & \\ \hline
13 & LFQ & cement & 2710 \\
14 & A military airport & cement & 2990 \\
15 & A military airport & cement & 2470 \\
16 & SHA (runway 1) & cement & 3290 \\
17 & SHA (runway 2) & asphalt & 3290 \\
18 & PVG (runway 1) & cement & 3410 \\
19 & PVG (runway 2) & cement & 3770 \\
20 & PVG (runway 3) & cement & 3390 \\ \hline \hline \end{tabular}
\end{table}
Table 1: _Cont._  
system differential equation becomes a time-varying differential equation, leading to a non-stationary response of the aircraft system. For a non-stationary random vibration system, the vibration response cannot be calculated directly using the frequency response function \(H(\omega)\). Therefore, this paper employs the method of instantaneous frequency response function with spatial frequency [30] combined with the PEM to solve the transient power spectrum of the vibration response during aircraft takeoff and landing taxiing.  
[PERSON] et al. demonstrated through field measurements and theoretical research that aircraft takeoff and landing taxiing on the runway can be regarded as a uniformly accelerated linear motion process. The corresponding velocity \(v\) and displacement \(s\) can be expressed as described in Equations (25a) and (25b) [31].  
\[v=v_{0}+at \tag{25a}\] \[s=v_{0}t+at^{2}/2 \tag{25b}\]  
where \(v_{0}\) is the initial velocity of uniform variable motion, which is taken as 0 during the aircraft takeoff taxiing, and the landing speed during aircraft landing taxiing; \(a\) is the acceleration, which is positive during the aircraft takeoff taxiing and negative during the aircraft landing taxiing.  
The time-space frequency relationship of the uniformly varying motion during aircraft takeoff and landing taxiing can be obtained by substituting Equations (25a) and (25b) into Equation (18), as shown in Equation (26).  
\[f=nv=n\sqrt{(v_{0}^{2}+2 as)} \tag{26}\]  
Based on the characteristic that the frequency response function \(H(\omega)\) of the aircraft system in non-stationary random vibration is constant at temporal frequency, the instantaneous frequency response function \(H(n,t)\) at spatial frequency can be derived from the relationship between temporal frequency and spatial frequency. Therefore, by substituting Equation (26) into Equation (22), the instantaneous spatial domain frequency response function matrix of the aircraft system can be expressed as Equation (27).  
\[H(n,t)=\frac{[K_{\rm t}]+i2\pi n\sqrt{({v_{0}}^{2}+2 as)}[C_{\rm t}]}{[K_{\rm a }]-4\pi^{2}n^{2}({v_{0}}^{2}+2 as)[M_{\rm a}]+i2\pi n\sqrt{({v_{0}}^{2}+2 as)[C_ {\rm a}]}} \tag{27}\]  
After obtaining Equation (27), the instantaneous spatial PSD of the aircraft vibration response can be calculated following the PEM in Section 3.1.