# Estimation of Aircraft-Dependent Bumpiness Severity in Turbulent Flight
## 2 Methodology
### Severity Estimation of Bumpiness  
Compared to the RMS-g by directly obtaining the root mean square of acceleration data, this paper develops a more theoretically accurate method to estimate the variance of vertical acceleration, which functions as the indicator of aircraft-dependent bumpiness severity. In turbulent flight, the intensity and spatial frequency of turbulence are measured by the flying-through aircraft, inducing the change of vertical acceleration. The Welch spectrum estimation is used to obtain the power density spectrum of measured acceleration data. Furthermore, as the aircraft bumpiness indicator, the variance of vertical acceleration is obtained by Parseval theorem. Firstly, a Tukey-Hanning window is used to decrease the spectrum leakage due to finite-length time series of vertical acceleration. The \(m\)-point Tukey-Hanning window is formulated as  
\[\tau(t_{k})=\left\{ \begin{array}{c}\frac{1}{2}\Big{(}1-cos\Big{(}\frac{t_{k }\pi}{M+1}\Big{)}\Big{)},0\leq t_{k}\leq M\\ 1,M<t_{k}\leq m-M-2\\ \frac{1}{2}\Big{(}1-cos\Big{(}\frac{(m-t_{k}-1)\pi}{M+1}\Big{)}\Big{)},m-M-2 \leq t_{k}\leq m-1\end{array} \right. \tag{15}\]  
where \(M=floor(0.1m-0.2)\), which corresponds to a taper factor of 0.2 in traditional formulation of Tukey-Hanning window. The power normalized Tukey-Hanning window is  
\[\widetilde{\tau}(t_{k})=\frac{\tau(t_{k})}{\sqrt{\frac{1}{m}\sum_{j=0}^{m-1} \tau^{2}\big{(}t_{j}\big{)}}} \tag{16}\]  
Accordingly, the windowed vertical acceleration data series is  
\[a_{z}^{w}(t_{k})=\widetilde{\tau}(t_{k})a_{z}(t_{k}) \tag{17}\]  
Furthermore, by data segmenting and overlap setting, Welch spectrum estimation is used to obtain the power spectrum of \(a_{z}^{w}(t_{k})\)[36]. Given that aircraft maneuvers and aero-elastic vibration would also induce the change of vertical acceleration, a band-pass filter is designed in EDR estimation to eliminate these adverse effects. Similarly, in the estimation of bumpiness severity, the high-end cutoff frequency \(f_{h}\) lies below the aero-elastic response mode of target aircraft, while the lower cutoff frequency \(f_{l}\) is set higher than the phugoid frequency of target aircraft. According to Parseval theorem, the variance of vertical acceleration in time domain equals the power spectrum in the band-pass of the frequency domain  
\[\partial_{a_{z}}^{2}=\sum_{k=k_{l}}^{k_{h}}\delta_{k}^{2} \tag{18}\]  
To summarize, an integrated computation flow for the severity estimation of aircraft bumpiness is shown in Figure 7. The grid on the wing-tail assembly of target aircraft is generated beforehand. According to real-time in-situ EDR indicator, a time series of longitudinal and vertical turbulence components is generated by Equation (6). In the algorithm loop, the generated turbulence and flight parameters are used to compute local velocity by Equation (11) firstly. After that, the aerodynamic force in turbulence is obtained by UVLM. Vertical acceleration and related flight parameters necessary for computing local velocity are obtained by solving the flight dynamics model. For each EDR value, the time series of vertical acceleration data is generated based on the input of turbulence components. If each time step of turbulence components is processed, the estimated aircraft bumpiness indicator \(\hat{\sigma}_{a_{z}}^{2}\) is obtained by Welch spectrum estimation. Section 2 provides a severity estimation method of aircraft bumpiness based on in-situ EDR indicator. The estimation method is dependent on the geometry of the wing-tail assembly and several flight parameters of specific aircraft. Compared to the aircraft-dependent RMS-g and DEVG indicators, which are computed according to recorded in-situ acceleration data, it provides an effective way of estimating the bumpiness severity in advance. Furthermore, since the adverse effects of aircraft maneuvers are eliminated in computing turbulence-induced acceleration response, the computing accuracy of acceleration response can be improved.