## 2 Vertical Acceleration and Turbulence Severity Estimation
### Vertical Acceleration Computation in Turbulent Flight
#### 2.3.3 Solution of Vertical Acceleration  
Since the lifting surface of the aircraft was simplified to the wing-tail combination, there may exist a gap between computing results and actual aerodynamic force. We try to develop the incremental aerodynamic derivatives induced by turbulence and add them into existing aerodynamic model. The incremental derivatives function as a supplement to the aerodynamic model, rather than replacement. For each collocation point, with the aerodynamic force in turbulence \(\mathbf{F}_{i}\) and the force without turbulence effects \(\mathbf{F}_{i}^{0}\), the force increment is obtained by \(\mathbf{F}_{i}-\mathbf{F}_{i}^{0}\). After summing the whole vortex rings, the turbulence-induced incremental aerodynamic force \(\Delta\mathbf{F}\) is obtained by  
\[\left[\begin{array}{c}\Delta F_{x}(t_{k})\\ \Delta F_{y}(t_{k})\\ \Delta F_{z}(t_{k})\end{array}\right]=\left[\begin{array}{c}\sum\limits_{i=1 }^{N}\left(F_{x,i}(t_{k})-F_{x,i}^{0}(t_{k})\right)\\ \sum\limits_{i=1}^{N}\left(F_{y,i}(t_{k})-F_{y,i}^{0}(t_{k})\right)\\ \sum\limits_{i=1}^{N}\left(F_{z,i}(t_{k})-F_{z,i}^{0}(t_{k})\right)\end{array}\right] \tag{13}\]  
Furthermore, the incremental pitching moment around the gravity center is  
\[\Delta M_{y}(t_{k})=\sum\limits_{i=1}^{N}\left(x_{cg}-x_{i}\right)\left(F_{z,i }(t_{k})-F_{z,i}^{0}(t_{k})\right)+\sum\limits_{i=1}^{N}\left(z_{i}-z_{cg} \right)\left(F_{x,i}(t_{k})-F_{x,i}^{0}(t_{k})\right) \tag{14}\]  
The turbulence-effected incremental aerodynamic derivatives are described by Equation (15). Since plunge and pitching motion are the main cause of aircraft bumpiness, the lift incremental derivative \(c_{L_{W}}\) and pitching moment incremental derivative \(c_{m_{W}}\) are computed by  
\[\left\{\begin{array}{c}c_{L_{W}}(t_{k})=\frac{\Delta F_{z}(t_{k})}{\hat{q} ^{5}_{S}}\\ c_{m_{W}}(t_{k})=\frac{\Delta M_{y}(t_{k})}{\hat{q}^{5}_{S}c}\end{array}\right. \tag{15}\]  
Figure 5: Neighboring rings for a general vortex ring.  
where \(\overline{q}\) is the dynamic pressure, \(c\) is the mean chord length. \(c_{LW}\) and \(c_{mw}\) function as one of the derivative terms in lift and pitching moment coefficients, respectively. Due to the limitation of panel method, the incremental aerodynamic derivatives including side force, rolling and yawing moment cannot be computed accurately. However, it is of less importance to compute lateral motion accurately in the vertical-acceleration estimation. The plunge and pitching motion which mainly lead to aircraft bumpiness can be solved by VRM with a satisfactory precision. Moreover, CFD method can be adopted to obtain more accurate solutions. However, it cannot be solved in real time and needs a complete set of aircraft's geometric and airfoil parameters. With the nonplanar and unsteady VRM, the real-time change of aerodynamic force caused by turbulence can be acquired accurately. Based on the modified aerodynamic model, flight parameters required in Equation (4) can be obtained by numeric integration [29] or states estimation in flight dynamic model [30]. Here only the differential equation of vertical acceleration \(a_{n}\) and pitching moment are listed:  
\[\left\{\begin{array}{c}a_{n}(t_{k})=q(t_{k})V_{Gx}^{b}(t_{k})-p(t_{k})V_{Gy}^ {b}(t_{k})+g\cos\theta(t_{k})\cos\phi(t_{k})+F_{z}(t_{k})/m\\ I_{y}\dot{q}(t_{k})=M_{y}(t_{k})+(I_{z}-I_{x})p(t_{k})r(t_{k})+I_{xz}(p^{2}(t_ {k})-r^{2}(t_{k}))\end{array}\right. \tag{16}\]  
where \(\theta(t_{k})\) and \(\phi(t_{k})\) are the pitching and rolling angle at \(t_{k}\), \(\left[\begin{array}{ccc}I_{x}&0&I_{xz}\\ 0&I_{y}&0\\ I_{xz}&0&I_{z}\end{array}\right]\) stands for the inertia matrix. Vertical acceleration at any position on the longitudinal plane is obtained by  
\[{a^{\prime}}_{n}(t_{k})=a_{n}(t_{k})+\sqrt{(x-x_{cg})^{2}+(z-z_{cg})^{2}}q^{2} (t_{k}) \tag{17}\]  
where \((x_{cg},z_{cg})\) represent the location of gravity center.