2.1 Aerodynamic database development

Aerodynamic database development includes data manipulation, compatibility check, input information of aircraft main geometry and moment of inertia data, equations of motion, and unsteady thrust model. The flowchart for development is presented in Fig. 1. In references of Chang et al. [Reference Chang, Lv, Shi and Chen9] & Jiang et al. [Reference Jiang, Chang, Yang and Mingwei Ding10] have detail descriptions of data manipulation, compatibility check, equations of motion, and unsteady thrust model.

Figure 1.

Flowchart of aerodynamic database development.

2.2 Development of root caused analysis in aviation accident

This section describes the development of root caused analysis in aviation accident, as shown in Fig. 2. The flowchart of Fig. 2 is based on the data of the nonlinear unsteady aerodynamic database. The fuzzy-logic modeling technique(Reference Lan and Chang11,Reference Chang, Ye, Lan and Guan12) is used to setup aerodynamic models. The aerodynamic models can provide the stability and controllability derivatives to be used for the accident analyses. The derivative indices can help to pinpoint the major cause more efficiently while proceeding event or accident investigation to judge about how difficult it was for the pilot or the autopilot system to control the aircraft in loss of control conditions.

Figure 2.

Flowchart for aviation accident root cause assessment.

2.3 Nonlinear unsteady aerodynamic models

For unsteady aerodynamic modeling, in the late 2009, two preferred modeling methods emerged based on neural network [Reference Yang, Ho, Lan and Hsiao7] and fuzzy-logic algorithm [Reference Chang and Tan13]. For the aerodynamic modeling using fuzzy-logic algorithm, the accuracy of modeling is adjusted by studying the selection of membership functions. The number of modeling parameters without limit setting is the advantage of this fuzzy-logic modeling method. Modeling means to establish the numerical relationship among certain variables of interest. In the fuzzy-logic model, more complete necessary influencing flight variables can be included to capture all possible effects on aircraft response to specify applications.

The longitudinal main aerodynamics is assumed to depend on the following ten flight variables [Reference Jiang, Chang, Yang and Ding3]:

(1) \begin{align}{C_x},{\rm{ }}{C_z},{\rm{ }}{C_m} = f(\alpha,\dot \alpha, q,{k_1},\beta,{\delta_e},M,p,{\delta_s},\bar q)\end{align}

The coefficients on the left-hand side of Equation (1) represent the coefficients of axial force (C _x ), normal force (C _z ), and pitching moment (C _m ), respectively. The variables on the right-hand side of Equation (1) are defined as follows: angle-of-attack ( $\alpha$ ), time rate of angle-of-attack (dα/dt, or $\dot \alpha$ ), pitch rate (q), longitudinal reduced frequency (k ₁ ), sideslip angle ( $\beta$ ), control deflection angle of elevator ( $\delta$ _e ), Mach number (M), roll rate (p), stabiliser angle ( $\delta$ _s ), and dynamic pressure ( $\bar q$ ). These variables are called the influencing variables. The roll rate is included here because it is known that an aircraft encountering hazardous weather tend to develop rolling which may affect longitudinal stability. The variable of dynamic pressure is for estimation of the significance in structural deformation effects.

For the lateral-directional aerodynamics to depend on the following 11 flight variables [Reference Jiang, Chang, Yang and Ding3],

(2) \begin{align}{C_y},{C_l},{C_n} = f(\alpha,\beta,\phi, p,r,{k_2},{\delta_a},{\delta_r},M,\dot \alpha, \beta \end{align}

The coefficients on the left-hand side of Equation (2) represent the coefficients of side force (C _y ), rolling moment (C _l ) and yawing moment (C _n ), respectively. The variables on the righthand side of Equation (2) are defined as follows: angle-of-attack ( $\alpha$ ), sideslip angle ( $\beta$ ), roll angle ( $\phi$ ), roll rate (p), yaw rate (r), lateral-directional reduced frequency (k ₂ ), control deflection angle of aileron ( $\delta$ _a ), control deflection angle of rudder ( $\delta$ _r ), Mach number (M), the time rate of angle-of-attack ( $\dot \alpha$ ), and the time rate of sideslip angle ( $\dot \beta$ ).

2.4 Model-based derivatives

The basic definition of a derivative is that the tangent slope of a point on a curve is the derivative of that point. In this study, the difference-in-center method is used to calculate the derivative value, that is, the abscissa value at the point, and add and subtract a minor abscissa value of the same value, which corresponds to the slope of the line connecting the two points on the curve. Fuzzy-logic modeling is very important in the aerodynamic performance analysis of real flight data. The effectiveness criteria for derivative values of stability, damping and controllability are shown in Table 1. The root caused analysis for aircraft accident due to the pilots in loss of control are based on the assessment of those derivatives [Reference Jiang, Chang, Yang and Ding3].

Table 1.

The effectiveness criteria for derivative values of stability, damping, and controllability [Reference Jiang, Chang, Yang and Ding3]

Remarks: The effectiveness is judged based on the positive or negative of derivative value.

2.5 Flight simulation of flight handling quality

The guidelines of flying quality in civil airplane are expressed in damping ratios and natural frequencies, The time to double or halve the amplitude of initial disturbances is described in Ref. [Reference Jiang, Chang, Yang and Mingwei Ding10], defined as:

(3) \begin{align}{T_2} = \frac{{\ln 2}}{{ - \varsigma {\omega _n}}}\end{align}

where ζ is the damping ratio and ω _n is the natural frequency. Note that -ζω _n is the real part of the eigenvalues. If it is positive, the system is unstable and T ₂ is also positive, representing the time to double the amplitude. On the other hand, if it is negative, the system is stable, and T ₂ is negative, representing the time to halve the amplitude. In this latter case, T ₂ is replaced by T _1/2 and ‘2’ in Equation (3) is replaced by ½. For simplicity in the computer output, T ₂ will be used in all cases, with the understanding that if it is negative, it should be the time to halve the amplitude.

From the above considerations, the first step in evaluating the flying qualities is to determine the eigenvalues of an airplane along the flight trajectory. In the present approach, one will numerically integrate the 6-DOF dynamic equations of motion to determine the eigen-modes of motion at the same time at every instant. The matrix for the eigenvalues consists of the first derivatives evaluated at the instant under consideration, not about the trim points as done conventionally. Note that in abnormal flight conditions there may not be trim points.

It is easier to extract the stability characteristics by formulating the equations in the stability axes. Therefore, in the present study of flight simulation, the general equations of motion in the following are integrated:

(4) \begin{align}\frac{{d\alpha }}{{dt}} = {f_1} &= \left\{ \left[\left( - \frac{X}{m} + g\sin \theta \right)/V - r\sin \beta \right]\sin \alpha + \left[\left(\frac{Z}{m} + g\cos \theta \cos \varphi \right)/V - p\sin \beta \right]\cos \alpha \right\}\nonumber \\[5pt] &\quad /\cos \beta + q\end{align}

(5) \begin{align}\frac{{d\beta }}{{dt}} &= {f_2} = \left[ - \left(\frac{X}{m} - g\sin \theta \right)\sin \beta /V - r\right]\cos \alpha + \left(\frac{Y}{m} + g\cos \theta \sin \varphi \right)\cos \beta /V\nonumber \\[5pt]&\quad \left[ - \left(\frac{Z}{m} + g\cos \theta \cos \varphi \right)\sin \beta /V + p\right]\sin \alpha \end{align}

(6) \begin{align}\frac{{dp}}{{dt}} = {f_3} = \left[L + \frac{{{I_{xz}}N}}{{{I_{zz}}}} + {I_{xz}}\left(1 + \frac{{{I_{xx}} - {I_{yy}}}}{{{I_{zz}}}}\right)pq + \left({I_{yy}} - {I_{zz}} - \frac{{I_{xz}^2}}{{{I_{zz}}}}\right)qr\right]/(A{I_{xx}})\end{align}

(7) \begin{align}\frac{{dq}}{{dt}} = {f_4} = [M + {I_{xz}}({r^2} - {p^2}) + ({I_{zz}} - {I_{xx}})rp]/{I_{yy}}\end{align}

(8) \begin{align}\frac{{dr}}{{dt}} = {f_5} = \left[\frac{{{I_{xz}}L}}{{{I_{xx}}}} + N + \left({I_{xx}} - {I_{yy}} + \frac{{I_{xz}^2}}{{{I_{xx}}}}\right)pq + \left(\frac{{{I_{yy}} - {I_{zz}}}}{{{I_{xx}}}} - 1\right){I_{xz}}qr\right]/(A{I_{zz}})\end{align}

(9) \begin{align}\frac{{d\theta }}{{dt}} = {f_6} = q\cos \varphi - r\sin \varphi \end{align}

(10) \begin{align}\frac{{d\varphi }}{{dt}} = {f_7} = p + \tan \theta (q\sin \varphi + r\cos \varphi )\end{align}

(11) \begin{align}\frac{{dV}}{{dt}} = {f_8} = \left(\frac{X}{m} - g\sin \theta \right)\cos \alpha \cos \beta + \left(\frac{Y}{m} + g\cos \theta \sin \varphi \right)\sin \beta + \left(\frac{Z}{m} + g\cos \theta \cos \varphi \right)\sin \alpha \cos \beta \end{align}

where $A = 1 - \frac{{I_{xz}^2}}{{{I_{xx}}{I_{zz}}}}$ , and X, Y, Z, L, M, N are the forces and moments acting on the transport and are estimated from the fuzzy-logic aerodynamic models.

To determine the flying qualities, one needs the eigenvalues of the linearised equations. Instead of using an ‘average’ linearised system, linearisation is done at every time instant, i.e. the so-called time linearisation, as indicated earlier. In other words, a system of matrix (A) consisting of $\partial {f_i}/\partial {x_j}$ , i = 1,…,8 and j = 1,…,8, is determined while the time integration of the dynamic equation is performed, where x _j stands for $\alpha$ , $\beta$ , p, q, r, θ, $\phi$ and V, respectively, for j = 1, …, 8. Note that the inertial effects from the moments of inertia are incorporated. The inertial effects could become essential in recovering a flight vehicle from stall, or when the conventional control surfaces are not effective. To determine the eigenvalues of A, the QR transformation technique is employed. Unfortunately, it is difficult to identify the individual modes of motion from these eigenvalues from one instant to another because of the rapid changes of aerodynamic forces and moments in turbulence. One approach to solve this problem is to use the approximate modes of motion obtained from decoupled longitudinal and lateral-directional Equations (14) as guidance.

(1) The decoupled linearised longitudinal equations of motion are as follows:

(12) \begin{align}\dot u = - g\theta \cos {\theta _1} + {X_u}u + {X_\alpha }\alpha + {X_{\delta e}}\delta e\end{align}

(13) \begin{align}{U_1}\dot \alpha - {U_1}\dot \theta = - g\theta \sin {\theta _1} + {Z_u}u + {Z_\alpha }\alpha + {Z_{\dot \alpha }}\dot \alpha + {Z_q}\dot \theta + {Z_{\delta e}}\delta e\end{align}

(14) \begin{align}\ddot \theta = {M_u}u + {M_\alpha }\alpha + {M_{\dot \alpha }}\dot \alpha + {M_q}\dot \theta + {M_{\delta e}}\delta e\end{align}

where the $\dot u$ is accelerations in X axis direction; ${U_1}\;$ is disturbance introduced speed in X axis direction of atmosphere turbulence; $\theta$ and ${\theta _1}$ are pitch angle and disturbance introduced pitch angle in atmosphere turbulence, respectively; ${X_u}$ , ${X_\alpha }$ , and $\;{X_{\delta e}}$ are the dimensional alteration of force along X axis with the speed, angle-of-attack, and elevator angle, respectively; ${Z_u}$ , ${Z_\alpha }$ , ${Z_{\dot \alpha }}$ , ${Z_q}$ , and ${Z_{\delta e}}\;$ are the dimensional alteration of force along Z axis with the speed, angle-of-attack, time rate of angle-of-attack, pitch rate and elevator angle, respectively. ${M_u}$ , ${M_\alpha }$ , ${M_{\dot \alpha }}$ , ${M_q}$ , and ${M_{\delta e}}\;$ are the dimensional alteration of moment about Y axis with the speed, angle-of-attack, time rate of angle-of-attack, pitch rate and elevator angle.

(2) The decoupled lateral-directional equations of motion are:

(15) \begin{align}{U_1}\dot \beta + {U_1}\dot \psi = g\varphi \cos {\theta _1} + {Y_\beta }\beta + {Y_p}\dot \varphi + {Y_r}\dot \psi + {Y_{\delta a}}\delta a + {Y_{\delta r}}\delta r\end{align}

(16) \begin{align}\ddot \varphi - {\bar A_1}\ddot \psi = {L_\beta }\beta + {L_p}\dot \varphi + {L_r}\dot \psi + {L_{\delta a}}\delta a + {L_{\delta r}}\delta r\end{align}

(17) \begin{align}\ddot \psi - {\bar B_1}\ddot \varphi = {N_\beta }\beta + {N_p}\dot \varphi + {N_r}\dot \psi + {N_{\delta a}}\delta a + {N_{\delta r}}\delta r\end{align}

where ${\bar A_1} = {I_{xz}}/{I_{xx}}$ and ${\bar B_1} = {I_{xz}}/{I_{zz}}$ ; ${Y_\beta }$ , ${Y_p}$ , ${Y_r}$ , ${Y_{\delta a}}\;$ and $\;{Y_{\delta r}}\;$ are the dimensional alteration of force along Y axis with the sideslip angle, roll rate, yaw rate, aileron angle and rudder angle, respectively; ${L_\beta }$ , ${L_p}$ , ${L_r}$ , ${L_{\delta a}}$ , and ${L_{\delta r}}\;$ are the dimensional alteration of moment about X axis with the sideslip angle, roll rate, yaw rate, aileron angle and rudder angle, respectively. ${N_\beta }$ , ${N_p}$ , ${N_r}$ , ${N_{\delta a}}$ , and ${N_{\delta r}}\;$ are the dimensional alteration of moment about Z axis with the sideslip angle, roll rate, yaw rate, aileron angle and rudder angle, respectively.

The detail dimensional derivatives of X, Y, Z, M, L and N are described and given in reference of Roskam 2018 [Reference Roskam14]. The characteristic equations of Equations (12)–(14) and (15)–(17) are quadratic polynomials. Their roots are solved by the quadratic factorisation method of the Lin-Bairstow algorithm based on the reference of Hovanesian and Pipes 1969 [Reference Hovanessian and Pipes15].

The 4th degree polynomial of the longitudinal eigen equation has four roots; they are two complex conjugates [Reference Yang, Ho, Lan and Hsiao7]. The short-period mode is one of the two complex conjugation modes. Another one is phugoid mode (long-period mode). Each pattern has the same real part, but the imaginary part is equal in size and with reverse sign. The 4th degree polynomial of the transverse eigen equation also has four roots; they are a pair of complex conjugations and two real numbers. This pair of composite conjugates represent the Dutch-rolling pattern. One of the two real values represents the spiral mode and another one represents the rolling mode.

3.1 Basic aircraft data required for modeling

The force and moment coefficients in equations of motion require to input the information of main geometry and moment of inertia data [Reference Hovanessian and Pipes15] for the jet transports. The data of main geometry and moments of inertia for these two transports are presented in Table 2:

Table 2.

Main geometry and moment of inertia data for Aircraft A & B

The moment (or product) of inertia in Table 2 is estimated by Weight Sizing Module, Advanced Aircraft Analysis (AAA), aircraft design software (DARcorporation 2018) [16]. The Weight Sizing Module allows determination of mission segment fuel fractions and moment (or product) of inertia. Aircraft A and B are symmetrical about the XZ plane; the case it automatically follows that: I _xy = I _yz = 0; both parameters are not shown in Table 2. Both values of I _xz in Table 2 are zero due to the clean configuration in cruise flight. The value of I_xz is not zero, when transport aircraft in takeoff and landing phases with landing gears and high-lift devices as the configuration (Lan and Chang, 2018) [Reference Lan and Chang11].

3.2 Analysis of model predictions

Aircraft A and B encountered serious atmosphere turbulence at the altitude around 10,050 m and 11,277 m in transonic flights, respectively. As a result, several passengers and cabin crews sustained injuries, because of which these two events were classified as the aviation accidents. The corresponding flight data of Aircraft A and B are extracted from FDR in time spans t = 3,910– 3,990s and t = 6,805 $\sim$ 6,895s, respectively.

The aerodynamic coefficient variations of vertical force C _z , pitching moment C _m , rolling moment C _l , and yawing moment C _n are worthy of attention; those aerodynamic coefficients are regarded as main aerodynamic coefficients in the present study.

The final main aerodynamic models of aerodynamic coefficients consist of many fuzzy rules for each coefficient as described from Tables 3–6. In Tables, the numbers below each input variable represents optimum structure in membership function. The total number of fuzzy cells (n) in each model is the product of each number which presented in column 3. The last column shows the final multiple correlation coefficients (R ² ). The accuracy of the established aerodynamic model through the fuzzy-logic algorithm can be judged by the multiple correlation coefficients (R ² ).

Table 3.

Final models of longitudinal aerodynamics for Aircraft A

Table 4.

Final models of lateral-directional aerodynamics for Aircraft A

Table 5.

Final models of longitudinal aerodynamics for Aircraft B

Table 6.

Final models of lateral-directional aerodynamics for Aircraft B

The comparisons of model-predicted results with modeling input data for Aircraft A and Aircraft B are shown in Figs. 3 and 4, respectively. Vertical force C _z, pitching moment C _m, rolling moment C _l, and yawing moment C _n. The predicted data of C _z-data by the final models have good match with the input data before the modeling, as shown in Figs. 3(a) and 4(a). The C _m-data in Figs. 3(b) and 4(b) scattering is most likely caused by turbulence-induced buffeting on the structure, in particular on the horizontal tail. The C _l-data in Fig. 3(c) has acceptable comparison and Fig. 4(c) has good match with the input data before the modeling. The C _n-data in Figs. 3(d) and 4(d) have acceptable comparisons with the input data before the modeling. The predicted data by the final models have good match with the input data before the modeling. Once the aerodynamic models are set up, one can calculate all necessary derivatives to analyse the stability.

Figure 3.

Comparisons of model-predicted results with modeling input data for Aircraft A.

Figure 4.

Comparisons of model-predicted results with modeling input data for Aircraft B.

3.3 Flight environment analysis

Aerodynamic situations can reflect the quality of the flight environment. The corresponding flight data of Aircraft A and B are extracted from flight data record (FDR) in time spans t = 3,910–3,990s and t = 6,805–6,895s, respectively. The changes in nonlinear unsteady behaviour and major flight variables in the flight environment of Aircraft A and B are shown in Table 7. One can select the minimum and maximum values of each flight variable within the time span. The other flight variables use the same method to establish Table 7.

3.3.1 Aerodynamic scenario

(1) Aircraft A

Figure 5 shows the relevant flight data of Aircraft A, when it encountered serious atmospheric turbulence during its cruise. The analysis model was built using data from the flight data over a period of t = 3,898–3,990s, during which Aircraft A encountered a serious atmospheric turbulence. Figure 5(a) variations of the normal acceleration ${a_z}$ over time shows that ${a_z}$ reaches a maximum of 1.74g at about 3,934s and a minimum of 0.015 g at t = 3,936s, in which case the passenger is highly likely to be injured due to the rapid change in g value. Figure 5(b) shows that the maximum angle-of-attack α when encountering atmospheric turbulence is about 6.6 deg.

Figure 5.

Aerodynamic scenario of Aircraft A encountering atmospheric turbulence.

Figure 5(c) shows the variations of altitude h changing with time when encountering atmospheric turbulence. It shows that the highest vertical descent height of the aircraft in the process of fluctuating movement when encountering atmospheric turbulence is t = 3,922–3,934s, and the altitude reaches 67.056m. Figure 5(d) shows the Mach number M change over time, with the Mach number dropping from 0.8092 to 0.7721 when Aircraft A encountered turbulence.

(2) Aircraft B

Figure 6 shows the relevant flight data of Aircraft B when it encountered serious atmospheric turbulence during cruise. The sampling period of the research data was t = 6,805–6,895s. Figure 6(a) shows that the normal acceleration ${a_z}$ reaches a peak of 1.67 g with t about 6,853s, and drops to a minimum of 0g at t about 6,856s. Figure 6(b) shows that the change in the angle-of-attack α is roughly in sync with the change in $\;{a_z}$ , reaching a maximum of about 6 degrees, which is already significantly above the cruising phase standard value. Figure 6(c) shows that when the height h is the highest (t = 6,853s), the maximum instantaneous height drop is about 28.65m. Figure 6(d) shows that the Mach number M also drops from the stable value of 0.86 to about 0.844 at the same time.

Table 7.

The changes in nonlinear unsteady behaviour and major flight variables in the flight environment

Figure 6.

Aerodynamic scenario of Aircraft B encountering atmospheric turbulence.

It can be expected that the dynamic aerodynamic effect is very large in the case of instantaneous changes of α, h, and M in transonic flight. Since the angle-of-attack α of aircraft A and B in transonic flight reaches about 6.0 degrees, the effect of compression effect is very large. The maximum ${a_z}$ value of flight A is larger than that of flight B, and the change of altitude and angle-of-attack of flight A is also larger than that of flight B.

3.3.2 Aircraft responses in flight operations

(1) Aircraft A

Figure 7 presents the variations of dynamic characteristics and control variables for Aircraft A The changes of α and sideslip angle (β) including turbulence effects are shown in Fig. 7(a). The variation ranges of α are 6.5 deg. to −6 deg. in turbulence encounter; the time history of β values are about ±2 degrees during the period t = 3,930–3,950s, as indicated in Fig. 7(a). The time history of θ and $\phi$ is shown in Fig. 7(b). The θ does not vary as much as α, but the highest value reaches 5.1 deg. The variations of $\phi$ is large with the variations −18–10 deg. during the turbulence encounter. The magnitudes of q and p are shown in Fig. 7(c); the variation of roll rate p is large from - 9 to 20 deg/s; the variation of p is much larger than that of q.

Figure 7.

Aircraft responses in flight operations for Aircraft A encountering atmospheric turbulence.

(2) Aircraft B

Figure 8 shows the dynamic characteristics of Aircraft B and changes in the control variables. The variation of angle-of-attack α and sideslip β is shown in Fig. 8(a). The variation of α is about 6 deg to −6 deg, and the variation of β is small about 0 deg. In Fig. 8(b), the variations in θ and φ amplitude appear to be the same but differ in magnitude, especially with large variations in φ amplitude with −12 to 12 deg; θ changes only slightly, reaching only about 7.5 deg. The magnitude of pitch rate q and roll rate p are shown in Fig. 8(c). The variation of roll rate p is large from - 5 to 10 deg/s. The change of p is greater than that of q. The yaw rate r is not shown in the figure because the value is much smaller than p and q.

Figure 8.

Aircraft responses in flight operations for Aircraft B encountering atmospheric turbulence.

3.4 Eigenvalue analysis of nonlinear unsteady behaviour

In this article, the longitudinal and lateral-directional motion modes are analysed through the digital flight simulation based on the decoupled dynamic equations of motion. The eigenvalue equation is expressed and solved in polynomial form. The longitudinal motion has short-period and phugoid (long-period) modes; the lateral-directional motion has Dutch-roll, spiral and roll modes. The analysis of both longitudinal and lateral-directional motion modes is based on damping ratio and undamped natural frequency [Reference Chang, Ye, Lan and Guan12]. The roots of the complex conjugate are as follows:

(18) \begin{align}{\lambda _{r,i}} = - \zeta {\omega _n} \pm i{\omega _n}\sqrt {1 - {\zeta ^2}} \end{align}

where $- \zeta {\omega _n}$ is real part (i.e. in-phase) and $\pm i{\omega _n}\sqrt {1 - {\zeta ^2}}$ are imaginary (i.e. out-of-phase) parts. λ _r and λ _i represent eigenvalues of real and imaginary parts, respectively. If λ _r is positive, the positive real part of the eigenvalues is to indicate unstable motion of the related modes; if it is negative, the motion is stable.

3.4.1 Eigenvalue analysis of motion modes for Aircraft A

(1) Longitudinal motion modes for Aircraft A

Figure 9 is the longitudinal eigenvalue of Aircraft A. Figure 9(a) and (b) are eigenvalue of short-period and phugoid (long-period) modes, respectively. In Fig. 9(a), the real part eigenvalue is positive, which is expressed as dynamic unstable. Figure 9(b) is the eigenvalues of the real part and imaginary part of the long period mode in the longitudinal motion. The real part eigenvalues in Fig. 9(b) are negative in most of the time, except in the period of 3,930.4–3,930.8 and 3,934.9–3,935.5s, which are positive. Their size change type is similar to the continuous mountain vein with peak in the negative direction. The imaginary part eigenvalues in Fig. 9(b) are mostly 0 from beginning to end.

Figure 9.

Eigenvalue analysis of longitudinal motion modes for Aircraft A.

(2) Lateral-directional motion modes for Aircraft A

Figure 10 is the lateral-directional eigenvalue for Aircraft A. Figure 10(a), (b), and (c) are eigenvalues of Dutch-roll, spiral and roll modes, respectively. It can be seen from Fig. 10(a) that the first half of the time shows a relatively dynamic unstable situation. Figure 10(b) is the eigenvalue of the spiral mode. In the figure, the real eigenvalue is positive in most of the time, and only a few small parts are negative. Figure 10(c) is the eigenvalues of the roll mode, and the eigenvalues of the real part in Fig. 10(c) are negative.

Figure 10.

Eigenvalue analysis of lateral-directional motion modes for Aircraft A.

3.5 Analysis of dynamic stability characteristics

The comparative analyses of the nonlinear unsteady behaviour for those two aircraft are similar through reliability assessment. Aircraft A has higher dropped-off altitude during the ups and downs motion between the two, so it is chosen to construct the movement mechanism study.

The main longitudinal and lateral-directional oscillatory derivatives of Aircraft A are along the flight path to associate with the derivatives of time rate of angle-of-attack ( $\dot \alpha$ ) and sideslip angle ( $\dot \beta$ ). The main longitudinal and lateral-directional oscillatory derivatives for Aircraft A are presented in Fig. 13. Note the oscillatory derivatives in Fig. 13 are defined as:

(19) \begin{align}{\left( {{C_{mq}}} \right)_{osc}} = {\rm{ }}{C_{mq}} + {C_{m\dot \alpha }}\end{align}

(20) \begin{align}{\left( {{C_{zq}}} \right)_{osc}} = {\rm{ }}{C_{zq}} + {C_{z\dot \alpha }}\end{align}

Figure 11.

Eigenvalue analysis of longitudinal motion modes for Aircraft B.

Figure 13(a) presents the variations of C _mq , (C _mq ) _osc , and C _mq with respect to time. Figure 13(b) presents the variations of ${C_{m\dot \alpha }}$ (i.e. C _mαd ) and ${C_{z\dot \alpha }}$ (i.e. C _zαd ) with respect to time. The damping oscillatory derivatives in Fig. 13(c) are defined as:

(21) \begin{align}{\left( {{C_{lp}}} \right)_{osc}} = {\rm{ }}{C_{lp}} + {C_{l\dot \beta }}\end{align}

(22) \begin{align}{\left( {{C_{nr}}} \right)_{osc}} = {\rm{ }}{C_{nr}} - {C_{n\dot \beta }}\;cosa\end{align}

Figure 12.

Eigenvalue analysis of lateral-directional motion modes for Aircraft B.

Figure 13.

The main longitudinal and lateral-directional oscillatory derivatives for Aircraft A.

Equations (19)–(22) show that the oscillatory derivatives are the combination of static damping and dynamic derivatives. The meaning of symbols describes as follows:

(1) For longitudinal aerodynamics

(C _mq ) _osc : Oscillatory derivative of C _m with respect to q; (C _zq ) _osc : Oscillatory derivative of C _z with respect to q; C _mq : Damping derivative of C _m with respect to q; C _zq : Damping derivative of C _z with respect to q; ${C_{m\dot \alpha }}$ (i.e. C _mαd ): Dynamic stability derivative of C _m with respect to ${\dot \alpha}$ and ${C_{z\dot \alpha }}$ (i.e. C _zαd ): Dynamic stability derivative of C _z with respect to ${\dot \alpha}$ .

(2) For lateral-directional aerodynamics

(C _lp ) _osc : Oscillatory derivative of C _l with respect to p; (C _nr ) _osc : Oscillatory derivative of C _n with respect to r; C _lp : Damping derivative of C _l with respect to p; C _nr : Damping derivative of C _n with respect to r; ${C_{l\dot \beta }}$ (i.e. C _lbd ): Dynamic stability derivative of ${C_l}$ with respect to $\dot \beta$ and ${C_{n\dot \beta }}$ (i.e. C _nbd ): Dynamic stability derivative of ${C_n}$ with respect to $\dot \beta$ .

The effectiveness criteria for derivative values are shown in Table 1. When analysing the stability characteristics of an aircraft encountering serious atmosphere turbulence, it is more realistic to use damped oscillation derivatives instead of static damping in the analysis of dynamic stability characteristics. Regarding to nonlinear unsteady behaviour, if the behaviour is unstable, the motion will be divergent in oscillatory motions; to be stable, (C _zq ) _osc > 0, (C _mq ) _osc < 0, (C _lp ) _osc < 0, and (C _nr ) _osc < 0 [Reference Chang, Ye, Lan and Guan17].

The main longitudinal and lateral-directional oscillatory derivatives for Aircraft A are presented in Fig. 13. Figure 13(a) is oscillatory derivatives of C _m and C _z with respect to q. Figure 13(b) is variations of ${C_{z\dot \alpha }}\;$ and $\;{C_{m\dot \alpha }}$ with respect to time. Figure 13(c) is main lateral-directional oscillatory derivatives. Figure 13(d) is $\;{C_{l\dot \beta }}$ and ${C_{n\dot \beta }}$ with respect to time. The values in the period of ups and downs motion have some differences between oscillatory and static damping derivatives in Figs of 13(a) and (c) for Aircraft A due to the effects of the dynamic derivatives (i.e. $\dot \alpha \;$ and $\;\dot \beta$ -derivatives). The magnitudes in the period of ups and downs motion have some differences between oscillatory and static damping derivatives in Figure of 13(a) and (c) for Aircraft A due to the effects of the dynamic derivatives.

Figures of 13(b) and (d) show dynamic derivatives of stability for Aircraft A. To be stable, ${C_{z\dot \alpha }}$ > 0, ${C_{m\dot \alpha }}$ < 0, ${C_{l\dot \beta }}$ < 0, and ${C_{n\dot \beta }}$ > 0 referred to Table 1. In Fig. 9(a), the eigenvalues of the real part are positive at the beginning, indicating instability because there is not enough damped oscillation in pitching moment (C _mq ) _osc > 0, as shown in Fig. 13(a). The eigenvalues of the real and imaginary parts of the long-period mode in the longitudinal direction, the eigenvalues of the real part in Fig. 13(b) are negative most of the time, except for a small part of 3,930.4–3,930.8s and 3,934.9–3,935.5s which are positive value, because there is sufficient damped oscillation in pitching moment, as shown in Fig. 13(a).

In Fig. 13(b), the values of $\;{C_{z\dot \alpha }}$ and $\;{C_{m\dot \alpha }}$ have significant variations in the period of t = 3,910–3,990s, as shown in Fig. 13(b). The most parts of ${C_{z\dot \alpha }}\;$ are in a nominal negative value. The value of $\;{C_{z\dot \alpha }}$ represents the virtual mass effect(Reference Sheu and Lan18) and is particularly large in transonic flow to affect the ups and downs motion.

The magnitudes of ${C_{z\dot \alpha }}$ is in a nominal negative and $\;{C_{m\dot \alpha }}$ is in a positive in the period of t = 3,928.5–3,929.5s; then, ${C_{z\dot \alpha }}$ approaches to zero and $\;{C_{m\dot \alpha }}$ becomes negative value in the period of t = 3,930–3,931.5s, as shown in Fig. 9(b). The effect of $\dot \alpha$ -derivative on (C _mq ) _osc is to improve the longitudinal motion modes in pitching moment after t = 3,929.5s in Fig. 13(a) through the magnitude comparisons of (C _mq ) _osc with C _mq .

From Fig. 10(a) of Dutch-roll mode, it can be seen that the relatively unstable situation is due to the damped oscillation derivative in yawing moment is not enough (C _nr ) _osc > 0 as shown in Fig. 10(c). The spiral mode in Fig. 10(b) is dynamic unstable in most of the time due to insufficient damped oscillation in yawing moment, as shown in Fig. 10(c). The roll mode is stable because the value of (C _lp )_osc is mostly negative and thus has sufficient damped oscillation in rolling moment, as shown in Fig. 10(c). The magnitudes of ${C_{l\dot \beta }}$ are positive and some parts of ${C_{n\dot \beta }}$ are negative with unstable conditions in the period of t = 3,923–3,955s. The effects of $\;\dot \beta$ -derivative on (C _lp ) _osc and (C _nr ) _osc are to cause the directional stability more unstable in Fig. 10(c) through the magnitude comparisons of (C _lp ) _osc and (C _nr ) _osc with C _lp and C _nr . It implies that the effects of $\dot \beta$ -derivatives of ${C_{l\dot \beta }}$ and ${C_{n\dot \beta }}$ in Fig. 10(d) are to cause the spiral mode more unstable.

Based on the analysis of dynamic stability characteristics for Aircraft A, the tentative conclusion is as follow:

The effect of $\dot \alpha$ -derivative on (C _mq ) _osc is to improve the stability in pitch; the effects of $\;\dot \beta$ -derivative on (C _lp ) _osc and (C _nr ) _osc are to cause the lateral-directional stability more unstable. It is especially important for the movement mechanism of the spiral mode.

Regarding to nonlinear unsteady behaviour, if the behaviour is unstable, the motion will be divergent in oscillatory motions; to be stable, (C _zq ) _osc > 0, (C _mq ) _osc < 0, (C _lp ) _osc < 0, and (C _nr ) _osc < 0 [Reference Chang, Ye, Lan and Guan17].

3.6 Influence of crosswind on loss of control

Wind direction in the QAR or FDR is based on cardinal direction of navigation. It refers to the horizontal angle between the clockwise direction and the target direction from the true north direction of a certain point. The angle between heading and track is known as the drift angle [19]. The crosswind component is computed by multiplying the wind speed by the sine of drift angle. For example, a 10-knot wind coming at 45 degrees of drift angle will have a crosswind component of 10 knots × sin (45°) or about 7.07 knots. If the magnitude of drift angles is large and with variations, the pilots should be educated that the varying crosswind encounter will occur [Reference Jiang, Chang, Yang and Ding3].

The flight data of Aircraft A is extracted from FDR in time spans t = 3,910 $\sim$ 3,990s. Figure 14 is rolling motion in crosswind and the corresponding to rolling control for Aircraft A. One takes the time spans t = 3,927–3,950sec in Fig. 14 for the purpose to study the crosswind before the ups and downs motion. Figure 14(a) is both horizontal wind shear (tailwind) and crosswind. The rolling motion in crosswind and the corresponding to rolling control for Aircraft A are presented in Figures of 14(b) and (c). The roll control power derivative is presented in Fig. 14(d).

Figure 14.

Rolling motion in crosswind and the corresponding to rolling control for Aircraft A.

Aircraft A is subjected to crosswind with considerable magnitude before the ups and downs motion; the magnitude of drift angles is in the range of +10.0 ∼ +7.4, which is larger than usual. It is well-known that varying crosswind, would induce rolling motion [Reference Jiang, Chang, Yang and Ding3]. The values of roll angles ( $\phi $ ) are changed from positive to negative (t = 3,932–3,942s.) and $\phi$ reaches −19 deg. at t = 3,939s in Fig. 14(b), when the crosswind is abruptly increased. In Fig. 14(c), $\phi$ changes rapidly in the negative value before t = 3,932s; after the aileron angle ( $\delta$ _a ) becomes more and more positive, reaching 12 degrees at t = 3,939s. Look at it from a different side, the opposite change occurs around t = 3,935s. The results show that the corresponding aileron input value does not control it effectively. The roll control power derivative should be positive in the validity criteria. In up-and-down motion, some values of C _lδa are negative, as shown in Fig. 14(d).

The large variations of roll angles ( $\phi$ ) obviously are induced by the influenced of crosswind, the corresponding aileron angle will not be effective to control the roll motion. It implies the large variations are due to the effects of-derivatives of ${C_{l\dot \beta }}$ and ${C_{n\dot \beta }}$ . It is the reason why that the spiral mode is dynamic unstable in most of the time. The most loss of control problems happen due to the corresponding control surface angle cannot be effective to control the required flight motion. It is clear that the effects of crosswind cause large changes in the roll angle ( $\phi$ ). The corresponding aileron angle will not effectively control the roll motion. Most loss of control problems occur because the corresponding control surface angles are not effective to control the desired flight motion.

3.7 Loss of control prevention in flight motion

The atmosphere turbulence in cruise flight at high altitude is difficult to detect and not easy to predict, but this kind turbulence on the flight route with a certain local area can be specified as a suspected turbulence area. If the drift angle is large in the suspected turbulence area, large crosswind may approach. The varying crosswind will induce rolling motion. The corresponding aileron angle will not be effective to control the roll motion. Most loss of control problems occur because the corresponding control surface angles are not effective for controlling the desired flight motion. Therefore, the rapidly changes of angles of attack (α), flight altitude and Mach number during the ups and downs motion obviously are caused by the aircraft response to the turbulence.

The numerical results and discussions of application to prevent the loss of control are described as follows:

1. (1) A large drift angle indicated by the instrument is equivalent to an indication that the aircraft will suffer a large crosswind. The crosswind before the turbulence encounter will easily induces rolling motion and then initiates the abrupt ups and downs motion during the turbulence encounter.

2. (2) The roll rate will increase the oscillatory rolling motion, if the rolling damping is insufficient. The drop-off altitude will be enlarged by the oscillatory rolling motion during the abrupt ups and downs motion.

3. (3) If the drift angle is large, to remove the autopilot of yaw control first and to stabilise the rudder by the pedal. When passing through the atmosphere turbulence area, the pilots do not need to amend the heading angle urgently.

4. (4) To provide the mitigation concepts and formulate preventive actions, the situation awareness for the operational pilot before and during severe atmospheric turbulence encounter is consolidated in the present article.

In the future research, one can consider to have more issues of other types of aircraft. It is expected to provide a valuable lecture for international training courses for IATA – Loss of Control In-flight (LOC-I) programme after this paper being published.