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The flight dynamic equations in mathematics for aircraft response to the adverse weathers, such as wind shear, atmospheric turbulence, and in-flight icing, are nonlinear and unsteady. To effectively analyze the performance degradation and variations in stability of commercial aircraft that encountered these weather hazards, the nonlinear and dynamic (i.e., time dependent) aerodynamic models based on flight data would be needed. In the present paper, a numerical modeling method based on a fuzzy-logic algorithm will be presented to estimate the aerodynamic models for a twin-jet transport by using the flight data from the flight data recorder (FDR). The aerodynamic models having the capability to generate continuous stability derivatives, especially for sensitivity study of unknown factors in adverse weather conditions, will be demonstrated in this paper.

Aircrafts in flight are frequently subject to atmospheric disturbances. The hazards due to these disturbances may be in the form of wind shear, turbulence (both clear-air and convective), thunderstorms, in-flight icing, and so forth. The severity of aircraft response to the disturbance is related to the dynamic aerodynamics that results from the instantaneous changes of aircraft flight attitudes. To provide the mitigation concepts and promote the understanding of aerodynamic responses of the commercial transport aircraft in adverse weather conditions, the nonlinear and dynamic (i.e., time-dependent) aerodynamic models based on flight data would be needed.

Regarding the flight data of commercial transport aircraft, the flight data recorder (FDR) is ICAO-regulated devices. The FDR, popularly referred to as a “black box”, is a mandatory device used to record specific aircraft parameters for event investigations. Although the data resolution and accuracy in the FDR are not as well defined as those in flight test, the predicted aerodynamic derivatives correlated well with the nominal values of transport aircraft in normal flight [

In setting up the aerodynamic models to predict the required derivatives in aerodynamics, the traditional method of using the tabulated data derived from such methods as the maximum likelihood method (MMLE) [

Among the different adverse weathers, clear-air turbulence is all the more important in flight safety since it is hard to detect and predict. Clear-air turbulence is the leading cause of serious personal injuries in nonfatal accidents of commercial aircraft. One main type of motion that causes flight injuries in clear-air turbulence is the sudden plunging motion with the abrupt change in altitude. This paper presents FLM technique to establish unsteady aerodynamic models with six aerodynamic coefficients based on the datasets from the flight data recorder (FDR) of a twin-jet transport. The aerodynamic derivatives extracted from these aerodynamic models are then used to evaluate the variations in stability. The sensitivity study of unknown factors during the sudden plunging motion in severe clear-air turbulence will be demonstrated in this paper.

The general idea of the FLM technique is to set up the relations between its input and output variables of the whole system. The internal functions, membership functions, and outputs are three basic elements for the FLM approach. Two main tasks are involved in the FLM process. One is the identification of the coefficients of the internal functions, which is called parameter identification. The other one is structure identification to identify the optimal structure of fuzzy cells of the model. Details of the FLM technique are described in the followings [

The fuzzy-logic model uses many internal functions to cover the defined ranges of the influencing parameters. Although the form of internal functions is very simple, it can represent a highly nonlinear relationship between the input and output for the whole system. These internal functions are assumed to be linear functions of input variables [

The recorded data in FDR, such as flight altitude

The values of each fuzzy variable, such as the angle of attack, are divided into several ranges; each of which represents a membership function with

The membership functions partition the input space into many fuzzy subspaces, which are called the fuzzy cells. The total number of fuzzy cells is

In the present application, triangular membership functions are used throughout. Let

In each fuzzy cell, the contribution to the outcome (i.e., the cell output) is based on the internal function (

Given a set of membership functions for each input variable, the unknown coefficients of the internal functions are adjusted by using the Newton gradient-descent method. The accuracy of the established aerodynamic model through the fuzzy-logic algorithm is estimated by the sum of squared errors (SSEs) and the multiple correlation coefficients (

In (

In usual, a large value of

For

The iteration during the search sequence stops when one of the following three criteria [

In the above criteria,

Given membership functions and the training data, this parameter identification procedure can be applied to establish a fuzzy-logic model. Although the procedure is well understood, to obtain a good model with the appropriate membership functions that are quite challenging, it requires numerical experimentation.

In the fuzzy-logic model, the model structure is indicated by the number of membership functions. For a fuzzy-logic model with multiple variables, the structure is the combination of the numbers and forms of the membership functions assigned to all input variables. Since the sequence defines the one-to-one relationship between the numbers and the forms for each variable, the structure can be uniquely described by the numbers.

The model structure is optimized by maximizing (

Best model structure searching flow.

The step-by-step searching flow is summarized as follows [

Specify the input variables

Assume an initial structure, also called parent structure, as

Begin at the search stage number

Select the top 5 child structures among all calculated values of

Go back to step (

Pick out the maximum value of

In the structure identification, parameter identification to determine the

The twin-jet transport of the present study encountered clear-air turbulence in cruise flight at the altitude around 10,050 m. As a result, several passengers and cabin crews sustained injuries, because of which this event was classified as an accident. The present study was initiated to examine possible concepts of accident prevention in the future. The datasets used for the modeling are extracted from the FDR during turbulence encounter lasting for 92 seconds.

The main aircraft geometric and inertial characteristics are taken to be

^{2} (2798.7 ft^{2}),

^{2} (7,899,900 slugs-ft^{2}), and ^{2} (10,978,000 slugs-ft^{2}),

^{2} (18,648,470 slugs-ft^{2}) and ^{2}.

The required operational parameters in FDR dataset for generating aerodynamic data files are time (

Typically, the longitudinal, lateral, and vertical accelerations (

The force and moment coefficients are obtained from the following flight dynamic equations [

The above equations are used to determine all aerodynamic coefficients based on accelerometer readings (

The reduced frequency is a parameter to indicate the degree of unsteadiness in unsteady aerodynamics and is estimated in this paper by fitting the local trajectory with a harmonic motion. In the static case, the reduced frequency is 0. Large values of the reduced frequency imply the importance of unsteady aerodynamic effect. For longitudinal aerodynamics, the equivalent harmonic motion is the one based on the angle-of-attack variation following the classical unsteady aerodynamic theory of Theodorsen [

For the longitudinal motion, the time history of the angle of attack (

In (

The local equivalent reduced frequency in the longitudinal motion is defined as

The lateral-directional equivalent reduced frequency is defined as

As shown before, the thrust terms appear in the force equations and the pitching moment equations (

For a commercial aircraft, most likely only the axial force and the pitching moment are affected by thrust. This assumption will be made in this paper. Theoretically, clear-air turbulence (i.e., random change in

For this purpose, data from the flight manual for the fuel flow rates (

For GE turbofan engines, the rpm of the low-pressure compressor (

In the present study, the P & W turbofan engines powering one twin-jet transport will be illustrated. The actual thrust in operation is obtained by using the recorded variables in the FDR, in particular the fuel flow rates.

In the present study, the P & W engines powering the twin-jet commercial aircraft will be illustrated.

The following climb equation [

All these equations are still valid in descent with negative climb angles (

Once the thrust model is generated as a function of

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 atmospheric disturbances. For longitudinal aerodynamics, the models are assumed to be of the form [

For the lateral-directional aerodynamics [

In the present study, the accuracy of the established unsteady aerodynamic models with six aerodynamic coefficients by using FLM technique is estimated by the sum of squared errors (SSEs) and the square of multiple correlation coefficients (_{m}

Predicted aerodynamic coefficients in normal force and moments for a twin-jet transport encountering severe clear-air turbulence at cruise altitudes around 10,050 m.

The fuzzy-logic aerodynamic models are capable of generating the continuous derivatives for the static and dynamic stability study of a twin-jet transport in turbulence response. Firstly, how the fuzzy-logic prediction is achieved will be illustrated with one numerical example in the

For the first cell

Assume that in the following flight conditions

These values of variables are converted to

Other variable values are converted in the same way. It follows that the cell internal function becomes

The membership functions for the first cell are exactly equal to

The total output from all cells can be calculated to be 5.9962; while the denominator in (

To examine the stability characteristics, it is imperative to understand the flight environment in detail. The corresponding flight data are presented in Figure

The time history of flight variables for a twin-jet transport in sever clear-air turbulence at the altitude around 10,050 m in transonic flight.

The aerodynamic derivatives extracted from the unsteady aerodynamic models can be calculated with two approaches, first one being the central difference method [

The time period between 3927.5 seconds and 3932.5 seconds is emphasized in evaluating the stability characteristics, because of the plunging motion that affects the flight safety the most. In order to evaluate the variations in characteristics, the units of all aerodynamic derivatives are converted to rad^{-1}. The main longitudinal and lateral-directional stability derivatives along the flight path are presented in Figure

The time history of main longitudinal and lateral-directional of the static stability derivatives along the flight path.

Figure

The time history of main longitudinal and lateral-directional oscillatory derivatives along the flight path.

In Figure

During the plunging motion, the values have some differences between oscillatory and damping derivatives in Figures

The main objective in this paper was to illustrate the nonlinear unsteady aerodynamic models based on the FLM technique having the capability to evaluate the variations in stability of commercial aircraft with adverse weather effects. It was shown that the FLM technique was capable of handling nonlinear and unsteady aerodynamic environment exhibited for a twin-jet transport in severe clear-air turbulence with sudden plunging motion in transonic flight. The predicted results showed that the models could produce relatively accurate aerodynamic coefficients and several derivatives for the assessment of stability characteristics, especially for the sensitive study of unknown factors in adverse weather conditions.

This research project is sponsored by a grant, NSC 98-2221-E-157-005, from National Science Council (NSC). The accomplishment in this project is part of the requirements set by the Aviation Safety Council (ASC), Taiwan.