^{1}

^{2}

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The oscillation susceptibility of the ADMIRE aircraft along the path of longitudinal flight equilibriums is analyzed numerically in the general and in a simplified flight model. More precisely, the longitudinal flight equilibriums, the stability of these equilibriums, and the existence of bifurcations along the path of these equilibriums are researched in both models. Maneuvers and appropriate piloting tasks for the touch-down moment are simulated in both models. The computed results obtained in the models are compared in order to see if the movement concerning the landing phase computed in the simplified model is similar to that computed in the general model. The similarity we find is not a proof of the structural stability of the simplified system, what as far we know never been made, but can increase the confidence that the simplified system correctly describes the real phenomenon.

Frequently, we describe the evolution of real phenomena by systems of ordinary differential equations. These systems express physical laws and geometrical connections, and often they are obtained by neglecting some influences and quantities, which are assumed insignificant with respect to the others. If the obtained simplified system correctly describes the real phenomenon, then it has to be topologically equivalent to the system in which the small influences and quantities (which have been neglected) are also included. Furthermore, the simplified system has to be structurally stable. Therefore, when a simplified model of a real phenomenon is build up, it is desirable to verify the structural stability of the system.

Interest in oscillation susceptibility of an aircraft is generated by crashes of high-performance fighter airplanes, such as the YF-22A and B-2, due to the oscillations that were not predicted during the aircraft development [

Oscillation susceptibility analysis means the evaluation of the oscillation potential of a given aircraft: identify characteristics of the pilot-aircraft interaction that may result in oscillation, demonstrate the potential for oscillation by analysis and simulations using appropriate piloting tasks and test maneuvers, distinguish aircraft configurations that are less susceptible to oscillations from those that have high oscillations potential, and suggest to reduce and/or eliminate oscillation susceptibility [

As an example in [

Our aim in this paper is to analyze numerically the oscillation susceptibility of the ADMIRE aircraft in a longitudinal flight in a quasilinear (simplified) and a nonlinear (general) flight model in landing phase, when the Automatic flight Control System (AFCS) is decoupled. The equations governing such a flight and the conditions which assure the existence of such a flight are presented. The equilibriums flights are analyzed numerically, from the point of view of bifurcations which can appear due to the changes of the elevator deflection. Appropriate piloting tasks and maneuvers for the touch-down moment are established. The behavior of the aircraft is simulated in both models. The computed results obtained in the models are compared in order to see if the simplified model correctly describes the real flight. This is not a proof of the structural stability of the simplified system but can increase the trust that the simplified system correctly describes the real phenomenon.

The system of differential equations [

A flight with constant forward velocity

In a flight with constant forward velocity

Replacing

If in a flight with constant forward velocity

Replacing

If in a flight with constant forward velocity

Equation (

If

System (_{1}, (_{2}, and (_{3} with the above obtained

A longitudinal flight is defined as a flight for which

A longitudinal flight is possible if and only if

This result is obtained from (

The explicit system of differential equations which describes the motion of the aircraft in a longitudinal flight is

This result is obtained from (

In system (

The explicit system of differential equations which describes the motion of the aircraft in a longitudinal flight with constant forward velocity

This system is obtained from (

A longitudinal flight with constant forward velocity is possible if the following equalities hold:

This result is obtained from Proposition

Notice that in (

In system (

The system (

The ADMIRE aircraft is an Aero Data Model In a Research Environment. To describe the flight of this vehicle with constant forward velocity

System (

The proof is given by computation.

The simplified system which governs the longitudinal flight with constant forward velocity

System (

In system (

The values of the parameters used in the simplified system (

Parameter | Value | Units |
---|---|---|

rad/s | ||

rad/s | ||

1.72514652738 | rad/s^{2} | |

rad/s^{2} | ||

— | ||

11.964 | ||

84.5 | m/s | |

9.81 | m/s^{2} | |

rad/s^{2} | ||

— | ||

— | — | — |

According to the simplified nonlinear model the equilibriums in a longitudinal flight with constant forward velocity are the solutions of the nonlinear system of equations:

System (

Equation (_{1} and (_{2} and eliminating

For the numerical values given in Table

The computed

The

The

The equilibrium manifold

The eigenvalues of the linearized system at an equilibrium are the solutions of the equation:

The proof is given by computation.

For

These results were obtained computing the eigenvalues of the linearized system at the equilibriums of

At the equilibriums, which correspond to

The proof is given by computation.

Transfers between two equilibriums which belong to a conex part of

A transfer between two equilibriums, which belong to

A transfer of an equilibrium, which belongs to

The behavior of the ADMIRE aircraft changes when the maneuver

Oscillations when

Oscillations when

Since the nonhyperbolic equilibriums at

The nonhyperbolic equilibrium at

Let us pose

Along the path of longitudinal flight equilibriums only saddle-point bifurcation exists. There is no Hopf bifurcation.

In both cases, that is, decrease of the parameter

Resetting

It is important to remark that the saddle-point bifurcation phenomenon occurring at

In a longitudinal flight with constant forward velocity an equilibrium is a descending flight if

Such an equilibrium is, for instance,

Figure

Transfer from

It is important to underline that the transfer period

If

For instance, when

In the simplified model it can be shown numerically that there exists a range

The general constant forward velocity longitudinal flight model of the ADMIRE aircraft is defined by the system of differential equations:

System (

The simplified model (

According to the general nonlinear model, the equilibriums in a longitudinal flight with constant forward velocity

System (_{1} can be written in the form

In the case of the longitudinal flight with constant forward velocity of the ADMIRE aircraft the equilibrium manifold

The eigenvalues of the linearized system at equilibrium are the solutions of the equation:

The proof is given by computation.

Computing the roots of (

Computing the roots of (

We conclude the following.

An equilibrium is stable in the general model if and only if it is stable in the simplified model. Moreover,

The same transfer maneuver between the same equilibriums which belong to a conex part of

A transfer between two equilibriums, which belong to

The same transfer maneuver from an unstable equilibrium which belongs to

A transfer of an equilibrium, which belongs to

The above simulations show that from the point of view of this type of transfer, the results obtained in general model are similar to those obtained in the simplified model. More about transfer maneuvers can be found in [

The behavior of the ADMIRE aircraft, when the maneuver

Oscillations when

Oscillations when

Moreover, the simulation in general model shows that the loss of stability, due to maneuver, is noncatastrophic (from mathematical point of view), because if

Resetting

The bifurcation phenomenon at

Transfer from

The flare maneuver has to be made also when the aircraft is at 840 [m] altitude, but the transfer period, computed in the general model can be different, as it is shown in Table

The transfer periods of the transitions corresponding to the maneuver

Transfer period [s] | 333 | 170 | 70 | 30 | 20 | 4 |

(i) Numerical computation shows that there exists a range

(ii) An equilibrium is stable in the simplified model if and only if it is stable also in the general model.

(iii) If

(iv) If

(v) In both models there is no Hopf bifurcation.

(vi) The similarity of the computed results increases the confidence that the simplified model correctly describes the real longitudinal flight, but it is not a proof of the structural stability of the simplified model. Such a proof would be welcome.

The authors thank the anonymous referees for their valuable comments, which led to the improvement of the manuscript. St. Balint and A. M. Balint thank the Romanian National Authority for Research for supporting the research under the Grant ID 354 no.7/2007.