#### Abstract

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.

#### 1. Introduction

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 [1]. Flying qualities and oscillation prediction are based on linear methods and their quasilinear extensions [2]. These analyses cannot, in general, predict the presence or the absence of oscillations, because of the large variety of nonlinear interactions that have been identified as factors contributing to oscillations. Some of these factors include pilot behavioral transitions, actuator rate limiting [3–5], and changes in aircraft dynamics caused by transitions in operating conditions [6], gain scheduling, and switching [7]. The oscillation susceptibility analysis in a nonlinear model involves the computation of nonlinear phenomena including bifurcations (Hopf or fold bifurcations) that leads sometimes to large changes in the stability of the aircraft.

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 [1].

As an example in [1] the X-15 aircraft oscillation caused by the rate limiting and an F/A-18 aircraft oscillation caused by nonlinear category III triggers are presented. The limit cycle amplitudes are computed for the longitudinal flight equations of motion and large jump in limit cycle amplitude indicating a significant change in the vehicle stability is revealed.

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.

#### 2. The General Nonlinear Model

The system of differential equations [8, 9], which describes the motion around the center of gravity of a rigid aircraft, with respect to an body-axis system, where is the plane of symmetry, is

The state parameters of this system are forward velocity , angle of attack , sideslip angle , roll rate , pitch rate , yaw rate , Euler roll angle , Euler pitch angle and Euler yaw angle . The constants , , and are the moments of inertia about the -, -, and -axis, respectively, is the product of inertia, is the gravitational acceleration, and is the mass of the vehicle. The aero dynamical forces and moments are functions of the state parameters and the control parameters: is the aileron deflection, is the elevator deflection, and is the rudder deflection (the body flap, speed break, , are available as additional controls but, for simplicity, they are set to in the analysis to follow).

*Definition 2.1. *A flight with constant forward velocity is defined as a flight for which (i.e., ).

Proposition 2.2. *In a flight with constant forward velocity the following equalities hold:
*

*Proof. *Replacing by 0 in the system (2.1), then (2.2) is obtained.

Proposition 2.3. *If in a flight with constant forward velocity one has , then the following equalities hold:
*

*Proof. *Replacing and in (2.2), then (2.3) is obtained.

Proposition 2.4. *If in a flight with constant forward velocity one has , then the following equality holds:
*

*Proof. *Equation (2.4) is the solvability (compatibility) condition of system (2.2) when .

Proposition 2.5. *If and equality (2.4) holds, then the system (2.2) can be solved with respect to , , obtaining the explicit system of differential equations, which describes the motion of the aircraft in a flight, with constant forward velocity :
*

*Proof. *System (2.5) is obtained solving system (2.2) with respect to , and replacing in system (2.1), then (2.1)_{1}, (2.1)_{2}, and (2.1)_{3} with the above obtained and .

*Definition 2.6. *A longitudinal flight is defined as a flight for which

Proposition 2.7. *A longitudinal flight is possible if and only if for and .*

*Proof. *This result is obtained from (2.1) taking into account Definition 2.6.

Proposition 2.8. *The explicit system of differential equations which describes the motion of the aircraft in a longitudinal flight is
*

*Proof. *This result is obtained from (2.1) taking into account Definition 2.6.

*Remark 2.9. *In system (2.7) depend only on , and . These dependences are obtained replacing in the general expression of the aerodynamic forces and moments: and .

Proposition 2.10. *The explicit system of differential equations which describes the motion of the aircraft in a longitudinal flight with constant forward velocity is
*

*Proof. *This system is obtained from (2.7) taking into account .

Proposition 2.11. *A longitudinal flight with constant forward velocity is possible if the following equalities hold:
*

*Proof. *This result is obtained from Proposition 2.7 and system (2.7), taking into account the fact that is equal to zero.

*Remark 2.12. *Notice that in (2.8) and (2.10) depend on and . Taking into account (2.10), the system (2.8) can be written as

*Remark 2.13. *In system (2.11), the functions and are considered known; and are parameters.

The system (2.11) describes the motion around the center of gravity of an aircraft in a longitudinal flight with constant forward velocity and defines the general nonlinear model.

#### 3. The Simplified Model of the ADMIRE Aircraft

The ADMIRE aircraft is an Aero Data Model In a Research Environment. To describe the flight of this vehicle with constant forward velocity , the following explicit system of differential equations is employed:

Proposition 3.1. *System (3.1) can be obtained from (2.5) substituting the general aero dynamical forces and moments with those corresponding to the ADMIRE aircraft [10], assuming that and are small and making the following approximations:*()_{1}*; ; ; ; ,*()_{2}

*Proof. *The proof is given by computation.

Proposition 3.2. *The simplified system which governs the longitudinal flight with constant forward velocity of the ADMIRE aircraft is
*

*Proof. *System (3.3) is obtained from the system (3.1) for and and defines the simplified nonlinear model of the motion around the center of gravity of the ADMIRE aircraft in a longitudinal flight with constant forward velocity .

In system (3.3) are considered constants (see Table 1).

*Remark 3.3. *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 (3.4) defines the equilibriums manifold of the longitudinal flight with constant forward velocity .

Proposition 3.4. *System (3.4) implies that satisfies
**
where , , , are given by
*

*Proof. *Equation (3.5) is obtained replacing in (3.4)_{1} and (3.4)_{2} and eliminating between the so-obtained equations.

For the numerical values given in Table 1, (3.5) has real solutions if and only if where and .

The computed , , , and solutions are represented on Figures 1 and 2.

The equilibrium manifold is the union of the following two pieces:

Proposition 3.5. *The eigenvalues of the linearized system at an equilibrium are the solutions of the equation:
**
where .*

*Proof. *The proof is given by computation.

Proposition 3.6. *For the equilibriums of are exponentially stable and those of are unstable.*

*Proof. *These results were obtained computing the eigenvalues of the linearized system at the equilibriums of and at the equilibriums of . More precisely, it was obtained that the eigenvalues are negative real numbers at the equilibriums of and two of the eigenvalues are negative and the third is positive at the equilibriums of .

Proposition 3.7. *At the equilibriums, which correspond to and , two of the eigenvalues are negative and one eigenvalue is equal to zero. Consequently, and are nonhyperbolic equilibrium points (turning points).*

*Proof. *The proof is given by computation.

*Remark 3.8. *Transfers between two equilibriums which belong to a conex part of are possible by small changes of the elevator deflection (Figure 3). On the other hand, a small change of the elevator deflection transfers an equilibrium which belongs to into an equilibrium which belongs to (Figure 4).

**(a)**

**(b)**

**(c)**

**(a)**

**(b)**

**(c)**

*Remark 3.9. *The behavior of the ADMIRE aircraft changes when the maneuver is so that and . Computation shows that after such a maneuver and oscillate with the same period and tends to or (Figures 5 and 6).

**(a)**

**(b)**

**(c)**

**(a)**

**(b)**

**(c)**

Since the nonhyperbolic equilibriums at and seem to be fundamental for the above behavior, we are going to prove that is a saddle point bifurcation for the system (3.3). An analogous proof holds for .

Proposition 3.10. *The nonhyperbolic equilibrium at is a saddle point bifurcation.*

*Proof. *Let us pose and write the system (3.3) as . Moreover, set the corresponding equilibrium point. As it has been already checked that is a nonhyperbolic equilibrium point, it is sufficient to verify that and , where and are right and left eigenvalues corresponding to the zero eigenvalues, respectively, and the derivatives are computed at , [11, page 148]. From (3.3) it follows that and are right and left eigenvectors, respectively, corresponding to the zero eigenvalue and
For the considered numerical data, given in Table 1, we have and . Therefore, is a saddle-node bifurcation.

*Remark 3.11. *Along the path of longitudinal flight equilibriums only saddle-point bifurcation exists. There is no Hopf bifurcation.

*Remark 3.12. *In both cases, that is, decrease of the parameter under or increase of the parameter over , we have a loss of steady state and occurrence of an oscillatory flight. Computation shows that this loss is not catastrophic, because if is reset, then a stable equilibrium is recovered, as it is illustrated in Figure 7.

**(a)**

**(b)**

**(c)**

*Remark 3.13. *It is important to remark that the saddle-point bifurcation phenomenon occurring at is of practical interest. Using this phenomenon it is possible to transfer the vehicle from a stable descending longitudinal flight equilibrium with constant forward velocity ( and ) into a state which is appropriate for the touch-down moment.

*Definition 3.14. *In a longitudinal flight with constant forward velocity an equilibrium is a descending flight if .

Such an equilibrium is, for instance, ; ; and corresponds to . Moreover, it is stable descent flight equilibrium. By the maneuver with the above state is transferred into a state which is appropriate for the touch-down moment and ( small) (Figure 8).

Figure 8 shows that the maneuver made 20.16 [s] before the touch-down moment, transfers the aircraft in 20.16 seconds from the stable descending flight equilibrium ; ; into the state , which is appropriate for the touch-down moment. This flare maneuver has to be made when the aircraft is at [m] altitude.

**(a)**

**(b)**

**(c)**

**(d)**

*Remark 3.15. *It is important to underline that the transfer period depends strongly on the initial value and on the final value of the elevator deflection.

If is fixed and decreases, then decreases and increases.

For instance, when , we have
Moreover, if the time interval between the moment of the elevator deflection change (flare) and the real touch-down moment is larger than the above presented periods of transitions, then the touch-down moment could be catastrophic, due to the oscillations.

*Conclusion*

In the simplified model it can be shown numerically that there exists a range of the values of the elevator deflection such that to a value from a set of equilibriums corresponds. Some of these equilibriums are stable and some of them are unstable. If a change of the elevator deflection transfers the vehicle into a stable equilibrium corresponding to . If and or with , then the change of the elevator deflection leads to an oscillatory movement of the vehicle , which is noncatastrophic from mathematical point of view. This is due to a saddle node bifurcation and can be useful for preparing the touch-down. Hopf bifurcations are not present.

#### 4. General Constant forward Velocity Longitudinal Flight Model of the ADMIRE Aircraft

Proposition 4.1. *The general constant forward velocity longitudinal flight model of the ADMIRE aircraft is defined by the system of differential equations:
*

*Proof. *System (4.1) is obtained from the system (2.11), substituting the general aero dynamical forces and moments with those corresponding to the ADMIRE aircraft.

*Remark 4.2. *The simplified model (3.3) is obtained from (4.1), approximating with 1 (small angle of attack ). If the simplified system (3.3) describes correctly the real phenomenon, then it has to be topologically equivalent to the system (4.1) [12]. Furthermore, the simplified system (3.3) has to be structurally stable. As far we know, the structural stability of the system (3.3) never was proved. What we intend to prove numerically in this paragraph is that the steady states (stable and unstable) of system (4.1) are exactly the same as those of system (3.3). Moreover, we show that the behavior of the aircraft described by (4.1) is similar to that described by (3.3). This is not a proof of the structural stability of the simplified system, but it is an increase of the believe that the simplified system (3.3) describes correctly the real phenomenon and we are not in the case reported in [13]. In [13] it was shown that the simplified system of differential equations which governs the motion of the automatic—landing flight—experiment (ALFLEX) reentry vehicle, is neither structurally stable, nor topologically equivalent to the general system governing the same motion. In other words, the general and the simplified mathematical models of ALFLEX give different images of the same reality.

*Remark 4.3. *According to the general nonlinear model, the equilibriums in a longitudinal flight with constant forward velocity are solutions of the nonlinear system of equations:
System (4.2) defines the equilibrium manifold of the longitudinal flight with constant forward velocity in the general model.

System (4.2) implies that , satisfy
Since (4.3)_{1} can be written in the form
the system (4.3) has the same solutions as the system
It follows in this way.

Proposition 4.4. *In the case of the longitudinal flight with constant forward velocity of the ADMIRE aircraft the equilibrium manifold in the general model is the same as in the simplified model.*

Proposition 4.5. *The eigenvalues of the linearized system at equilibrium are the solutions of the equation:
**
where
**
or
**
and , , and are the coordinates of the equilibrium.*

*Proof. *The proof is given by computation.

*Remark 4.6. *Computing the roots of (4.6) at the equilibriums of we find that these are negative real numbers for and for or one of the roots is equal to zero.

Computing the roots of (4.6) at the equilibriums of we find that two of them are negative real numbers and one is strictly positive for and for or one of the roots is equal to zero.

We conclude the following.

Proposition 4.7. *An equilibrium is stable in the general model if and only if it is stable in the simplified model. Moreover, , are saddle-points in both models.*

*Remark 4.8. *The same transfer maneuver between the same equilibriums which belong to a conex part of , simulated already in simplified model, is simulated now in the framework of the general model in Figure 9.

**(a)**

**(b)**

**(c)**

The same transfer maneuver from an unstable equilibrium which belongs to into a stable equilibrium which belongs to , simulated already in simplified model, is simulated now in the framework of the general model in Figure 10.

**(a)**

**(b)**

**(c)**

*Remark 4.9. *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 [14].

*Remark 4.10. *The behavior of the ADMIRE aircraft, when the maneuver is so that and , simulated in general model, is represented in Figures 11 and 12 and shows that this is similar to that obtained in the simplified model.

**(a)**

**(b)**

**(c)**

**(a)**

**(b)**

**(c)**

*Remark 4.11. *Moreover, the simulation in general model shows that the loss of stability, due to maneuver, is noncatastrophic (from mathematical point of view), because if is reset, then a stable equilibrium is recovered; see Figure 13.

**(a)**

**(b)**

**(c)**

*Remark 4.12. *The bifurcation phenomenon at , present also in the general model, can be used for the transfer of the aircraft from a stable descending longitudinal flight equilibrium into a state which is appropriate for the touch-down moment. The simulation of such a transfer is presented in Figure 14.

**(a)**

**(b)**

**(c)**

**(d)**

*Remark 4.13. *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 2.

#### 5. Conclusions

(i) Numerical computation shows that there exists a range of the elevator deflection values , such that to a value from the same set of equilibriums corresponds in both models.

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

(iii) If , a change of the elevator deflection transfers the vehicle into a stable equilibrium corresponding to in both models.

(iv) If and or with , then the change of the elevator deflection leads to a noncatastrophic oscillatory movement of the vehicle, in both models. This is due to the saddle node bifurcation in both models and can be used for preparing the touch-down in both models.

(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.

#### Acknowledgments

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.