М.Г.Гоман, А.В.Храмцовский «Методика численного исследования нелинейной динамики самолёта», Advances in Engineering Software 39 (2008), стр..167-177 M.G.Goman, A.V.Khramtsovsky "Computational framework for investigation of aircraft nonlinear dynamics", Advances in Engineering Software 39 (2008) pp.167-177 A computational framework based on qualitative theory, parameter continuation and bifurcation analysis is outlined and illustrated by a number of examples for inertia coupled roll maneuvers. The focus is on the accumulation of computed results in a special database and its incorporation into the investigation process. Ways to automate the investigation of aircraft nonlinear dynamics are considered. Описана и проиллюстрирована на ряде примеров (связанных с инерционным вращением) методика численного анализа, основанная на теории качественного анализа, методе продолжения решений, зависящих от параметра и на бифуркационном анализе. Особое внимание уделяется накоплению результатов расчетов в специальной базе данных и интеграции этой базы в процесс исследований. Рассмотрены способы автоматизации исследований нелинейной динамики самолёта.

1. Computational framework for investigation of aircraft
nonlinear dynamics
M.G. Goman *, A.V. Khramtsovsky
De Montfort University, Leicester, England LE1 9BH, United Kingdom
Received 15 October 2006; received in revised form 8 February 2007; accepted 12 February 2007
Available online 24 May 2007
Abstract
A computational framework based on qualitative theory, parameter continuation and bifurcation analysis is outlined and illustrated
by a number of examples for inertia coupled roll maneuvers. The focus is on the accumulation of computed results in a special database
and its incorporation into the investigation process. Ways to automate the investigation of aircraft nonlinear dynamics are considered.
Ó 2007 Published by Elsevier Ltd.
Keywords: Nonlinear aircraft dynamics; Qualitative theory; Continuation method; Bifurcation analysis; Object oriented model; Database; Roll coupling
problem
1. Introduction
The increasing requirements for aircraft performance
and maneuverability and the demand for more reliable pro-
vision of flight safety have led to the implementation of
more complex and adequate nonlinear models for modern
aircraft. These models in their turn require an efficient set
of computational tools for aircraft dynamics simulation
and nonlinear analysis, for clearance of flight control laws
with account of inherent model nonlinearities and control
constraints [1].
An application of qualitative theory and bifurcation
analysis methods for investigation of nonlinear aircraft
dynamics is now a well established approach. Various
kinds of aircraft dynamic instabilities and loss of control
have been effectively investigated using continuation tech-
nique and bifurcation methods. The critical flight regimes
such as high incidence departures, wing rock behavior, spin
and jump-like instabilities during intensive roll-coupled
maneuvers have been effectively investigated within a uni-
fied framework.
The advanced computational methods in flight dynam-
ics were first introduced in [2,3]. Several further contribu-
tions presented in [4 12] proved the efficiency of the
proposed methods in a number of applications for an
open-loop and closed-loop aircraft dynamics. Global sta-
bility and bifurcation analysis methods, continuation and
eigenstructure assignment techniques have been success-
fully applied to several control law design problems in
[13,14]. Bifurcation diagrams for high incidence flight
turned out to be a useful tool for planning of piloted sim-
ulations [15,16]. A review of these research results can be
found in [17,18].
Recently the bifurcation analysis method has been
extended for evaluation of aircraft trim at straight-and-
level flight and at level turn by adding to the problem a
set of kinematical constraint equations [19,20]. This exten-
sion generalizes aircraft performance evaluation using sta-
bility and controllability analysis typical for one-parameter
continuation techniques. A review of the latest results in
the application of bifurcation and continuation methods
to flight dynamics problems is given in [21].
Although considerable progress in this area has been
achieved during the last two decades, there still remains a
significant gap between the academic research and the
acceptance of the new methods in aeronautical engineering
0965 9978/$ see front matter Ó 2007 Published by Elsevier Ltd.
doi:10.1016/j.advengsoft.2007.02.004
*
Corresponding author.
E mail address: mgoman@dmu.ac.uk (M.G. Goman).
www.elsevier.com/locate/advengsoft
Available online at www.sciencedirect.com
Advances in Engineering Software 39 (2008) 167 177

2. practice. The main reason for that is the lack of efficient
and reliable software tools which can be adopted by users
who are not experts in application of qualitative and bifur-
cation analysis methods [18].
Qualitative analysis of aircraft nonlinear dynamics
incorporates a number of diverse computationally-inten-
sive procedures, which generate different types of data for
further consideration in planning of numerical simulations,
in computation of regions of attractions and in construc-
tion of phase portrait representation. This paper describes
the authors’ experience in developing a computational
framework for the investigation of aircraft nonlinear
dynamics. The framework is based on qualitative theory,
a continuation method and bifurcation analysis techniques.
The set of software tools outlined in this paper is an exten-
sion and further development of the KRIT package [22] in
a MatlabÒ
environment. Some results for this computa-
tional framework have been presented earlier in [23].
There was no intention in this paper to provide a user
guide; it is mostly focused on some methodological aspects
of the developed computational framework. A brief outline
of the aircraft motion equations along with the elements of
qualitative theory adopted in flight dynamics are presented
in Section 2. This section also includes a description of
interactive capabilities for aircraft dynamics investigation
using coherent parameter continuation and phase portrait
construction. The object-oriented model of the autono-
mous nonlinear dynamic system used as a phase portrait
database structure is outlined in Section 3. The continua-
tion database features and outline of some new classes of
algorithms are presented in Section 4. And finally a few
computational examples for the inertia roll-coupling prob-
lem are given in Section 5.
2. Goals of the qualitative investigation
The motion equations suitable for bifurcation and qual-
itative methods are represented in the form of an autono-
mous nonlinear system of first order differential equations
(the explicit form is given in the Appendix):
dx
dt
¼ Fðx; dÞ; x 2 X & Rn
; d 2 U & Rm
ð1Þ
where the state vector includes velocity vector parameters,
rigid body angular rates, the pitch and bank angles
x (V,a,b,p,q,r,h,/)T
, the control vector includes
elevator, aileron, rudder and thrust throttle deflections
d (de,da,dr,dT)T
. This system of equations is normally
used for aircraft performance evaluation and investigation
of spin dynamics. For maneuverability evaluation when
aircraft perform an intensive velocity vector roll maneuver
a simplified system (1) with the reduced-order state vector
x (a,b,p,q,r)T
and control vector d (de,da,dr)T
can be
considered [31,33,34].
All equilibrium states of system (1) are defined by solu-
tions of the following system of nonlinear algebraic
equations:
Fðx; dÞ ¼ 0; x 2 X & Rn
; d 2 U & Rm
ð2Þ
The eigenvalues and eigenvectors of the Jacoby matrix
J ¼ oF
ox
, calculated at equilibrium point under consideration,
are used for evaluation of local stability and reconstruction
of invariant manifolds of trajectories associated with
equilibrium.
Any equilibrium solution of system (1) with constant-
control input d corresponds to steady coordinated flight
along a helical trajectory with a vertical axis. Note that
straight-and-level flight and the level turn belong to this
class of equilibria. An aircraft in steady spin also follows
a helical trajectory with small, comparable with the wing-
span (R/l % 1), turn radius, with high angle of attack
a % 40 80° and practically vertical flight path angle
c % Àp/2. The reduced 5th-order system is used for investi-
gation of the roll-coupling problem. The equilibria of this
simplified system of motion equations are considered as
the pseudosteady state (PSS) solutions in comparison with
the full system equilibria, because the effect of the gravita-
tional force is ignored and speed is assumed to be constant
[31,34].
Different time-varying control inputs (1) allow one to
simulate a variety of aircraft maneuvers (note that in this
case system (1) should be extended by equations for posi-
tion in space X, Y, Z and yaw angle w). For illustration
purposes Fig. 1 shows the visualization of a simulated flight
trajectory with aircraft departure at high incidence, entry
into developed spin and further successful recovery from
spin. Fig. 2 shows the visualization of a flight trajectory
which includes intensive velocity roll.
Using only direct numerical simulation the identification
of maneuver limits due to onset of instability or loss of con-
trol is an extremely difficult and time-expensive task. The
qualitative theory, continuation and bifurcation analysis
Fig. 1. High incidence departure, steady spin and spin recovery.
168 M.G. Goman, A.V. Khramtsovsky / Advances in Engineering Software 39 (2008) 167 177

3. methods facilitate this task providing estimates for critical
control inputs and dangerous maneuver boundaries. Nor-
mally these estimates are further verified in numerical
and piloted simulation using the full mathematical model.
The basic principals of the qualitative theory and bifur-
cation analysis methods can be found, for example, in [24
26]. An introduction to continuation methods is given in
[27]. In this paper these methods and techniques are dis-
cussed in the context of their application to flight dynamics
problems.
At constant-control input the state space of system (1)
incorporates all the trajectories representing aircraft
dynamics for different initial conditions. The topology of
the system state space, or the phase portrait, is defined by
a set of its special trajectories, namely equilibrium points,
closed orbits, toroidal manifolds and chaotic attractors.
The investigation in flight dynamics is mostly focused on
aircraft equilibria or trim conditions and on closed orbits
representing aircraft limit cycle oscillations (LCO).
A special interest is given to bifurcational parameters
when the phase portrait experiences qualitative transfor-
mations like changes in a number of equilibrium points,
onset or disappearance of a closed orbit. Local bifurcations
can be analyzed entirely via migrations of equilibrium
eigenvalues and periodic orbit multipliers across the imag-
inary axis or the unit circle, respectively.
The most important bifurcations arising in flight
dynamics applications are related to the limit point or sad-
dle-node bifurcation (Fig. 3), signifying aircraft jump-like
departure, and with the Hopf bifurcation or oscillatory
instability, associated, for example, with the onset of wing
rock at high incidence flight or flutter in aeroelastic systems
(Fig. 4).
Aircraft critical regimes such as spin, autorotation in
roll or wing rock oscillations can coexist with the normal
flight conditions at the same control inputs. Entry to one
of these critical regimes depend on prehistory of control
actions and also on external disturbances. This multiattrac-
tor nature of aircraft dynamics requires one to perform
analysis of regions of attraction (or domains of attraction)
in addition to local stability for each stable regime of
motion. For evaluation of regions of attraction in the pre-
sented framework the method outlined in [11,13] is imple-
mented. This method automates the computation and
visualization of two-dimensional cross-sections of the mul-
tidimensional region of attraction.
Two closely interconnected tasks of qualitative analysis
of aircraft nonlinear dynamics consist of:
• Identification of all qualitatively different types of air-
craft nonlinear behavior and specification of separating
Fig. 2. Intensive velocity vector roll maneuver.
Fig. 3. Equilibrium limit point or saddle node bifurcation.
Fig. 4. Hopf bifurcation for equilibrium point, onset of limit cycle.
M.G. Goman, A.V. Khramtsovsky / Advances in Engineering Software 39 (2008) 167 177 169

4. surfaces between corresponding regions in the parame-
ter space. Analysis of their sensitivity to variation of sys-
tem parameters is the important part of such an
investigation. This allows the researcher to determine
the safe boundaries for normal flight conditions and to
isolate critical flight regimes.
• Investigation of aircraft dynamics for each mode of
behavior considering the phase portrait representation
at different constant-control inputs and evaluation of
regions of attraction for every important stable solution.
The continuation procedure is the main computational
tool for performing numerical bifurcation analysis [27
29,22]. It is used to study the dependence of steady-state
solutions on variation of system parameters. The main fea-
ture of this algorithm is the introduction of a general arc-
length for the solution curve in the extended state space,
which is the cartesian product of the state space and one-
parameter axis (i.e. the selected bifurcation parameter).
The predictor-corrector algorithm is implemented for com-
putation of this solution curve.
In continuation of aircraft equilibria the saddle-node
bifurcation points may have a form of sharp kinks, because
functions in aircraft aerodynamic models are normally
made by the linear interpolation of tabulated experimental
data. The continuation algorithm in this case needs special
adjustment to guarantee a reliable passing through these
critical points [22].
For computation of the closed orbits different versions
of mapping technique are used. The presented computa-
tional framework supports the Poincare´ mapping and the
time-advance mappings with fixed and free periods. Vari-
ous checks are performed during continuation of equilib-
rium and closed orbit solutions to allow localization and
classification of the encountered bifurcation points. Special
procedures automatically locate, or at least try to locate, all
the branches emanating from the bifurcation and branch-
ing points, in order to capture the complete set of solutions.
There is a number of interactive user interfaces for man-
aging the investigation process using a broad spectrum of
computational procedures based on qualitative methods.
They include the tools for storage and retrieval of com-
puted data and also special algorithms guiding the investi-
gation by processing already accumulated data.
Figs. 5 and 6 show the continuation and the phase por-
trait GUI’s (Graphical User Interfaces), respectively. They
both are used for managing the process of qualitative and
bifurcation analysis. The continuation GUI includes three
windows where the continuation branches in different pro-
jections with local stability information can be displayed. It
is possible to save the computed data in a database and to
reload previously saved data. There is also a number of
tools and control panels facilitating computational investi-
gation, in particular, for initiation of continuation of equi-
libria and closed orbits, for numerical simulation, for
calculation of two-dimensional cross-sections of stability
region, for opening a new or existing continuation or phase
portrait database, for report generation.
The phase portrait can be composed using the phase
portrait GUI from the steady-state solutions available in
the continuation database. This set of solutions can be sup-
plemented by a number of trajectories from stable and
unstable invariant manifolds associated with the equilib-
rium point or the closed orbit. All this may be added by
a number of regular trajectories specified for computation
in the simulation panel.
Fig. 5. Control panel for continuation and bifurcation analysis.
170 M.G. Goman, A.V. Khramtsovsky / Advances in Engineering Software 39 (2008) 167 177

5. 3. Object-oriented model of the autonomous nonlinear
dynamic system
Qualitatively different trajectories of dynamical system
(1) can be considered as objects grouped in a number of
classes. The object-oriented approach to system modeling
provides a useful basis for development of a database struc-
ture for storing computational results from nonlinear
dynamic analysis. An attempt to model the autonomous
nonlinear dynamical system using the class diagram
adopted from the Unified Modeling Language (UML)
[30] is presented in this section.
The system is modelled using three classes of objects.
The first class includes invariant manifolds of steady trajec-
tories (IMST). It has four children classes, namely, the
class of equilibrium points with dimension n 0, the class
of closed orbits with dimension n 1, the class of toroidal
invariant manifolds with dimension n 2,3,N À 1, where
N is the order of the dynamical system, the class of strange
attractors with fractional dimension p (Fig. 7).
The second class includes invariant manifolds of tran-
sient trajectories (IMTT) approaching or departing from
the invariant manifolds of steady trajectories (IMST).
For example, any invariant manifold of steady trajectories
(i.e. equilibrium, closed orbit, etc.) is associated with a sta-
ble invariant manifold of transient trajectories approaching
it and an unstable invariant manifold of transient trajecto-
ries departing from it. If IMST is stable IMTT has a full
dimension of the state space and coincides with the IMST
region of attraction (RA). Dimensions of a stable IMTT
Fig. 6. Control panel for phase portrait investigation.
Fig. 7. Object oriented model of the autonomous nonlinear dynamical system (1): the UML class diagram.
M.G. Goman, A.V. Khramtsovsky / Advances in Engineering Software 39 (2008) 167 177 171

6. and an unstable IMTT coincide respectively with the num-
ber of stable and unstable eigenvalues for an equilibrium
and the number of stable and unstable multipliers for a
closed orbit. A typical example is a separatrix surface
formed by a stable manifold of trajectories approaching
an aperiodically-unstable equilibrium or a closed orbit.
The third class includes all regular trajectories (RT),
which are different from the first and the second class of
objects. They do not have association with IMST and
IMTT objects, but can approach them with any required
accuracy.
The attributes of the objects from the first class include a
state space location, eigenvalues and eigenvectors of the
Jacoby matrix. The attributes of a stable object from the
first class can include estimates for its region of attraction.
There are two main operations with the objects. The first
operation is the time advance mapping along the trajectory
i.e. numerical integration of the dynamical system on some
time interval, and the second operation is continuation of
the object due to variation of some system parameter.
4. Database for computational results and data processing
algorithms
Keeping and reusing the already computed results can
significantly enhance the continuation algorithm’s perfor-
mance. The novelty of this approach is in the tight integra-
tion between the database and the continuation algorithm.
For example, at each continuation step a newly computed
solution is checked against the database to determine
whether such a solution has been already computed, or
whether some other steady-state solution is approaching
the current branch. Such a feature resulted in significant
time savings, especially in the presence of complex branch-
ing points or during continuation of periodic solutions.
The existing databases can be used as a source of initial
information for a number of algorithms. For example, a
one-parameter continuation database can provide limit
points for computation of two-parameter bifurcation
diagram. Another example, all steady-state solutions at
constant-control input can be extracted from the continua-
tion database and used for phase portrait representation
and if some new attractors are found using the phase por-
trait investigation they can be returned back to the contin-
uation database.
A database structure for the one-parameter continua-
tion algorithm is the most thoroughly tested one. It con-
tains information about various kinds of bifurcation
points and also about error points, where computation
failed to converge to a steady solution, etc. Descriptive
information is given for each computed and stored contin-
uation branch with an outline of relationships between dif-
ferent branches.
The phase portrait investigation requires special compu-
tational activities. Collecting data for phase portrait repre-
sentation has an iterative and incremental nature and the
main goal was to automate this process. A special database
structure was developed for accumulating results of com-
putational analysis (see Section 3) and a graphical user
interface was created for managing the computational
process.
The phase portrait GUI, shown in Fig. 6, unifies a num-
ber of different algorithms while the database is used for
data exchange and storage. To create a particular phase
portrait view, the following data are usually computed:
• steady-state solutions and their stability (the data may
be extracted from an existing continuation database or
alternatively a number of various methods are provided
in the phase portrait GUI for computation of steady-
state solutions);
• special trajectories, incoming to or outgoing from a
steady-state solution along eigenvectors, corresponding
to stable and unstable eigenvalues, respectively;
• numerical simulation of system’s dynamics at fixed and
time-varying control inputs providing time histories and
visualization of aircraft spatial trajectory;
• two-dimensional cross-sections of the regions of attrac-
tion for all stable steady-state solutions.
There is a number of auxiliary computational tools such
as random search for equilibrium and periodic solutions,
procedure for automatic specification of the system’s tran-
sition after meeting the saddle-node bifurcation point, etc.
A procedure for search of points on a surface separating
two stable attractors allows finding additional unstable
equilibrium and periodic solutions.
A low-level compatibility between the phase portrait
and the continuation databases allows a quick generation
of the phase portrait representation from data already
obtained by continuation. Conversely, the data collected
in the phase portrait database can be exported back to
the continuation database allowing continuation of new
branches of solutions.
5. Computational examples
A classical aircraft nonlinear problem for inertia-cou-
pling roll maneuvers outlined in [31 34,21] will be consid-
ered in this section to present the computational
framework in action. The 5th order mathematical model
for a hypothetical swept-wing fighter at high altitude super-
sonic flight conditions presented and analyzed in [17] (pp.
553 559) is used for computational analysis. There is a rep-
lication of some results from [17] and also some new forms
such as phase portrait views, two-parameter continuation
diagrams and root-loci for separate continuation branches
are given and accompanied by comments on reliability and
convergence of the algorithms employed, an indication of
computation time and of data storage requirements.
The aircraft dynamics during intensive rotation in roll is
affected by inertia coupling between the longitudinal and
lateral motions leading to a multitude of nonlinear
phenomena such as bifurcation-induced departures,
172 M.G. Goman, A.V. Khramtsovsky / Advances in Engineering Software 39 (2008) 167 177

7. multiattractor behavior, apparent loss of controllability,
hysteresis-type responses to control inputs, etc. All these
phenomena can be effectively investigated using the pre-
sented computational framework with a variety of tools
for analysis, data storage and interactivity.
The continuation of equilibrium solutions is performed
as a single procedure, which automatically generates sets
of initial solutions for several values of the continuation
parameter and after that sequentially carries out continua-
tion from these starting points. At every selected value of
the continuation parameter there may be a number of start-
ing points, which generate different branches of solutions.
All new branches are stored in a continuation database.
Special protection against recontinuation of the same
branch is made by means of synchronous comparison of
the current point with the data already stored in a data-
base. The final set of equilibrium branches computed in
an extended state space and stored in a database can be
visualized in different projections. To give information
about equilibrium eigenvalues the branches are drawn
using different line types, for example, a solid line means
that all eigenvalues are stable, a dashed line means that
there is one positive real eigenvalue, a dashed-dotted line
corresponds to one complex pair with positive real part,
etc. All bifurcation points (i.e. the limit and the Hopf bifur-
cation ones) are specially marked using information stored
in a database.
The obtained branches of equilibrium solutions in two
projections for angle of attack vs aileron deflection and
for roll rate vs aileron deflection are presented in Fig. 8.
There is one branch of equilibrium solutions corresponding
to normal flight conditions with small angles of attack.
Two other branches correspond to aircraft autorotation
regimes, which even exist at zero aileron deflection. At
two limit points on these branches at da 5.5° and
da À7.5°, respectively for negative and positive rotation,
transitions take place to the normal branch leading to
decrease in intensity of rotation (see arrows in Fig. 8). After
return to the normal branch the roll rate returns to zero
when aileron input is removed.
At the next step the user can initiate the continuation of
limit cycle oscillations (LCO) or closed orbit solutions
starting from the already computed Hopf bifurcation
points. The two computed families of periodical solutions
are shown in Fig. 8 for positive values of aileron deflection.
The amplitudes of limit cycles are superimposed on the
equilibrium branches (note that similar families of period-
ical solutions can be computed for negative aileron deflec-
tions). The limit cycle oscillations on the main equilibrium
branch become unstable after meeting the secondary Hopf
bifurcation point at da 18°. Further increase of aileron
input produces departure to stable limit cycle oscillations
settled on the ‘‘autorotation’’ equilibrium branch with very
high values of angle of attack.
Based on these continuation and bifurcation analysis
results the engineer can draw two main inferences. The first
one is that there is an apparent loss of aileron efficiency at
da > 8° (practically zero increase in angular velocity). This
happens due to increasing counter-rotative aerodynamic
moment produced by sideslip, which is generated by inertia
coupling. The second conclusion is that at 12° < da < 18°
the velocity vector roll maneuver becomes very agitated
due to existence of stable limit cycles and there is also a
critical aileron deflections da > 18°, when aircraft can
depart to an agitated autorotation regime with very high
angles of attack (see arrow in Fig. 8). Transition to high
angles of attack at supersonic speeds is very dangerous as
the normal and side load factors can easily exceed the
structural limit. A more accurate evaluation of the aerody-
namic loads during transition to high angles of attack
requires inclusion into the simplified 5th-order mathemati-
cal model of the nonlinear dependencies of aerodynamic
characteristics and additional dynamic equation for flight
velocity.
For more thorough consideration of multiattractor
dynamics the phase portrait views for different aileron
deflections are constructed using information stored in
Fig. 8. Continuation of equilibria and closed orbits: angle of attack and
roll rate vs aileron deflection.
M.G. Goman, A.V. Khramtsovsky / Advances in Engineering Software 39 (2008) 167 177 173

8. the continuation database. At small aileron deflections the
system has five different equilibria, three stable and two
aperiodically unstable. The phase portrait for da 0 is pre-
sented in Fig. 9 (top plot). In addition to equilibrium points
taken from the continuation database several special trajec-
tories are integrated using the phase portrait GUI and
added to the phase portrait. Four unstable trajectories
associated with two saddle points approach three stable
points. For each equilibrium point a trajectory from the
associated stable invariant manifold is reconstructed by
reversed time integration from its close vicinity. The stable
invariant manifolds for unstable points are most important
as they form the boundaries of regions of attraction for
three stable equilibrium points.
The phase portrait at positive aileron deflection da 14°
is shown in Fig. 9 (bottom plot). There are two unstable
equilibrium points and two stable solutions, a stable limit
cycle on the normal equilibrium branch and a stable equi-
librium regime on the autorotation branch. Unstable tra-
jectories associated with saddle equilibrium point
approach two stable attractors, the stable invariant mani-
fold for saddle equilibrium point separates the regions of
attraction for autorotation regime at high angle of attack
and agitated roll maneuver at small angle of attack.
Along with steady state solutions the continuation data-
base stores eigenvalues and multipliers for equilibrium and
closed orbits, respectively. The trajectories of eigenvalues
displayed in the continuation GUI (Fig. 5) for a specified
branch of solutions can be saved in a separate figure for
report generation. For example, Fig. 10 shows the trajecto-
ries of eigenvalues for two different branches of equilibrium
solutions presented in Fig. 8.
Every locally stable equilibrium and closed orbit shown
in Fig. 8 has a bounded region of attraction (RA). For
evaluation of the multidimensional RA a number of its
two-dimensional cross-sections are generated by direct
integration of dynamical system (1). Initial conditions are
taken from a grid of points on the considered two-dimen-
sional cross-section plane. Final destinations of integrated
trajectories are identified by their entering into a close
proximity of one of the available attractors stored in the
Fig. 9. Phase portrait views for a, b, p projection at da = 0 (top plot) and
for da = 14° (bottom plot).
Fig. 10. Root loci vs aileron deflection for two branches of continuation
diagram in Fig. 8.
174 M.G. Goman, A.V. Khramtsovsky / Advances in Engineering Software 39 (2008) 167 177

9. database [11,17]. For three stable equilibrium points at
da 0 two cross-sections of regions of attraction in the
plane (b,a) at p q r 0 and in the plane (r,p) at
q a b 0 are shown in Fig. 11. Trajectories attracted
by equilibrium with zero roll rate are marked by solid dia-
monds. Trajectories attracted by equilibrium with positive
roll rate are marked by ‘‘·’’ and trajectories attracted by
equilibrium with negative roll rate are marked by ‘‘+’’.
The computed cross-sections, for example, allow one to
specify critical disturbances in the state space leading to
departure from the normal flight conditions with zero rota-
tion. A locally stable equilibrium with very small size of
region of attraction should be considered as practically
unstable form of motion.
The saddle-node bifurcations on the continuation dia-
gram (8) separate on parameter axis da segments with dif-
ferent number of equilibrium points. The location of
saddle-node bifurcation points in the plane of two control
parameters, for example, da and dr, are defined by continu-
ation curves of the extended bifurcation problem [2]:
Fðx; da; dr; deÞ ¼ 0; det
oFðx; da; dr; deÞ
ox
¼ 0 ð3Þ
The procedure for continuation of the saddle-node bifurca-
tion in two-dimensional plane of (da,dr) (3) takes the initial
points from the database for one-parameter continuation
results. Fig. 12 shows the bifurcation diagrams in the plane
of aileron and rudder deflections for two different values of
elevator deflection de 0 (left plot) and de À5° (right
plot). The computed results specify regions with different
number of equilibria and thus allow one to assign the crit-
ical boundaries in the parameter space associated with an
aircraft jump-like departure.
Size of the database containing the results presented in
this section is about 2 Mb. The computation time for con-
tinuation of three equilibrium branches is about 3 min on
PC (Pentium IV, 2.4 GHz), the closed orbits require about
7 min per branch, and the computation time for two-
parameter continuation diagram in Fig. 12 is about
2 min. Only the computation of two-dimensional cross-sec-
tions of region of attraction with a fine grid of points
remains rather time-consuming it takes about 20 min to
Fig. 11. Cross sections of regions of attraction for three equilibrium
points in (b,a) and (r,p) planes.
Fig. 12. Two parameter bifurcation diagrams in the plane of aileron and rudder deflection for a saddle node bifurcation point: regions with different
number of equilibria (left plot for de = 0, right plot for de = 5°).
M.G. Goman, A.V. Khramtsovsky / Advances in Engineering Software 39 (2008) 167 177 175

10. compute one cross-section. The developed computational
tools were tested during investigations of nonlinear dynam-
ics for a number of realistic aircraft models covering the
full flight envelope. Acceptable convergence and robustness
properties of the algorithms were demonstrated, as well as
the possibility to restart computations from a checkpoint in
case of hardware or software failure.
6. Conclusions
The presented computational framework provides a full
suite of algorithms and methods required for investigation
of aircraft nonlinear dynamics by a systematic analysis of
aircraft equilibrium and periodical forms of motion. The
computational tools for one- and two-parameter continua-
tion, bifurcation analysis, phase portrait/time histories,
evaluation of regions of attraction, root loci, systematic
starting solution solver, etc. are linked together in a logical
manner that promotes a structured approach to problems
by engineers who are not necessarily highly specialized in
the field of qualitative methods and continuation software.
The developed databases for accumulation of computed
results, their integration with computational algorithms
and an interactive way of managing the investigation pro-
cess via a number of Graphical User Interfaces provide a
productive and user-friendly environment. The way in
which computed information is accessed and processed is
a significant step forward in making such software user-
friendly and suited to an engineering practice.
Acknowledgements
The presented computational framework for aircraft
nonlinear dynamics investigation was developed during
the research project funded by DERA/QinetiQ Ltd., Bed-
ford, UK. During various years it was monitored by Phill
Smith, Darren Littleboy and Yoge Patel. Special thanks
go to the reviewers and also to Daniel Walker for their
valuable and constructive comments and suggestions,
which helped to improve the presentation. The authors
acknowledge the contribution to the computational system
of Evgeny Kolesnikov.
Appendix. Aircraft equations of motion
The autonomous set of equations for rigid aircraft flight
dynamics [2]:
_V ¼ ð1=mÞ½ððX þ TÞ cos a þ Z sin aÞ þ Y sin bŠ Á Á Á
þ gðsin b cos h sin h À cos a cos b sin h
þ sin a cos b cos h cos /Þ
_a ¼ q À ðp cos a þ r sin aÞ tan b þ ð1=mV Þ
 ½Z cos a À ðX þ TÞ sin aŠ Á Á Á
þ ðg=V cos bÞðsin a sin h þ cos a cos h cos /Þ
_b ¼ p sin a À r cos a þ ð1=mV Þ
 fY cos b À ½ðX þ TÞ cos a þ Z sin aŠ sin bg Á Á Á
þ ðg=V Þðcos a sin b sin h þ cos b cos h sin /
À sin a sin b cos h cos /Þ
_p ¼ ððIYY À IZZÞ=IXX Þqr þ L=IXX
_q ¼ ððIZZ À IXX Þ=IYY Þpr þ M=IYY
_r ¼ ððIXX À IYY Þ=IZZÞpq þ N=IZZ
_h ¼ q cos / À r sin /
_/ ¼ p þ ðq sin / þ r cos /Þ tan h
The aerodynamic coefficients CX,Y,Z
1
and Cl,m,n
2
are repre-
sented as functions of motion parameters a, b, p, q, r and
control deflections de, da, dr. Note that at high angles of at-
tack the aerodynamic dependencies are nonlinear with a
strong effect of prehistory of motion [35].
The reduced set of equations for analysis of aircraft
intensive rolling maneuvers [34]:
_a ¼ q À pb þ zaa þ zde de
_b ¼ pa À r þ ybb þ ydr
dr
_p ¼ ððIYY À IZZÞ=IXX Þqr þ lbb þ lpp þ lrr þ lda da þ ldr dr
_q ¼ ððIZZ À IXX Þ=IYY Þpr þ maa þ mqq þ ma _a þ mde de
_r ¼ ððIXX À IYY Þ=IZZÞpq þ nbb þ npp þ nrr þ nda da þ ndr dr
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