ABSTRACTThis paper deals with the methods of physical modelling of FlightControl Systems (FCS) by means of Dynamically Similar free-flyingModels (DSM) for the investigation of stability and controllability ofaircraft at subsonic flight speeds. The subsonic flight regime allowsus to avoid Mach number similarity considerations. The large scaleof the DSM meets autosimilarity of Reynolds numbers, whilstFroude similarity is assured during the development and manufac-turing of the DSM. This paper proves the presence of necessary andsufficient conditions of similarity for the FCS of an aircraft, and thatof its DSM. The existence of necessary conditions have been provedboth mathematically (by means of the π − Theorem from the theoryof dimensions), and with equations involving physical quantities.The generalised scale coefficients for transitioning from the FCS’saircraft gain factors, to those of the FCS of the DSM have beenobtained, and it is shown that the coefficients depend only on thelinear scale of the DSM.

NOMENCLATUREc mean aerodynamic chordC1...C5 coefficient of linearised equationsCD drag coefficientCLa aeroplane lift coefficientCM pitching momentCMq ∂CM / ∂qCMα ∂CM / ∂α

CMα ∂CM / ∂αCMδ ∂CM / ∂δg acceleration due to gravityGq gain of FCS by pitch rateGθ gain of FCS by pitch angleGθ gain of FCS by integral of pitch angleIY moment of inertia about the Y body axisKi scale coefficient ( i = C1 ...C5, L , q,θ , θ, ρ)L length, m m massq pitch rate, deg/sq dimensionless pitch rate qc / 2Vs variable of the Laplace transformation; second Equation (10)S wing surface, m2

t time, sV velocity, ms–1

Greek letters

α angle-of-attack, degδ surface deflection, deg πi nondimension coefficients of similarityθ pitch angle, degρ mass density, kgm–3

ξ damping ratio

[ ] designates the dimensions of a quantity

THE AERONAUTICAL JOURNAL MAY 2003 249

Paper No. 2717. Manuscript received 28 August 2001, revised version received 25 February 2002, 2nd revised version received 01 November 2002 accepted 16 December 2002.

Physical modelling of a flight control system

S. SadovnychiyMexican Petroleum InstituteMexico CityMexico

··

Subscripts

H altitudeM modelN nature

Superscripts

• derivative with respect to time

1.0 INTRODUCTIONThere are several methods known for modelling the Flight ControlSystem (FCS) of a new aircraft: mathematical modelling, partialsystem test and by means of a flying simulator.

Mathematical simulation is usually applied at an early stage of FCSdesign. It has considerable advantages, such as: the results of asimulation are obtained in a very short time period, the possibility toinvestigate large number of variants, and its low cost. Nevertheless,together with these advantages, mathematical simulation has alsodrawbacks: the preparation of the mathematical model for such acomplex object as an aircraft is a quite difficult problem.

Computational methods in aerodynamics may produce large errorsduring the calculation of an aircraft’s performance, whilst wind tunneltests can not reproduce many of the complex manoeuvres of anaircraft; moreover, the physical boundaries of the walls that surroundthe model, the presence of a suspension and attachments, all produceadditional errors in the flight parameters investigated.

Assumptions and limitations introduced in the preparation of themathematical model do not allow for an adequate representation of thereal system ‘Aircraft-FCS’. In particular, discrepancies arise duringthe development of tests for an aircraft of a new generation or whenexploring new flight regimes. In these cases new physical phenomenamay be involved for which a mathematical description is not yetavailable; for example, the aircraft’s behaviour at angles of incidencebeyond stall, which is very difficult to describe.

A partial system test uses real FCS hardware together with themathematical model of the controlled object. This method, excludeserrors derived from the various assumptions in deriving the mathe-matical model of the FCS. In this case, the model of the aircraftremains without change and it contains all the errors introduced duringthe development of the mathematical modelling method.

The DSM test is a physical simulation. This simulation is differentfrom the other kinds of simulation (mathematical, partial system etc.).The model has the same physical nature as a full-scale object and it isthe cardinal distinction. The method of physical simulation allows toestablish a quantitative ratio among parameters of a full-scale objectwithout the solution of differential equations which ones describe aprocess (flight).

Flight simulation by means of a flying simulator is widely used forthe investigation of new aircraft. For example, the flying simulatorNT-33A used for the F-15, YF-16, YF-17, YF-18 aeroplane(1). Short-comings in the flight performance of these aeroplanes were detected atan early stage of flight-testing. The general principles of design of aflying simulator are the following: the basic aircraft configuration isselected; additional modules are introduced in the control system forexamining the modelling characteristics of the FCS. These units havesoftware, which may change the control laws and dynamics of thebasic aircraft. This flying simulator is very good for crew training andfor the estimation by test pilots of the flight performance of a newaircraft. However, as for the addition of FCS modules that have amathematical model of the investigated system, one must be awarethat all errors of mathematical model also appear in the flyingsimulator.

In scientific and research centres of England and USA, DSM areused since 1940. In 1940 in England, the tests of freely flying modelfor research of the aerodynamic characteristics were conducted. In

1941-1946 in USA researches on models of seaplanes was held(2). TheLockheed Company has received a significant part of the flightcharacteristics of an aeroplane XF by means of models started in freeflight on a high altitude. The part of the flight characteristics ofaeroplanes XF-91, XF-92A, XF4D-1 and Bell X-2 was obtained bymeans of DSM.(3) The models were used for development ofaeroplanes F-104 and F-15. In 2000, the tests of model X-36 were heldin USA.(4) In USSR and later in Russia DSM were applied for researchof newest aeroplanes. So, on the exhibition MAKS-92 and MAKS-93the models of aeroplanes MiG-29, Su-27 and Su-35 making about 50exploratory flights everyone were shown.

Any simulation (similarity of a model and an aeroplane) is approxi-mated, so the presentation of investigated phenomenon in its entiretyis possible only for full-sized aeroplane flight conditions. On the otherhand, the fully exact presentation of all aeroplane parameters on itsmodel does not always provide more exact data. It is necessary to haveprecise understanding about allowable differences of the model andthe aeroplane for the suitable experiment realisation.

For example, two cases can be investigated. A subject of investi-gation is the aeroplane, which is flying on a certain regime (the givenaltitude and speed of flight). The questions are:

1. If the aeroplane flying on the determined regime (altitude andspeed) is a subject of research, is it possible to consider as itsmodel the same aeroplane but which is flying on the other regime(at other altitude, or at the same altitude, but with other speed)?

2. Can the aeroplane of other type (other weight and the form), flyingon the same or other regime, be considered as a model?

The unequivocal answer for these two questions does not exist,because they do not include enough information for decision-making.It is impossible to speak about similitude of two objects (twoaeroplanes or two regimes of their flight), without defining the type ofsimilitude of the phenomena that is being investigated. Model andconditions, in which it is situated, can reproduce some authenticallyproperties of the full-scale object and at the same time, it does notcorrespond to a nature of a lot of other characteristics.

In the first case, for example, if radar reflection power of anaeroplane is investigated, practically, it has no value, considering theaeroplane flies at speed of 150 or 200ms–1. If the stall or spin processare investigated, the flight speed has basic and defining value. With aflight speed about 150ms–1, the aeroplane can have enough stabilityand controllability, but to fly with a speed about 50ms–1 is unable.With a flight speed of about 150ms–1 near to the Earth surface, theaeroplane is stable and controlled, but the same speed for a flightaltitude above 10,000m, is close to critical. In this case, despite ofapparent full conformity of a nature and model, it is observed thatflight of the aeroplane with another speed, as its own model isinadmissible for researching of such phenomena.

In the second case, accurate definition of a modelling problem isrequired too. For example, for researching of a man-autopilot systemand also for training of pilots of heavy aeroplanes, more lightaeroplanes are used as simulators. But, into their flight control system,the special units, which change dynamic characteristics of aeroplane-model and make it similar to a heavy aeroplane, are added. In thiscase, the full-scale object is simulated by model, which does not havethe geometrical similitude with it, but precisely enough and authenti-cally represents its behaviour.

So, to assert that one object is or not model of other object and todetermine conditions of similitude, it is possible to precise only withreference to certain class of modelled phenomena. DSM should corre-spond to a full-sized aeroplane by the parameters, which provide theessential effect on researched process. In this case, the necessity to getthe maximum conformity of the model to the aeroplane by the otherparameters (for example, to be aimed to make its sizes as much aspossible) is absent. On the contrary, the capability of the optionalselection of some scales and parameters of the model, which are notessential for the researched process, considerably facilitates the modeldesign and the flight test program development.Dynamically similar free-flying models (DSM) may be utilised in

250 THE AERONAUTICAL JOURNAL MAY 2003

practice in each step of the process of generating new aircraft, whendata and performances are entirely different from a previous generation.

A DSM is an autonomous, pilotless, research and developmentmultiple mission aircraft. It can carry out automatic or remotelycontrolled flights, according to a present program, and has thecapability to record all flight relevant information during a test. TheDSM is geometrically similar to the aircraft under investigation. Itsmass, moment of inertia and other parameters ensure exact perfor-mance similarity, including the case when it is necessary to ensure anadequate simulation of aeroelastic processes. DSM is a physical modelof an aircraft, and it excludes all errors that the non-linear or linearisedmathematical model of that aircraft may have. It is of most importancein the investigation of new flight regimes: for example, manoeuvresuch as the well know ‘Cobra’ (reaching an angle of attack of morethan 90° without stalling). The development of a mathematical modelfor predicting behaviour of aircraft is as mentioned above a verydifficult problem; however, a DSM may solve this problem readily.Regimes which attain an of angle of attack equal to 90° are pre-set atthe FCS of the DSM, and after flight tests, better control laws may beselected. The DSM is launched from a carrier (aircraft or helicopter)or from ground ramps with the help of a solid-propellant ascent stage.

After stage separation, the DSM accomplishes pre-programmedmanoeuvres; in case of an emergency, or after flight programcompletion, a parachute and soft landing system performs a properrecovery. It may also recover the DSM from critical or uncontrolledflight regimes, can stabilise and decelerate its motion and at thelanding moment, the same system will decrease vertical speed fortouchdown and counteracts ground wind effects. After processing theflight’s information, performing turn-round servicing and on-boardsystems readjusting, in accordance with a new flight program, furtherflight experiments may be conducted. Thus, dictated by issues of testpilot safety as well as by economic considerations, there is a clearneed for the most realistic simulation of the dynamics of flight,specially in the exploration of critical flight regimes for new aircraft,by means of such type of dynamically similar free-flying models.

At an early stage in the development of studies applying free-flyingmodels, investigators were only interested in the aerodynamic charac-teristics of the aircraft, without taking much into account the influenceof the FCS on these characteristics, and as is well known(5,6) modellingof FCS through DSM have not been utilised. The more frequent devel-opment and the growth of research efforts on active control systemshas provided a need to further simulate FCS, specially since asimilarity transfer function between the FCS of the real aircraft, withthat of its DSM, is either incomplete or simply absent(4,7-9).

2.0 SIMILARITY CONDITIONS OF THE FCS OF AN AIRCRAFT AND ITS DSM

Physical phenomena, processes or systems are similar if in theanalogous time moments and in the analogous space points, thevariable value which characterise the state of one system, are propor-tional to the corresponding variable values of other system. Theproportionality factors for each of these values are named as scales ofsimilarity.

The mandatory condition of a simulation realisation on DSM is theproviding of identical flow around the surfaces of the model and theaeroplane. It means a providing of a more exact correspondence of theangles of attack, control surface displacements, bending of mainstreamflow, stalled and vortex flow, i.e. the providing of the kinematicssimilarity of airflow around the model and the aeroplane.

The motion of two systems is kinematically similar if both systemsare geometrically similar and if it is possible to establish thecontinuous sequence of the corresponding time moments. At thesemoments the speed vectors of all corresponding points of systems areproportional to the value and are equally oriented in the space.

One of the conditions of a kinematics similarity is a geometrical

similarity. The geometrical similarity is the concept, which defines thepresence of the identical configuration of bodies irrespective of theirsizes. Two bodies are named geometrically similar it is possible toestablish the mutual unique correspondence among their points. In thiscase, the ratio of the distances between any pairs of the correspondingpoints is equal to the same constant. This constant is named as asimilarity scale and it is the scale of linear sizes KL. The requirementof a geometrical similarity propagates not only for the configuration ofthe streamlined body but also for the stream boundaries. The geomet-rical similarity is a condition of adequate mapping of aerodynamic coefficients.

The concept of a kinematics similarity only fixes certain propertiesof two systems (similarity of their motion). It states the fact of acertain ratio of the system parameters (proportionality of timeintervals, proportionality of speeds and accelerations of moving bodiesor fragments of streams). This concept is formal and does notdetermine the reasons of appearance of such properties.

The reason of linear and angular displacements, body speeds andbody accelerations is the acting of the forces and moments. Therefore,the support of a kinematics similarity of the processes which takesplace during flight of an aeroplane and DSM, requires the realisationof certain ratio between the forces acting on the aeroplane and on themodel, i.e. the dynamic similarity.

To establish similarity conditions between the FCS of an aircraftand that of its DSM, it is necessary to confront two phenomenon: first,the behaviour of the FCS of the real aircraft, and second, thebehaviour of the FCS of the free flying model. Furthermore, it isnecessary to examine how it is possible to corroborate similarityconditions for both systems.

Peter Smith (Aerodynamic Theory, Volume 1, p 27) states that: “π − theorem plays so important a part in many problem involvingdimensions and kinematics similitude. It is the equivalent in mathe-matical language of the statement that any algebraic formexpressing arelation between and among a related series of physical quantitiedmaybe adequately and fully represented as a series of terms involving π −functions.” Where π −functions are the criteria of a similarity. In otherwords, the π −theorem may be formulated in the following way: Thefunctional relation among the process parameters can be presented as arelation among the criteria, taken from them.

Necessary and sufficient conditions must be met for the comparisonof such phenomenon to be valid. Necessary conditions include: thepresence of a similarity criterion which must be numerically equiv-alent for similar phenomenon, sufficient accuracy in the geometricalsimilarity of the aircraft and its model, and finally, similarity of theinitial and boundary conditions of the processes to be compared.

3.0 NECESSARY CONDITIONSThe purpose of this paragraph is the obtaining of similarity criteria,which must be numerically equivalent for similar phenomena. Thesecriteria should be obtained by means of stringent mathematical trans-formations of differential equations which describe the ‘Aeroplane −FCS’ system.

The method presented in this section is valid for nth order andnon-linear equations. The simplicity of the example does notinfluence the results, but allows the obtaining of it in a faster andmore obvious way. Besides, it is possible to use linearised equationsbecause it is not fundamental for getting the technique of the criteriaobtaining. For a clear demonstration of necessary conditions forsimilarity between of the FCS of an aircraft and that of its DSM it ispossible to examine the simple longitudinal, short-period motion ofthe system ‘Aircraft-FCS’, described by the following equations(11,

12):

(1)

SADOVNYCHIY PHYSICAL MODELLING OF A FLIGHT CONTROL SYSTEM 251

0äá)(è)( 32512 =++++ CCsCsCs

0á)(è 4 =++− Css

èè1èèä G

ssGG q ++= θ

Let us write the system of Equations (1) in the form;

Where;

. . . (3)

The symbols of differentiation and integration have not dimensionand do not influence conditions of proportionality. Therefore, themembers of the equations ϕi, which contain these symbols, it ispossible to replace with their analogues ϕ′i, which are named asintegral-analogue(13), i.e.

The rules of definition of the similarity criteria of processes, whichare described by equations with an inhomogeneous function,exist(13). For example, if ϕ = U⋅ simω ⋅ t, then π = U and theadditional criterion of a similarity πadd = ω ⋅ t is entered.

In order to comply with the similarity criterion by the integral-analogue method is possible to write;

. . . (5)

and

. . . (6)

As far as the similarity criterion these are non-dimensionalquantities, therefore the product of the scale coefficients of thiscriterion must be equal unity;

KC4⋅ ⋅ Κt = 1; KC1 ⋅ Κt = 1;KC1 ⋅ ΚC4 ⋅ Κ2

t = 1; KC5⋅ ⋅ Κ1 = 1;KC2 ⋅ Κ2

t = 1; KC3⋅ ⋅ ΚGq ⋅ Κt = 1;K3⋅ ⋅ Κ4 ⋅ ΚGq ⋅ Κ2

t = 1; . . . (7)KC3⋅ ⋅ ΚGθ ⋅ Κ2

t = 1;KC3⋅ ⋅ KC4 ⋅ ΚGθ ⋅ Κ3

t = 1;KC3⋅ ⋅ ΚGθ ⋅ Κ3

t = 1;KC3⋅ ⋅ KC4 ⋅ ΚGθ ⋅ Κ4

t = 1;

It is known that it is impossible to provide for full aerodynamicsimilarity in a single model test’s flight, in other words, simulta-neous equality of Reynolds, Froude and Mach numbers for the realaircraft and its DSM is not within reach(10, 14).

The Froude, Reynolds and Mach criteria impose the conflictingrestriction on the scales of linear sizes KL and speeds KV. Let usconsider, for example, the obtaining of the speed scale from the

Froude criterion: Fr = idem

where ‘idem’ is the latin word and means the same.The aeroplane and the model are in the Earth’s gravitational field,

therefore gN = gM and

then; or

. . . (8)

Analogously, from Reynolds criterion;

where v is the coefficient of kinematics viscosity, it is possible toreceive;

Moreover, the ratio;

varies in a narrow range (according to the table of standardatmosphere the v is equal v = 1⋅46 × 10−5 m2s–1 near the ground,and v = 3⋅9 × 10−5 m2s–1 at 11 kilometres altitude).For realisation of the similarity by the Mach criterion;

it is necessary to provide the speed scale,

The sound speed a varies as altitude function in the smaller v range.The ratio of sound speed a on difference altitude is equal.

Three ratios between the speed similarity scales and the linear sizes;

. . . (9)

can be executed simultaneously only in a trivial case,

i.e. at experiments on the aeroplane or on the model with the sameaeroplane sizes.

However, it is possible to neglect similarity of Mach numbers for thesimulation of stability and controllability, when the airspeed is less thanthe velocity of sound. In the case of a large-scale DSM model, it ispossible to sustain the self-similarity of Reynolds number. Therefore,for the exploration of stability and controllability behaviour it isnecessary observe first of all the Froude criterion.

In aerodynamic experiments, the Newton criterion;

252 THE AERONAUTICAL JOURNAL MAY 2003

0... 12321 =++++ ϕϕϕϕ

è41 s=ϕ è4

32 Cs=ϕ

è13

3 Cs=ϕ è412

4 CCs=ϕè5

35 Cs=ϕ 2

6 s=ϕ è2Cè3

37 qGCs=ϕ è43

28 qGCCs=ϕ

èè32

9 GCs=ϕ è4310 θϕ GCCs=èè311 GCs=ϕ èè4312 GCC=ϕ

nn

n

yx

yx →

dd ∫ → yxyx d

;;

;

;

;

;;

;

;

;

;

;

and

;è41 t

=′ϕ ;è

34

2 tC

=′ϕ ;è31

3 tC

=′ϕ

241

4è

tCC

=′ϕ

;è

2è3

9 tGC

=′ϕ ;èè43

10 tGCC

=′ϕ

;èè3

11 tGC

=′ϕ èè4312 GCC=′ϕ

;ð 41

21 tC=

′′

=ϕϕ ;ð 1

1

32 tC=

′′

=ϕϕ

;ð 241

1

43 tCC=

′′

=ϕϕ

;ð 3è43

1

109 tGCC=

′′

=ϕϕ

;3è3

1

1110 tGC=

′′

=ϕϕπ

4è43

1

1211ð tGCC=

′′

=ϕϕ

. . .

. . .

. . . (4)

;22

FrLg

VLg

VMM

M

NN

N ==

;22NMMN LVLV =

N

MV

N

M

LLK

VV ==2

2

Re==M

MM

N

NN LVLVνν

LN

MV K

K 1νν=

N

M

νν

M==M

M

N

N

aV

aV

N

MV a

aK =

1≈N

M

aa

LidemFrV KK ==

1== LV KK

LN

MidemV K

K 1Re ν

ν==

LV KK =i.e.

; ;

NeLmtF =

⋅⋅ 2

1≈==N

MidemMV a

aK

is always used. It is interpreted as a proportionality support of theacting aerodynamic and inertial forces of body. From this criterion,it is possible to receive;

by means of the above-stated transformations.The equation expressing the relationship between inertial and

weight forces (Fr = idem), follows:Length LM = LN ⋅ ΚL

Area SM = SN ⋅ K2L

Time tM = tN ⋅ Kt = tN √ΚL

Speed VM = VN ΚV = VN . √ΚL

Mass mM = mN ⋅ K3L

Density ρM = ρN ⋅ Κρ. (10)Acceleration aM = aN, Kα = 1Angle ϕΜ . ϕΝ , Kϕ = 1Angular velocity ωΜ = ωΝ / √KL

Moment of inertia IM = IN ⋅ K5L

In practice, after the linear scale is determined, the model mass isbeing obtained by an equation;

The model mass is adjusted by means of the selection of equipmentweight and, if it is necessary, by means of additional masses. Thefulfilment of the inertial scale;

is obtained by appropriate allocation of the equipment and by usingadditional masses in the model construction.With Equation (7) and;

from Equation (10) it is possible to calculate the scale coefficientsfor Equation (1)

The foregoing method of obtaining the scale coefficients is a formalmathematical method; therefore, it is necessary to verify by means ofother procedures, which do take into account the physical nature ofthe occurring process(13).

For this purpose it is possible to use the well-known relations forthe coefficients C1, C2, C3, C4, C5, of linearised Equations (1)(12)

and the expression for obtaining the gain of the pitch damper

With Equation (10) solved, the coefficients (12) of the linearisedequations for the system ‘model-damper’ will be:

And from this:

Therefore, with this equation, the gain of the pitch damper for theDSM will be:

Analogously, the transformation for roll and yaw dampers, have thesame relation between the gain of the aircraft and that of the DSM.

If take the control law for a longitudinal zero-constant-errorsystem as follows(15);

where: dimension of gain;

is the stabiliser deviation on one degree with aeroplane deviation onone degree on pitch angle, and dimension of factor;

is the stabiliser deviation with a speed of one degree per second withaircraft deviation on one degree on pitch angle (factor Gθ defines theconstant of integration, or speed of the stabiliser deviation, in theintegral control law) and making an analogous transformation, thenobtain;

SADOVNYCHIY PHYSICAL MODELLING OF A FLIGHT CONTROL SYSTEM 253

3Lm KK =

m m KM N L= ⋅ 3

I I KM N L= ⋅ 5

K Kt L=

K K K KC C C L1 4 5 1= = = /

K KC L2 1= /

1è =GK

K KGq L=

LG KK /1è =

;12

21

−=s

cSVI

CC

Y

qM ρ

;12

ñ 2

2á

2

−=s

cSVI

CCY

M

;12

ñC

;12

ñ

á4

2

2ä

3

+−=

−=

sSV

mCC

scSV

I

CC

DL

Y

M

,12

ñ 2á5

−=s

cSVI

CCY

M&

. . . (11)

. . . (13)

. . . (14)

. . . (15)

. . . (16)

. . . (17)

Gθ

Gθ

. . . (12)

. . . (18)

3

541412 )(î2C

CCCCCCGq

++−+=

CC

I KMMq

YN L1 5=

2LNN KVρ

=22 2LLN KKS Nc .1

1NL

CK

CK

CML

N2 21= ; C

KCM

LN3 3

1= ;

CK

CML

N4 41= ;

;qNLqM GKG =

∫⋅+⋅+⋅= èdtèèä èè GGGq &

anglepitch of degree

elevator of degree

anglepitch of degree

s elevator / of degree

CK

CML

N5 51= .

×

×

×

+

=N

L

NL

NL

NL

qM

CK

CK

CK

CK

G3

412

1

111î2

;1

111

3

541

NL

NL

NL

NL

CK

CK

CK

CK

+−×

NL

M

NMqNLqM

GK

G

GGGKG

èè

èè

1

; ;

=

==

In this case the scale coefficients obtained, by the first(expression 11) and second (expression 15 and 18) modes,coincide. The validity of this ratio ensures the fulfilment ofequalities(7); i.e. it makes the products of the scale coefficientsequal to the unit. However, following the order, it means theexistence of the dimensionless similarity criterion, which mustbe numerically equivalent for similar phenomena. Therefore, itis possible to consider that the presence of necessary similarityconditions for the FCS of the aircraft and that of the DSM hasbeen proven.

It is evident that the value and dimensions of the gains aredetermined by means of coefficients from the equations for thesystem ‘Aircraft-FCS’. Since these coefficients for the DSM andfor the aircraft have a certain relationship, therefore after alltransformations in the equations from aircraft to model’s gainsthey remain only as a matter of scale, which are defined bydimensions of its gains. Therefore, in all cases, the coefficientsto transfer from the aircraft’s to the model’s gains, depend onlyon the actual dimensions of both, and are defined by thefollowing equations:

. . . (19)

5.0 SUFFICIENCY CONDITIONSTo ensure sufficiency of conditions, at first the DSM must bemanufactured as an exact copy of the real aircraft, but on asmaller scale, this provides the first condition of sufficiency, thatis, geometrical parameter similarity between the model and thereal object. Second, it is necessary to provide a functionalsimilarity of the block diagram of the FCS, i.e. to keep in theDSM formulation the same control laws as on the real aircraft;which is not usually a problem. For the model, the precise sameview of the control law is selected. In the case of the FCS of thedesigned DSM, its control laws are not arbitrarily selected, butare taken directly from the real aircraft.

The numerical values of DSM FCS gains are received fromthe appropriate gains of the aeroplane FCS by means ofEquations (18). The physical adjustment of gains is made bymeans of the model autopilot programming (or by adjustment ofthe electrical parameters of the autopilot) in the same way assetting-up of gains on an aeroplane.

Third, ensuring similarity of the starting boundary conditionsof confronted phenomenon: the flight of the real object and itsmodel occur in a real atmosphere and at a definite altitude h. It isalso necessary for meeting similarity to comply with the nextrelationship:

If in the manufacturing of the model, it is not possible to provideequality of density coefficients i.e.:

then, to comply with Equation (18), it is necessary to ensure,

This is provided at the expense of adjusting the test flightaltitude of the model. For example, the flight of an aircraft at

8,000m (ρ = 0⋅536) may be simulated with a flight of the DSMat 5,000m (ρ = 0⋅751).

These are in essence the existing necessary and sufficientconditions of similarity for the simulation of the FCS of theaircraft by means of Dynamically Similar Models.

6.0 CONCLUSIONSThe physical modelling of a FCS by means of large scaledynamically similar free-flying models has a long-rangeperspective, which allow for a significant increase in the qualityof design and development of new types of aircraft, and theirFCS; with the advantage of drawing nearer the physical model tothe natural object being simulated.

Using two methods for proving the existence of necessarysimilarity conditions, on the one hand the formal mathematicalmethod, and on the other dealing with the physical sense of the phenomenon, we increase the reliability of the obtainedresults.

Such results show the possibility of physically modelling thedynamics of flight, taking into account all peculiarities ofaerodynamics, block diagram equivalence of the FCS, as wellas real atmospheric conditions. Scale coefficients for transi-tioning from the FCS’s aircraft gain factors, to those of theFCS of the DSM allow us to simplify the development processof the DSM and its FCS, because of the exclusion of sometraditional stages in design. Furthermore, it allows us totransfer more easily the flight-test data from the DSM’s FCS tothe aircraft FCS.

Finally, it is necessary to note that the question of similarity ofservos and the qualitative characteristic of the FCS still demandfurther investigation.

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4. DORNHEIM, M.A. X-36 To Test agility of tailless design. AviationWeek & Space Technology, March 1996, 4, pp 20-21.

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254 THE AERONAUTICAL JOURNAL MAY 2003

degrees / degreedimension timein the ,K1/=K

s / degreedegreedimension timein the ,K=K

degreedegreedimension timein the 1,=K

L

L

. . . (20)

. . . (22)

. . . (23)

. . . (21)

hN

hM

N

MKññ

ññ

ñ ==

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)1K where(ññ ≠= NM K

hNhM K ññ ⋅=

SADOVNYCHIY PHYSICAL MODELLING OF A FLIGHT CONTROL SYSTEM 255

14. SADOVNYCHIY, S., BETIN, A., RYSHENKO, A. and PERALTA, R.Simulation of aircraft flight dynamics by means of dynamicallysimilar models. Proceed. of AIAA Modelling and SimulationTechnology Conference-98, 1998, pp 64-69, Boston , USA.

15. MCRUER, D., ASHKENAS, I. and GRAHAM, D. Aircraft Dynamics andAutomatic Control, Princeton University Press, Princeton, NewJersey, 1973.