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UNCLASSIFIED
AD445594
DEFENSE DOCUMENTATION CENTERFOR
SCIENTIFIC AND TECHNICAL INFORMATION
CAMERON STATION. ALEXANDRIA, VIRGINIA
UNCLASSIFIED
NOTICE: When government or other drawings, speci-fications or other data are used for any purposeother than in connection with a definitely relatedgovernment procurement operation, the U. S.Government thereby incurs no responsibility, nor anyobligation whatsoever; and the fact that the Govern-ment may have formilated, furnished, or in any waysupplied the said drawings, specifications, or otherdata is not to be regarded by implication or other-wise as in any manner licensing the holder or anyother person or corporation, or conveying any rightsor permission to manufacture, use or sell anypatented invention that may in any way be relatedthereto.
REPORT 367
Oj ADVISORY GROUP FOR AERONAUTICAL RESEARCH AND 4T
64 RUE DE VARENNE, PARIS VII
APRIL 1961
~ATHEMATICAL MODELS
CFOR MISSILES
by
W. S. BROWN and D. I. PADDISON
DCREPORT. 367
DDC-IRA B
NORTH ATLANTIC TREATY ORGANISATION
367
ADVISORY GROUP AERONAUTICAL RESEARCH DEVELOPMENT
MATHEMATICAL MODELS FOR MISSILES
D)by
W.S. Brown and D.I. Paddison ,
This Report is one in the Series .34-374, inclusive, presenting papers, with dis-cussions, given at the AGARD Specialists' Meeting on 'Stability and Control', Training
Center for Experimental Aerodynamics, Rhode-Saint-Genbse, Belgium, 10-14 April 1961,sponsored jointly by the AGARD Fluid Dynamics and Flight Mechanics Panels
SUMMARY
This paper describes the use of a large analogue computer, supplemented by digitalcomputations, to study the effects of high-incidence aerodynamic non-linearities andcross-couplings on the performance of a hypothetical cruciform missile. Aerodynamicsdata, obtained from wind-tunnel tests of a suitable model, were simulated in detail,and linear aerodynamic characteristics were subsequently substituted and simulatedfor purposes of comparison.
The effects of the aerodynamics on response and on homing performance againsttargets turning at a constant rate are displayed as graphs, and the importance ofaccurately simulating the aerodynamics is demonstrated.
Techniques which proved useful in simulation are described, and the accuracy andlimitations of the analogue method are indicated by comparisons of the simulatorresults with others obtained digitally.
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CONTENTS
Page
SUMMARY i
LIST OF FIGURES v
NOTATION vt
1. INTRODUCTION 1
2. CHOICE OF AERODYNAMIC CONFIGURATION 2
3. AERODYNAMIC CHARACTERISTICS OF TEST VEHICLE 23.1 General 2
3.2 Wing-Body Tail-Fixed Characteristics 33.3 Control Surface Characteristics 43.4 Effect of Aerodynamic Characteristics 4
4. DETAILS OF SIMULATION 4
4.1 Wing-Body Tail-Fixed Aerodynamics 54.2 Function Generators 5
4.3 0 Servo 6
4.4 Control Surface Aerodynamics 7
4.5' Rotary Derivatives and Gravity 7
4.6 Mach number Variation 7
4.7 Altitude Variation 84.8 Centre of Gravity Vari~tion 84.9 Control System 8
4.10 Dynamical Equations 8
5. DIGITAL CHECK SOLUTIONS 9
6. RESPONSE OF TEST VEHICLE AS SIMULATED 9
6.1 Response in Single Plane 9
6.2 Response in Three-Dimensions 9
6.3 Effect of Linearising Aerodynamics 9
6.4 General Conclusions on Response 10
7. INVESTIGATION OF HOMING PERFORMANCE 10
7.1 System Adopted for Simulation 10
7.2 Details of.Simulation Extension 10
8. EFFECT OF AERODYNAMICS ON MISS DISTANCE 12
8.1 Homing in Single Plane 128.2 Homing in Three Dimensions 13
9. EFFECT OF ANGULAR NOISE ON MISS DISTANCE 14
10. PROBLEMS OF A LARGE SIMULATION 14
iii
Page
10.1 Simulator Errors 14
10.2 Comparison of Analogue and DigitalResults in Single Plane 14
10.3 Comparison of Analogue and Digital Results
in Three Dimensions 15
CONCLUSIONS 15
REFERENCES 15
FIGURES 16
DISCUSSION A-i
ADDENDUM: Complete List of Papers in Series
DISTRIBUTION
iv
LIST OF FIGURES
Page
Fig. 1 Diagrammatic representation of test vehicle 16
Fig.2 Normal force and pitching moment coefficient 17
Fig.3 Rolling moment coefficient 18
Fig.4 Typical trim curves for test vehicle 18
Fig.5 Silicon diode function generator 19
Fig.6 Block diagram of test vehicle simulation 20
Fig. 7 Generation of aerodynamic forces and moments 21
Fig.8 Responses in single plane with linear and non-linear aerodynamics 22
Fig. 9 Response in three dimensions to three demands; XGo 0 23
Fig. 10 Block Diagram of homing missile simulation 24
Fig. 11 Miss distance vs initial range in single plane homing 25
Fig. 12 Maximum lateral acceleration vs initial range in singleplane homing 25
Fig. 13 Miss distance vs initial angle of roll in 3-dimensional
homing; initial range fixed 26
Fig. 14 Diagrammatic representation of homing in three dimensions 26
Fig. 15 Effect of angular noise on miss distance in 3-dimensional homing 27
Fig. 16 Comparison of analogue and digital results in single planehoming 27
Fig. 17 Comparison of analogue and digital results in three dimensions 28
v
NOTATION
GXYZ principal axes in test vehicle (see Fig.1)
G centre of gravity
m mass of test vehicle
S representative area on test vehicle
representative length on test vehicle
A, B, C principal moments of inertia
X, Y, Z aerodynamic forces in principal axes
L,M,N aerodynamic moments about principal axes
V total linear velocity (relative wind velocity)
u,v,w component linear velocities
p q,r component angular velocities
b,c component lateral accelerations
9total body incidence to relative wind
azimuth angle of wind vector, tan d = v/w
a,8 13 partial incidences
a sin a ::w/V
8sin /8= v/V
0 sin 0, cos6= u/V
0- control panel deflection
S, 77, equivalent aileron, elevator and rudder angles
cx, cy, cZ aerodynamic force coefficients
clcmcn aerodynamic moment coefficients
lpmqfnr aerodynamic rotary derivatives of L,M,N
M Mach number
vi
y ratio of specific heats of air
p ambient air static pressure
G, XY'Z' rectangular axes obtained by rotating G, XYZ system about GX through angle
so that plane GXZ' contains wind vector
prime denotes quantity measured in G,XY'Zf axes
Demand rectangular axes with centre G and same X-axis as G,XYZ system, but not
Axes rotating with test vehicle. Demand always for rotation about Y-axis of
demand-axis system
XGo initial angle of rotation of G,XYZ axes relative to demand axes
R Range
Suffix D = demanded
d = in demand axes
m = in missile axes
o = initial value
i = indicated value
vii
MATHEMATICAL MODELS FOR MISSILES
A Simulator Study of the Effects of Aerodynamic Non-Linearities and
Cross-Couplings on the Performance of a Guided Missile
W.S. Brown and D.I. Paddison*
1. INTRODUCTION
As the science of aeronautics advances and aircraft performance steadily improves,
the problem of intercepting high-flying assault aircraft by anti-aircraft guided
missiles becomes increasingly difficult. Missiles which are not steered by such
means as thrust deflection depend for manoeuvre entirely upon the aerodynamic forces
and moments which they experience when their fixed or movable surfaces are inclined
to the relative wind. A simple approximate calculation shows that, in the stratos-
phere, the incidence necessary to achieve any desired rate of turn of the flight
path varies directly as the density of the missile relative to the air and inversely
as the Mach number of flight. Hence, for any particular missile and Mach number,
the required incidence increases with altitude approximately as the inverse of the
air density. The same law, of course, applies to the aircraft to be intercepted, but
the effect on the missile is likely to be more severe because its design is usually
determined not solely by aerodynamic considerations but also by those of logistics
which do not apply to the aircraft - questions of transport, ease of deployment,
suitability for loading on and firing from aircraft or launching platforms in alimited environment. Because of these special requirements, the designer is seldom
free to select the optimum aerodynamic configuration for his missile but must, in
general, restrict the size of its lifting surfaces. In consequence, the range of
incidence required for adequate manoeuvre under all conditions may be much larger in
the case of the missile than in that of the aircraft. It is important, therefore,that due attention be paid to the aerodynamic characteristics of winged missiles
required to operate at high altitudes.
Whatever its design, it is unlikely that the aerodynamic forces and moments acting
on a missile will vary linearly with incidence over more than a limited range. Thissuggests that, at high altitudes, when adequate manoeuvrability may require large
incidences, any aerodynamic non-linearities which may be present may have animportant effect on performance. If this is so, the customary assumption of linear
aerodynamic characteristics made in design studies, and commonly employed also in
more detailed investigations conducted with the aid of simulators or digitalcomputers, may lead to erroneous conclusions. It was such considerations that led us
to plan a programme of research combining flight experiments, using a test vehicle,with a detailed simulation which would include any aerodynamic non-linearities
which happened to be present in the design. One object of this dual programme was todetermine to what extent the observed performance of the test vehicle in flight, in
response to selected demands for manoeuvre, was reproducible on the simulator. Given
good agreement, we would then use the simulator to extend the investigation by adding
some form of homing system to the model and determining whether or not homing
* United Kingdom
2
performance was likely to be affected by aerodynamic characteristics of the typementioned. It is the latter work which we propose to describe in this paper. Weshall give some account of the aerodynamic features of the test vehicle, the mathema-tical model on which the simulation was based, the techniques which we saw fit toemploy in order to represent the aerodynamic characteristics with reasonable accuracyand the limitations of our methods. In discussing some of the results obtained we
shall note an instance in which the use of the customary approximate mathematical
model would have led to erroneous conclusions.
2. CHOICE OF AERODYNAMIC CONFIGURATION
For a variety of reasons, we selected a fixed wing cruciform design having four
movable tail surfaces in line with the wings as most suitable for our work. From the
point of view of aerodynamic non-linearities, fixed-wing designs are likely to bemore interesting than their moving-wing counterparts because, with the former, largewing incidence is associated with a large body incidence. Furthermore, a Cartesian
missile is often regarded as a combination of two 'single plane' missiles at rightangles and, from this standpoint, is a simpler proposition to analyse than a missile
of the twist and steer variety. This apparent simplicity might be deceptive however
if, in an actual missile, aerodynamic cross-coupling between the planes were presentin addition to the customary inertial and gyroscopic couplings. In any case, itwas of some interest and importance to determine the effect of such features on the
ideal performance.
Our choice was finally determined by the fact that the Royal Aircraft Establishment
already had a general-purpose test vehicle of this type which had been extensivelytested in flight and in the wind tunnels as part of a programme of basic research.However, as our investigation was primarily one of the effects of large incidences, itwas necessary to supplement the existing wind-tunnel data by further tests on the
model. These were conducted at a variety of supersonic Mach numbers over a range ofincidence up to 350 to the relative wind. In the meantime, the flight tests alreadycompleted and the wind-tunnel data already available served as a useful starting point
and check on the simulator model.
3, AERODYNAMIC CHARACTERISTICS OF TEST VEHICLE
3.1 General
A diagrammatic representation of the test vehicle is shown in Figure 1, which alsoindicates the system of co-ordinate axes adopted for the mathematical model and the
scheme of notation. G, XYZ are principal axes defining the longitudinal axis of thetest vehicle and the planes through the two pairs of wings. The centre of gravityis assumed to be located at G ; & is the angle between the velocity vector GV and
the longitudinal axis and, hence, represents the total incidence, and 0 is the
angle between the XZ plane and the plane containing the longitudinal axis and thevelocity vector. Hence, tan q)= v/w , where v and w are the sideslip velocities
in the directions of GY and GZ . We may also here define partial incidences aand g with sin a = a = w/V and sin 8 = 8 = v/V , V being the total velocity ofthe missile, so that w = Ocos O and v = gsin qT, where 0= sin 0 . We shall
3
have occasion to refer to these later. The angular deflections of the controlsurfaces are denoted by o with an appropriate suffix, and linear combinations of
these quantities are equivalent to the , , of the standard notation for
aileron, elevator and rudder angles.
Since the wind-tunnel model with its control surfaces undeflected had geometricalsymmetry in each wing plane, all the aerodynamic quantities measured were periodic
functions of 0 , a feature which was of considerable value in both the wind-tunnel
testing and the subsequent simulation. Only effects resulting from deflections ofthe controls disturbed the symmetry in the four quadrants and, even so, it was
possible to reduce the amount of wind-tunnel work by considerations of symmetry. Inorder to cover fully all contingencies, the effects of deflecting the control surfaceswere determined over ranges of incidence of the latter between 350 positive and
negative.
3.2 Wing-Body Tail-Fixed Characteristics
The general pattern of the results is indicated in Figures 2, 3 and 4, Figures 2and 3 relating to the model with controls undeflected. In Figure 2, the symbolswithout primes denote lift and pitching-moment coefficients in the principal plane
GXZ, and primed symbols the same quantities measured in the plane containing thewind vector and the X-axis, with the addition of C' as the lateral force at rightangles to this plane. Figure 2a shows that the Z force varies almost linearly withincidence, in the XZ plane, and Figure 2b that it varies little with p , thetrend at 250 incidence being plotted to indicate this. On the other hand, thepitching moment, as Figure 2c shows, is markedly non-linear with incidence in theXZ plane and, in the wind plane, changes so rapidly with 0 as to become negativearound 0 = 450 , as indicated by Figure 2d. The same figure indicates the presenceof a yawing moment C' at right angles to the pitching moment, which is zero only innplanes of symmetry. The non-linearity in the pitching moment is attributable to alongitudinal movement of the centre of pressure with change of incidence, which is
different in different wind planes; and the side force and moment are indicative of
aerodynamic cross-coupling.
The rolling moment about the X-axis, with the wind plane inclined at 150 to the
XZ plane, is shown in Figure 3a as a function of the total incidence, and itsvariation with k at 250 incidence is indicated in Figure 3b. The rolling moment,therefore, is markedly non-linear with respect to both incidence and angle of roll
and, although zero in planes of geometrical symmetry, that is, in either of the wingplanes or in the planes through the longitudinal axis of the model equally inclinedto the wings - planes referred to later as the 450 planes, is always of such a sign
as to rotate the model into the nearest 450 plane, where the roll equilibrium isstable in contrast to that in the wing planes, which is unstable. It may be noted inpassing that the rolling moment at any incidence is a maximum for a value of 0 inthe neighbourhood of 22V ° .
To sum up, therefore, our tests showed that, in all planes other than those ofgeometrical symmetry, the resultant force and moment on the model did not lie in the
plane of total incidence as defined above, and the pitching, rolling and yawingmoments were markedly non-linear with respect to both total incidence and angle ofroll. The stability in pitch was greatest in the wing planes and least in the 450
4
planes, the opposite being true of the stability in roll. These results are attribut-
able to aero-dynamic non-linearities and cross-couplings. Had such effects been
absent, the quantities C' and C' would have been small and C' and C' almost
independent of k . The rolling moment coefficient Cl would have been identically
zero. The effects noted varied to some extent with Mach number but their general
character was unchanged.
3.3 Control Surface Characteristics
Deflection of the control surfaces revealed a complicated situation. With the body
in any selected attitude, the tail moments were non-linear functions of the control-
surface deflections relative to the wings; but the shapes of the curves depended also
on the model's attitude to the wind, since part of the non-linearity was due to wing-
body interference. The method of investigation adopted in the wind-tunnel was first
to deflect only one tail fin at a time, and then to move two adjacent ones together
by varying amounts. This was done over the full range of attitudes of the model
covered in the tests. The results revealed that the mutual interference of the fins
was relatively small and that it was possible to find approximate mathematical
expressions for this and the other effects observed. Figure 4, which is a plot of
elevator trim angles in terms of the partial incidences a and )3 for a typical
Mach number and centre of gravity, is indicative of the overall non-linearities and
cross-couplings.
3.4 Effect of Aerodynamic Characteristics
These peculiarities in the aerodynamics revealed by a study of the model might be
expected to have important effects on the performance of the test vehicle. In
particular, the presence of aerodynamib cross-couplings implied that, if the vehicle
were required to manoeuvre under aerodynamic force in any plane other than a principal
plane of symmetry, the control surfaces would have to provide not only the requisite
force and moment in the desired plane of manoeuvre but also those necessary to
resist a couple tending to rotate the vehicle out of the plane. Hence, unless the
control system were designed to prevent this, the vehicle would tend to yaw and roll
away from the desired plane of manoeuvre. In particular, unless the roll control
system provided stabilisation of roll position, the vehicle would, in general, roll
under the application of a demand for incidence until it reached a position of
stable roll equilibrium; in other words, the incidence would be developed ultimately
in one of the 450 planes. This would apply equally to a vehicle without any roll
autopilot and to one which was stabilised by control of roll rate. For this reason,
it was decided to simulate roll rate stabilisation in the first place, although the
test vehicle in its earlier flight tests had been roll position stabilised. In any
case, an easy transition could be made to an idealised roll position stabilised system,
if desired, by suppressing the roll.
4. DETAILS OF SIMULATION
The simulation was undertaken on TRIDAC, a large analogue computer at the Royal12
Aircraft Establishment, which has been described elsewhere'
5
4.1 Wing-Body Tail-Fixed Aerodynamics
The generation of the complex aerodynamics on the simulator posed problems. At
the outset, two methods of approach were considered and tried. The first of these
involved the expansion of the aerodynamic forces and moments as polynomials in d
and j , a method which had been adopted previously. On this basis, the total
incidence 8 of the model is given by g2 = a2 + 42 ,where 0= sin 8 . TRIDAC is
well suited to such a method, for it is equipped with several sets of high-grade
linear potentiometers ganged together and driven by high-performance hydraulic servo-
motors. However, although the method of polynomials is extremely effective over a
limited range of incidence if the functions to be generated are reasonably linear, it
becomes rapidly less satisfactory if marked non-linearities occur at larger incidences;
for then it is necessary to include many more terms in the expansion. Invariably,
both positive and negative terms occur and the associated constant multipliers often
vary greatly in magnitude. A formidable scaling problem therefore arises, added to
which the successive multiplications necessarily introduce inaccuracies. In the end,
it usually happens that, while the value of the function generated is correct at
each of the points selected to define the polynomial, the accuracy of the representa-
tion at intermediate points is poor. It was found to be much better, when generating
the non-linear wing-body aerodynamics, to take advantage of the aforementioned
symmetry of the cruciform design when the tail control surfaces are undeflected.
This results in the forces and moments being periodic functions of the angle ,
which enables one to express these quantities as Fourier sine or cosine series in
the coefficients being functions of 8 .
The general form adopted to express the wing and body forces and moments with the
controls undeflected was, therefore,
F'(9, ) Z [Gn() sin 4n + Hn(O) cos 4n ]
n
the fundamental variable in the trigonometrical terms being 4q rather than q5 , since
the symmetry repeats in each of the four quadrants formed by the wings.
TRIDAC is well equipped to generate such expressions also, since it has several
highly accurate multiple sine and cosine resolvers driven by hydraulic servo-motors.
The general term in sin 4nq or cos 4n is, of course, easily obtained as a sum
of products of powers of sin 4q5 and cos 40 but it was, in fact, never found
necessary to include more than two terms of the series to obtain an adequate fit.
The wind-tunnel data had been obtained over series of values of 8 and 0 covering
the required ranges and were therefore, in a suitable form from the start. The
ordinates of the Gn(j) and Hn(O) functions at each experimental value of 8 were
determined by applying the method of least squares to the appropriate section of the
data at that incidence, i.e. to the set of values of each quantity at all the values
of 0 ,
4.2 Function Generators
The G and H functions were simulated by means of diode function-generators of
a pattern developed by the staff of TRIDAC. One of these generators is illustrated
6
in Figure 5. The basic unit has six sections and it is mounted on a card measuring
approximately 8.5 cm by 19.5 cm, which also carries the silicon diodes and the
miniature potentiometers used to fix the start and slope of each link of the chain of
tangents to the curve used as the first approximation to it. A printed circuit is
employed, and the card also carries input sockets for the signal, bias and a. c.
smoothing voltages, and output sockets for the connections to the associated
amplifiers. The smoothing voltage, which is the special feature of these units, has
a saw-tooth waveform, is adjustable in amplitude, has a frequency of 100 kc/sec, and
is superimposed on the input voltage. It modifies the output of the function
generator by a process which amounts to sampling the slopes of the tangents in
succession. As each junction of tangents is approached, the next slope is sampled to
an increasing extent. It may readily be shown that the process rounds off the
corners at the junctions, which are commonly referred to as 'break points', and,
since the amount of rounding is adjustable by varying the input summing resistor of
the smoothing voltage, it results in a close fit to the curve at all points. If
higher accuracy is required, two function generators may be used in series, thus
doubling the number of sections. In our experience, however, one six-section unit
with smoothing is as good as one of twelve-sections without smoothing. Invariably,
we have been able to generate the required functions with error everywhere less than
one per cent of the maximum value of the function in the range. This is certainly
as good as, and probably better than, the accuracy of the wind-tunnel data.
4.3 0 Servo
By such means, the aerodynamic forces and moments in any wind plane were expressed
as functions of 0 and 0 , the variables defining the position of the wind vector,
and were then resolved, as required, to form the forces and moments in the principal
planes of the model. The servo-resolver which provided sin 495 and cos 40 also
generated sin 0 and cos q5 for these resolutions, a gear box of ratio 4:1 having
been provided. The method of driving this servo may be of interest. In the course of
the simulation, the quantities v, w and V, defined earlier, were generated. By
feeding the voltages representing v and w to two resolvers driven by the servo,
the quantity
E = v cos 5- w sind
could be formed. If the shaft angle of the servo were correct, this quantity would
be zero. Hence, E could be employed as an error signal to drive the servo as a
position servo. The servo was also able to provide
V9 = v sin d+ w cos 0
and, therefore the j necessary to generate the G and H functions. Difficulty
arises with such a system, however, when 0 is small, because then both v and w
are small; hence, e is small even when is substantially in error. For this
reason, it was necessary to provide the servo with a form of automatic gain control
to increase the error signal when 0 was small. A simple arrangement utilising an
auxiliary servo driving a potentiometer to vary the gain of the error signal in
proportion to the inverse of 0 over a suitable range of the latter was found tobe completely effective.
7
It will have been noted also that, when d is 0 or 7T/2 , so that v or w iszero, E is zero for any value of the incidence. Hence, if the wind vector happensto lie in either. of the wing planes, where there is no rolling moment, there will beno error signal whatever the value of the incidence, large or small, positive ornegative. Hence, the incidence could, apparently, change sign without the servo beingaware of the fact. We feared this might be a fatal objection to the method but, inpractice, it was found that, with the a.g.c. control, the small electrical driftswhich were always present sufficed to cause the servo to swing rapidly through 1800in such instances. This also implies that the incidence, as defined in our system, isalways positive, the position of the wind vector being determined by the value of theangle k . A similar situation occurs when the wind vector is in either of the 450planes, in which case the component velocities v and w are numerically equal.Once again, however, no trouble was experienced in practice, ( being preciselydefined whenever 8 exceeded 0.20.
4.4 Control Surface Aerodynamics
As regards the generation of the contributions of the tail surfaces when deflected,it was found, as one might expect, that these consisted of a major term which was alinear function of the control angle relative to the appropriate wing, and which could,therefore, be expressed as a quantity multiplying f , W7 or , and a part whichcontained cross products of these with 5. and B or, alternatively, the equivalentfunctions of e and b . This secondary part of the effect of the tail was found tobe most easily expressed by polynomials in 5 , , , , 1, and the terms weregenerated by electric servo-multipliers driven by these quantities.
4.5 Rotary Derivatives and Gravity
No attempt was made in the simulation to generate unsteady aerodynamic effects, butprovision was made to include derivatives such as ZP I mq and nr , values of whichwere obtained by theoretical calculation. Their effect was found to be small.Gravity forces were not simulated, since their inclusion in the earlier stages ofthe work would have involved adding an earth-to-missile axis transformation to thesimulation. After the latter was installed for other reasons, the gravity terms inthe equations were still omitted since their effect seemed likely to be secondary andin the nature of a bias which was regarded as liable to obscure the main issue so faras homing performance was concerned. The effect of gravity was included, however, indigital studies which were undertaken to check the performance of the early flightrounds.
4.6 Mach Number Variation
A further point may be mentioned regarding the generation of the aerodynamics. Asalready noted, the curves of force and moment varied with Mach number. It was foundupon analysis that the overall variations of the G and H functions wereconsiderably less if the coefficients were multiplied by the Mach number. This wastherefore done, and the resultant forces and moments were obtained from the resultingexpressions by multiplying them by 1/2SypM,, or 1 SZYPM , in the usual notation, ratherthan these factors with M2 . The effect of varying Mach number was then studied byinterpolating linearly between the outputs of successive pairs of function generators.
8
4.7 Altitude Variation
In addition to the means described for varying the Mach number, provision was made
to alter the altitude. The interest being in large incidences, only altitudes
exceeding the height of the tropopause were considered. The simulation of height
variation was accomplished, therefore, simply by varying p , the ambient pressure,
in the factor multiplying the aerodynamic force and moment coefficients.
4.8 Centre of Gravity Variation
Provision was also made to vary the position of the centre of gravity in accordance
with changes which would be produced by the burning of the vehicle's sustainer rocket
motor. The effect is e:-ressible as a change in the leverage of the forces which
contribute to the moments about the cencre of gravity. The variation was assumed to
be linear in time and was, therefore, readily accomplished with the aid of a servo-
driven potentiometer which varied the gain of the force contributions to the moments.
4.9 Control System
It has already been mentioned that a scheme of roll rate stabilisation was selected
for the first phase of the investigation. A system utilising rate feedback was also selected
for the pitch and yaw control system, since this had been employed on the test vehicle.
4.10 Dynamical Equations
The simulation of the dynamical equations was straightforward, The equations ofmotion were formed in moving axes coincident with the principal axes of the vehicle,
and were integrated once to produce the linear and angular velocities in the same
system. The former were then employed to geaerate the aerodynamics after derivationof 8 and 4 , as already described. The equation of motion in u , the velocity in
the direction of the longitudinal axis of symmetry, was not simulated but was replaced
by the equation V = f(t), V being the total velocity and f(t) a function of time
which, for much of the work, was assumed constant. The value of u was then
obtained from the relation u = V cos 6 , V sin 0 being available from the 0 servo,
as explained earlier. The cosine was derived from the sine with the aid of a diode
function generator utilising the relationship
cos = ( - sin 28)
or u _
A block diagram of the simulation is shown in Figure 6, and the mode of generatingthe aerodynamics in Figure 7.
13
missile passes the target on the opposite side and that there is an optimum initialrange for which the miss distance is zero. The growth of the maximum lateral
acceleration with decrease of initial range is indicated in Figure 12.
8.2 Homing in Three Dimensions
Curve 1 of Figure 13 shows the effect of both non-linearities and cross-couplings
when the missile is roll-rate stabilised and, therefore, tends to roll in its progresstowards the target, seeking to complete its manoeuvre in a plane of stability in roll.
The abscissa here is the initial angle of roll of the missile relative to the plane
in which the target is turning aside at constant rate. All the engagements start fromthe same range, with the missile and target approaching each other in line. It will
be seen that the miss distance depends markedly on the initial angle of roll, being a
maximum when the angle is near 0 or 900, and a minimum near 650, when the demand plane
is initially nearer to a stable plane and the missile, therefore, does not roll muchthroughout its manoeuvre. That this minimum is the least of all may be explained bythe fact that, as was noted earlier, the stability in pitch is least in the 450 planeand, consequently, the response is more rapid in this neighbourhood. The asymmetry
of the curve is due to the fact that the dish gimbal system was necessarily asymmetric
since we assumed an outer and inner gimbal at right angles. The curve is also somewhat
idealised, since the end points correspond to single plane manoeuvres in planes of
unstable roll equilibrium.
Curve 2 shows the effect of suppressing the roll, i.e. of fitting a perfect systemof roll position stabilisation in place of the roll rate system of Curve 1. Naturally,
the two curves have the same ordinates at roll angles of 0 and 900 and cross near 450 .
The reduction in miss distance at intermediate angles in this second case is very
marked.
Curve 3 shows the effect of linearising the aerodynamics in pitch and yaw while
retaining the cross-coupling in roll. This, and Curve 4, are somewhat artificial forthe reason given above in discussing the single plane case. In particular, the
assumption made in linearising the pitching moment resulted in the static stability
in the 450 plane, and neighbouring planes, being somewhat greater than it was in fact,
and it is probably for this reason that the miss distance recorded is somewhat greater
than in the fully non-linear case. It is clear from the curve that the linearisingprocess has had almost as much effect on the miss distance as the suppression of the
roll, the additional effect of which is shown by Curve 4 to be relatively small.
This last curve shows the result which would be obtained by assuming the aerodynamicsof the missile to be completely linear. This is an assumption which is frequentlymade in theoretical analyses and in simulations, where investigations are oftenflirther restricted to single plane studies. It will be clear from Figure 13 that
such assumptions may lead to erroneous conclusions. It should be pointed out, however,that although the curves of Figure 13 differ so much, the r.m.s. values of the missdistance, averaged in each case over the whole range of the roll angle, do not differgreatly. Since the assumed manoeuvre is somewhat artificial, it might be argued
from this that aerodynamic non-linearities and cross-couplings are not of great
importance and that there is little to choose between the different systems of roll
control. It is important to note, however, that the curves of Figure 13 were allobtained for one value of the initial range and it does not follow that the effects
14
must be the same at other ranges. The whole problem is extremely complicated and somany factors are involved that it is almost impossible to reach general conclusions.
The type of manoeuvre discussed above is illustrated in Figure 14.
9. EFFECT OF ANGULAR NOISE ON MISS DISTANCE
The homing system which produced the results of Figures 11-13 was not afflicted by
noise, radome aberration, or any other of the troubles which beset real missiles. It
would have taken too long to investigate all these effects on the simulator, but wedid consider it worthwhile to obtain an estimate of the probable effect of angularnoise on the results. The conclusions are indicated in Figure 15. These results were
obtained digitally. It is fairly clear that the general trend of the curve without
noise is unaltered.
10. PROBLEMS OF A LARGE SIMULATION
10.1 Simulator Errors
By the time the simulation had reached the above stage, it had grown to an assemblyof some 500 amplifier units and had absorbed most of the available capacity and
special facilities of TRIDAC. The problem of keeping the computer 'on the rails' had bythen grown formidable, and a considerable portion of the working time had to be spentin tracing and eradicating intermittent faults in the equipment. This was a
difficult matter on account of the many loops in the system, and here the numerouscomplete and partial digital solutions mentioned earlier proved most useful. There
were certain shortcomings in the equipment, however, which we could not overcome,notably the limited resolving power of the sine and cosine potentiometers in the axis
transformations, and this feature was most serious when it came to generating the
sight line error angles in dish axes. Although the computer behaved remarkably well
on the whole, we were driven to rely more and more on digital methods where high
accuracy was essential, and, indeed, although TRIDAC confirmed the general trends ofall the curves, the results shown in Figure 11 and 13 were, in fact. obtained
digitally.
10.2 Comparison of Analogue and Digital Results in Single Plane
Figure 16 compares the digital and TRIDAC solutions in the case of single plane,
homing. The initial range was varied from 20,000 ft downwards. It will be seen that
the simulator confirms the general trend of the digital results, though there is somescatter in the experimental points. Four configurations were investigated, namely,target turning upwards or downwards in the vertical plane or to port or starboard in
the horizontal plane. Hence, four values of miss distance were obtained for each
value of the initial range. At values of the latter exceeding 10,000 ft, all four
results were valid, but at lesser ranges, some amplifiers in the system were overloaded.
The trouble could have been overcome by rescaling, but this was not considered worth-
while. Since the overload limit points of the amplifiers are higher forpositive thanfor negative output voltages, it was possible to obtain only two valid measurementsfor initial ranges less than 10,000 ft.
15
10.3 Comparison of Analogue and Digital Results in Three Dimensions
Figure 17 compares the digital and analogue results in the case of three-dimensionalhoming from a constant initial range. The digital result was given previously, on adifferent scale, in Figure 13, Curve 1. There are four results at each value ofinitial roll angle, namely, the two obtained by simulating the manoeuvre and its mirrorimage in the yaw plane, and the two obtained by repeating this process in the pitchplane. The discrepancies between the points so obtained are probably due to slightasymmetries and inaccuracies in the axis transformations which could not be removed.It is interesting to note, however, that over much of the range of roll angle, themean of the four solutions at each angle is close to the digital curve.
CONCLUSIONS
Aerodynamic non-linearities and cross-couplings may have important effects on theresponse of a guided missile, but these effects may be greatly modified if the missileis provided with a suitable autopilot.
A cruciform missile, which is roll-rate stabilised, has preferred planes ofmanoeuvre and will, in general, attempt to manoeuvre in these planes when homing. Therolling motion may affect miss distance adversely.
Within the limited field explored, roll-position stabilisation appeared to bepreferable to roll-rate stabilisation, although the r.m.s. miss distances, averagedover all angles of roll, may not differ widely in the two cases.
Incorrect and misleading results may be obtained from incomplete simulations, forexample, those which represent the aerodynamics by linear approximations, or which arecapable of simulating only single plane manoeuvres.
On the other hand, a large and complex simulation is fraught with many difficulties.Digital check solutions are indispensable in such cases.
REFERENCES
1. Spearman 7TIDAC, A Large Analogue Computing Machine. Proc. Inst.et alii Elect. Engrs., Vol. 103, Pt. 13, 1956, p. 375
2. Gait, J.J., ThIDAC. A Research Flight Simulator. Elect. Eng. Vol. 28,Nutter, J.C. 1956, pp.368 and 430.
16
1*0)
+ +-
+UE)SN0)
+.
CZ~
21
FUNCTIONS OF
M _\_ _ __1_ _ _ _ _ _ _ _ _ jI I I
MACH NUMBER INTERPOLATION
4 I~ SEV REOLE MIIEFUNCTIONS OF 1
FUNCTOS OFII4 CS TAIL C0NTRI BUTIONS
*SERVO RESOLVER]
WING AND BODYFORCE AND MOMENT
TOTAL FORCE ANDMOMENT COEFFICIENTS
SCALE FACTORS
ii P 0- -ALTITUDE VARIATIONpREF
m m C B A
Fig.7 Generation Of aerodynamic forces and moments
22
6
1-e
wIz
_ _ _ _ _ _3
_ _ _ _ _ ___ 'a
Ll 0
- -IQ
CY C)--4
I 0EM4
23
I I 1,
-A-
/ \, ' i
t0
, II
00
0
0"-4, , , '*e
I I o__' I I 0
,_ _..-- I
-d \ ia
24
Z 0
w-
wUJ w,
00
J fk4W & 0
w0w Q0
W.
z CL r w0tu4
bic/i LoX (D u
w~ Tk --
25
LINEAR AERO
miss NON - LINEAR AERODIST ANNCE
OL INITIAL RANqE
Fig. 11 Miss distance vs initial range in single plane homing
_____ _____ - 9 LIMIT
CMM _______
0 00020,000 30,000INITIAL RAN~qE RO
Fig. 12 Maximum lateral acceleration vs initial rapigd in single Plane homing
26
NON-LINEAR AERO
-- -LINEAR ACRO
DISTANCEWITH ROLLINCT
INITIAL ROLL ANCTLE
0 30* 6.00 eo*
Fig. 13 Miss distance vs initial angle of roll in 3-dimensional homing; initialrange fixed
INITIAL ROLLANqLE (o
Fig. 14 Diagrammatic representation of homing in three dimensions
27
-N0ISE FREE
DISTANCE
0 INTA OL NZE9
Fi.1 Efec ofaglrnieo isdsacpn3dmnilhmn
PR 0 0 U
Fig. 15 Effect ; of anuloie nm digtlrsulce in 3-imgensionl homing
28
xx
DISTANCExx TRIDAC,
xx x
0 INITIAL ROLL AN~qLE 90
Fig. 17 Comparison of analogue and digital results in three dimensions
DISCUSSION
B.E. Amsler (U.S.A.): I wish to agree strongly with Mr. Brown's conclusion that
aerodynamic non-linearities must be considered in the control design. Too often these
terms are ignored only in the interest of mathematical siinplicity even though they are
critically important. However, I do not feel a large 'all up' dynamic simulation is
required to predict flight performance, at least in the case of cruciform configurations.
Specifically, if control parameters are selected with a view to overpowering or
operating satisfactorily in conjunction with the complete range of non-linear aerodynamic
parameters and if interchannel coupling effects are considered and, where necessary,
parameters are chosen to ensure no cross-channel resonances (e.g. path spiraling), then
a single plane simulation has been found to give a very good prediction of flight
performance.
Let me however point out that fundamental non-linear effects (e.g. variations in
slope of CN vs a. with increasing a ) which cannnot be compensated out by special
control parameter selection must be included in performance studies. The significant
point is that interchannel coupling has not in this case been found to be important and
variations between manoeuver planes are made insignificant.
Reply by W.S. Brown: I would agree with Mr. Amsler to the extent of saying that a large
dynamic simulation is not desirable if one can avoid it; for the time required to
construct a complex simulation and keep it running smoothly may easily become
prohibitive. In missile design, where an approximate answer is required quickly, as
for example, when seeking to optimise the control system, a single plane simulation is
probably adequate as a means of predicting performance.
However, in the work described, we were not attempting to design a missile. We
merely adopted an existing arrangement and sought to reproduce its features as
accurately as possible in order to find out whether aerodynamic non-linearities and
cross-couplings, which were known to be present, affected performance significantly.
Our conclusions, as outlined in the paper, were that the effects could be important in
certain circumstances, for example, in a cruciform missile free to roll, which a single
plane simulation would not have revealed.
L.T. Prince (U.S.A.): It has been our experience at Honeywell that stability problem
areas resulting from non-linear aerodynamic and kinematic factors can ba predicted with
linear uncoupled analyses of airplanes. These simpler techniques are of great value
in control system design studies and are highly recommended. They involve an examina-
tion of control system performance at the worst expected values of stability derivatives
that may be encountered during maneuvering flight using fixed values of the derivatives.
A. Lightbody (U.K.): I would like to say that we at A.W.A. have carried out a three
dimensional analogue simulation of a Seaslug missile which is about the same size as that
of Mr. Browne's on TRIDAC. Ours uses electronic multipliers and runs on a 1 1 time
scale.
A-i
We have found many of Mr. Brown's conclusions borne out on our simulator; forexample, we also have found that for comprehensive simulation of non-linear aerodyna-mics the polynominal method becomes too expensive and complex in multiplier. We alsouse diode function generators.
However, the main point I have to make is that complex three-dimensionalsimulations such as have been described have two main functions: 1) as a tool for thestudy of the overall guided missiles system; 2) as a facility for performing pre andpost flight simulations. Flight trials can be planned better using such a facility,and the causes of flight failures can be thoroughly investigated after the trial.
I would like to ask Mr. Brown whether he has used TRIDAC in this capacity, i.e., for1 ,1 flight trial simulation. This would seem to me to be the logical outcome ofthe comprehensive work carried out on TRIDAC.
Reply by W.S. Brown: Our investigation was intended to be a piece of fundamentalresearch into aerodynamic effects, and was originally planned to include a programmeof flight trials. For various reasons, the flight work was abandoned. We did, how-ever, simulate a few of the earlier flights of the test vehicle, and obtained generalalthough not precise, agreement with the telemetered data. Certain discrepancies,however, led us to doubt whether a detailed simulation, such as we had constructed,would be justified in missile design and development work.
L.G. Evans (U.K.): I would like to support Mr. Lightbody' s point that the pay-offwhich justifies these rather extensive computing facilities is the saving in flighttrials that they allow, due to the improved ability to understand the meaning offlight results. It is therefore necessary to tie the results of the model back to trialresults, and here I would support Mr. Amsler' s point.
However, in my experience the designer may well be faced with a severe disagreementbetween the model and trial results, and in this case something more than a simpleoverlay system is necessary to help the designer to track down the source oZ errors inthe model.
S.H. Scher (U.S.): I was glad to hear the comments of Mr. Amsler. It indicates anawakening of people to a problem which does exist.
There are times when you can get away with two-dimensional studies using linearizedaerodynamics and simplified equations of motion. But, as I mentioned previously inmy comments on post-stall gyrations, there are other times when a study of the motionrequires that you have enough inputs to completely study the motion.
As Mr. Brown said, when one gets specific and attempts to deal with motions whichare completely beyond the realm of normal linear inputs and non-cross coupled inertiafactors, one encounters much difficulty.
The inputs needed depend on the motion you are trying to study and, as I see it, youeither try to predict what will happen or what you get in attempting to explain whatdid happen. In either case, you need a complete set of inputs.
A-i
ADDENDUM
AGARD SPECIALISTS' MEETING
on
STABILITY AND CONTROL
Complete List of Papers Presented
Following is a list of the titles and authors of the 41 papers presented at the
Stability and Control Meeting held in Brussels in April, 1960, together with the
AGARD Report number covering the publication of each. paper.
INTRODUCTORY PAPERS
The Aeroplane Designer's Approach to Stability and Control, by
G.H.Lee (United Kingdom) .. .. .. .. .. .. .. Report 334
The Missile Designer's Approach to Stability and Control Problems, by
M.W.Hunter and J.W.Hindes (United States) .. .. .. .. Report 335
DESIGN REQUIREMENTS
Flying Qualities Requirements for United States Navy and Air Force
Aircraft, by W.Koven and R.Wasicko (United States) .. .. .. Report 336
Design Aims for Stability and Control of Piloted Aircraft, byH.J.Allwright (United Kingdom) .. .. . .. .. Report 337
Design Criteria for Missiles, by L.G.Evans (United Kingdom) .. Report 338
AERODYNAMIC DERIVATIVES
State of the Art of Estimation of Derivatives, by H.H.B.M.Thomas
(United Kingdom) .. .. .. .. .. .. .. .. Report 339
The Estimation of Oscillatory Wing and Control Derivatives, by
W.E.A.Acum and H.C.Garner (United Kingdom) .. .. . .. Report 340
Current Progress in the Estimation of Stability Derivatives, by
L.V.Malthan and D E.Hoak (United States) .. .. .. .. Report 341
Calculation of Non-Linear Aerodynamic Stability Derivatives of
Aeroplanes, by K.Gersten (Germany) .. .. .. .. .. Report 342
Estimation of Rotary Stability Derivatives at Subsonic and TransonicSpeeds, by M.Tobak and H.C.Lessing (United States) Report 343
Calcul par Analogie Rhoelectrique des Dirivees Aeredynamniques d'uneAile d'Ehvergure Finie, by M.Enselme and M.O.Aguesse (France) Report 344
A Method of Accurately Measuring Dynamic Stability Derivatives inTransonic and Supersonic Wind Tunnels, by H.G.Wiley and A.L.Braslow(United States) Report 345
Mesure des D~rivges Agrodynamiques en Soufflerie et en Vol, byM.Scherer and P.Mathe (France) Report 346
Static and Dynamic Stability of Blunt Bodies, by H.C.DuBose(United States) Report 347
AEROELASTIC EFFECTS
Effects of Aeroelasticity on the Stability and Control Characteristicsof Airplanes, by H.L.Runyan, K.G.Pratt and F.V.Bennett (United States) Report 348
The Influence of Structural Elasticity on the Stability of Airplanesand Multistage Missiles, by L.T.Prince (United States) Report 349
Discussion de deux Mithodes d'Etude d'un Mouvement d'un MissileFlexible, by M.Bismut and C.Beatrix (France) Report 350
The Influence of Aeroelasticity on the Longitudinal Stability of aSwept-Wing Subsonic Transport, by C.M.Kalkman (Netherlands) Report 351
Some Static Aeroelastic Considerations of Slender Aircraft, byG.J.Hancock (United Kingdom) Report 352
COUPLING PHENOMENA
Pitch-Yaw-Roll Coupling, by L.L.Cronvich and B.E.Amsler (United States) Report 353
Application du Calculateur Analogique & l'Etude du Couplage desMouvements Longitudineaux et Transversaux d'un Avion, by F.C.Haus(Belgium) Report 354
Influence of Deflection of the Control Surfaces on the Free-FlightBehaviour of an Aeroplane: A Contribution to Non-Linear StabilityTheory, by X.Hafer (Germany) .. .Report 355
STABILITY AND CONTROL AT HIGH LIFT
Low-Speed Stalling Characteristics, by J.C.Wimpenny (United Kingdom) Report 356
Some Low-Speed Problems of High-Speed Aircraft, by A. Spence andD.Lean (United Kingdom) .. .. .. .. .. .. .. Report 357
Factors Limiting the Landing Approach Speed of an Airplane fromthe Viewpoint of a Pilot, by R.C.Innis (United States) .. .. Report 358
Post-Stall Gyrations and Their Study on a Digital Computer, byS.H.Scher (United States) .. .. .. .. .. .. .. Report 359
THE APPLICATION OF SERVO-MECHANISMS
The Place of Servo-Mechanisms in the Des'ign of Aircraft with GoodFlight Characteristics, by K.H.Doetsch (United Kingdom) .. .. Report 360
Effects of Servo-Mechanism Characteristics on Aircraft Stabilityand Control, by P.A.Gaynor (United States) .. .. .. .. Report 361
Les Commandes de Vol Considir'es comme Formant un Syst~me Asservi,by J.Grdmont (France) .. .. .. .. .. .. .. Report 362
Determination of Suitable Aircraft Response as Produced by AutomaticControl Mechanisms, by E.Mewes (Germany) .. .. .. .. Report 363
An Approach to the Control of Statically Unstable Manned FlightVehicles, by M.Dublin (United States) .. .. .. .. .. Report 364
THE USE OF SIMULATORS
The Use of Piloted Flight Simulators in General Research, byG.A.Rathert, Jr., B.Y.Creer and MSadoff (United States) .. .. Report 365
Simulation in Modern Aero-Space Vehicle Design, by C.B.Westbrook(United States) .. .. .. .. .. .. .. .. Report 366
Mathematical Models for Missiles, by W.S.Brown and D.I. Paddison(United Kingdom) .. .. .. .. .. .. .. .. Report 367
In-Flight Simulation - Theory and Application, by E.A.Kidd, G.Bulland R.P.Harper, Jr. (United States) .. .. .. .. .. Report 368
DEVELOPMENT TECHNIQUES
Application of Analytical Techniques to Flight Evaluations inCritical Control Areas, by J.Weil (United States) .. .. .. Report 369
Investigation on the Improvement of Longitudinal Stability of a JetAircraft by the Use of a Pitch-Damper, by R.Mautino (Italy) .. Report 370
Methodes Utilisees pour la Mise aou Point de l'Avion Bre'guet 940 aAiles Soufflees, by G. de Richemont (France) Report 371
TURBULENCE AND RANDOM DISTURBANCES
Theory of the Flight of Airplanes in Isotropic Turbulence; Reviewand Extension, by B.Etkin (Canada) Report 372
The Possible Effects of Atmospheric Turbulence on the Design ofAircraft Control Systems, by J.K.Zbrozek (United Kingdom) Report 373
L'Optimisation Statistique du Guidage par Alignement d'.un EnginAutopropuls' en Pr~sence de Bruit, by P.LePivre (France) Report 374
ADVISORY GROUP FOR AERONAUTICAL RESEARCH AND DEVELOPMENTOrganisation du Trait6 de I'Atlantique Nord
64, rue de Varenne - Paris 7 eme
August 1961
AGARD Distribution List
Category II: "Not for Sale" Publications
COUNTRY ADDRESS NO. OF COPIES
BELGIUM Centre National d'Etudes et 25de Recherches Aeronautiques11, rue d'Egmont, Bruxelles
CANADA T.I.L.-Ministry of Aviation 30Leysdown RoadMottinghamLondon, S.E.9Attn: Mr. F. G. Waite
DENMARK Danish Defence Research Board 10Risterbrogades KaserneCopenhagenO
FRANCE ONERA (Direction) 9025, avenue de la Division LeclercCh~tillon-sous-Bagneux, (Seine)
GERMANY Deutsche Gesellschaft fur Flugwissen- 90schaften
Zentralstelle fur Luftfahrtdokunmenta-tion una Information
Munchen 64, FlughafenAttn: Dr. H. J. Rautenberg
GREECE Greek Nat. Def. Gen. Staff 10B. MEOAthens
ICELAND Director of Aviation 3c/o FlugradReykjavik
ITALY Ufficio del Generale Ispettore del Genio 85Aeronautico
Ministero Difesa-AeronauticaRoma
LUXEMBOURG obtainable through Belgium
NETHERLANDS Netherlands Delegation to AGARD 35*Michiel de Ruyterweg, 10Delft
NORWAY Norway Defence Research Establishment 22KJeller per LillestromAttn: Mr. 0. Blichner
PORTUGAL Direccao de Servico de Material da Forca 5Aerea
Rua da Escola Politecnica, 42LisboaAttn: Coronel Joao A. de Almeida Viana
TURKEY Ministry of National Defence 30AnkaraAttn: AGARD National Delegate
UNITED KINGDOM T.I.L. 120Ministry of AviationLeysdown RoadMottinghamLondon, S.E.9Attn: Mr. F. G. Waite
UNITED STATES National Aeronautics and Space 400Administration
Langley Research CenterLangley Field, VirginiaAttn: Report Distribution and
Storage Unit
AGARD 64, rue de Varenne 45Paris 7 eme
*Netherlands meets demands of SHAPE Air Defence Technical Centre.
AGL (1) 7-2-IM-90U
