A Multi-Degree-of-Freedom Rig for the Wind Tunnel

Determination of Dynamic Data

J. Pattinson∗ and M. H. Lowenberg†

University of Bristol, Bristol, BS8 1TR England, United Kingdom

and

M. G. Goman‡

De Montfort University, Leicester, LE1 9BH England, United Kingdom

A wind-tunnel rig to provide data from manoeuvring aircraft is developed. The “Ma-noeuvre rig” is a horizontal pendulum type rig that is capable of large amplitude motionsin 5 degrees of freedom. This enables a large set of conventional and extreme aircraftmanoeuvres to be performed in the controlled environment of a wind tunnel. A mixeddifferential-algebraic mathematical model, with terms accounting for friction in the gimbalsis developed. A pilot rig has been fabricated and a preliminary experimental programmeundertaken. To illustrate some of the capabilities of the rig, a selection of 1,2,3 and 4degree-of-freedom experimental results is given. The procedure for the fitting of the fric-tion model is presented and a time-domain based filter-error method is used to fit a linearaerodynamic model to the longitudinal cases.

Nomenclature

A,G transformation matrixAR wing aspect ratioc wing mean aerodynamic chordC constraint vectorC force coefficientF force vector in body coordinatesFc Coulomb forceFs stiction forceg gravitational accelerationI moment of inertia tensorlcga length from aircraft yaw gimbal to aircraft c.g.lcgs length from 3-DOF gimbal to rig c.g.lf length from 3-DOF gimbal to aircraft yaw gimballr length from 3-DOF gimbal to compensator c.g.M mass matrixm component of mass matrixnb number of bodies in a systemnj number of joints in the rig systemq generalised system coordinate vectorQc conservative force vectorQe external force vector in generalised coordinatesQv quadratic velocity vector∗PhD Student, Department of Aerospace Engineering.†Senior Lecturer, Department of Aerospace Engineering, Senior Member AIAA.‡Professor, Department of Engineering, Senior Member AIAA.

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AIAA Atmospheric Flight Mechanics Conference10 - 13 August 2009, Chicago, Illinois

AIAA 2009-5727

Copyright © 2009 by The University of Bristol. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

Qnc non-conservative force vectorq joint angular positionR position coordinates of a bodyS Stribeck component of frictiont timeVt wind tunnel velocityx longitudinal coordinatey lateral coordinateZ average bristle deflectionz vertical coordinate

Symbolsα angle of attackαf transition coefficientΦ roll angleλ vector of Lagrange multipliersΘ pitch angleθ quaternion componentσ0 stiffness coefficientσ1 damping coefficientσ2 friction coefficientθ attitude of a bodyτf friction forceω angular velocity vectorΨ yaw angle

Subscriptsa aircraftD drag forcei indexj rotational joint indexL lift forceM pitching momentR positional componentr compensators armθ rotational component

Conventions• time derivative• body coordinates•T matrix transpose

I. Introduction

Many methods to determine the flight characteristics of a new aircraft configuration exist. These rangefrom empirical techniques, CFD analyses, wind tunnel experimentation and finally flight-test itself. To

date current aircraft development involves a combination of all of these methods.1

Wind tunnel techniques for deriving flight mechanics data for controllability and dynamic stability eval-uation are long established. A review of existing dynamic rig mechanisms can be found in Ref. 2. Themajority of small- and large-amplitude motion rigs are basic single degree-of-freedom(DOF) mechanisms,moving in heave, pitch or roll. For multiple DOF motion a wide range of various mechanisms with two andthree degrees of freedom have been used. As yet, dynamic tests using multiple-degree-of-freedom dynamicrigs have not become part of industrial practise. However their potential for the reliable prediction of dy-namic behaviour, the separation of dynamic derivatives, and the validation of CFD analyses such as Ref. 3,means that they cannot be ignored and new research into these devices continues.4–6 The wind tunnel test

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rig presented in this paper is a novel type of dynamic rig for the simulation of a variety of aircraft manoeu-vres. It is designed to implement important degrees of freedom of aircraft motion which affect the onset ofaerodynamic loads, such as heave and pitch, sway and yaw, and the velocity vector roll. A non-exhaustivelist of potential manoeuvres is presented in Table 1. These manoeuvres should allow the identification ofaircraft dynamic behaviour while excluding the effect of rig dynamics and interference. Furthermore datacould be obtained to populate more advanced aircraft models, that account for time-dependant dynamics,such as in Ref. 7.

Table 1. Manoeuvres permitted by the dynamic rig

ManoeuvreFree heave-pitch and sway-yaw oscillationsForced heave-pitch and sway-yaw oscillationsPull-up/push-overBank to bankVelocity vector rollHigh incidence departures

Compensator

3-DOF gimbal

2-DOF gimbal

Model

Support

Arm

Figure 1. The manoeuvre rig. The labels indicate the five available degrees of freedom.

The manoeuvre rig is illustrated in Figure 1. It consists of a flight vehicle model mounted via a two degree-of-freedom gimbal to an arm. This arm is connected via a three degree-of-freedom gimbal to the supportingstructure. To generate heave and sway aircraft model motions, compensate for the mass, inertia aerodynamiceffects of the supporting arm an aerodynamic compensation device is mounted on the downstream end ofthe arm. The weight of this compensator also means that the pendulum can be statically balanced in thehorizontal position. The 2-DOF gimbal at the model together with the 3-DOF gimbal at the support providethe model with up to 5 degrees of freedom. These axes of motion are indicated in the illustration (Figure 1).All axes of motion are lockable for those cases in which all 5 available degrees of freedom are not desired.

For use on this rig the aircraft model is required to have its own actuated control surfaces and shouldideally be dynamically scaled. This is so that it can be ‘flown’ on the arm in a manner that reflects thebehaviour of the full scale aircraft as closely as possible.

The motions of this type of rig can be highly non-linear and an advanced mathematical description isrequired to separate out the effects of the rig dynamics. To this end a mathematical model is described. It is

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based on the Lagrange formulation with constraints which results in a differential-algebraic equation system.A significant addition to this model are terms that account for friction losses in the gimbals. This frictionmodel is also described in this paper. A minor extension has also been made to account for the cases whenthe centre-of-gravity (c.g.) of the flight vehicle model does not lie on the pivot point.

A pilot rig has been commissioned to study the concept. A CAD representation of the rig can be seenin Figure 1. Large amplitude motions are permitted in all axes with roll (φ) and model-yaw (ψa) unlimited.For the purposes of this research an existing approximate BAe Hawk aircraft model is used.8 The 5 axes ofmotion are instrumented and a wireless system is used to transfer data to and from the aircraft model andthe compensator.

The time-domain based filter-error method9 has been used to fit a linear aerodynamic model to theresults from two of the rig’s longitudinal configurations: model-pitch only and 2-DOF model-pitch and heave(rig-pitch). Also presented in this paper are rig capability demonstrations in roll, a 3-DOF and a 4-DOFcase. This latter case uses all available axes but for heave.

II. Mathematical model

For the purposes of design, simulation and parameter estimation a mathematical model of the rig isdeveloped. The equations are derived using a Lagrange formulation for multibody systems as described byRef 10. As is necessary at this stage rather crude or ideal approximations of the various components of therig are used. Some of these assumptions are listed here:

• Flexibility of the arm or any other part of the rig is not taken into account.

• The effect of aerodynamic forces on the arm is not modelled.

For a system of bodies i = 1, 2, ..., nb the following set of differential-algebraic equations can be written:10

M iqi +CTqiλ = Qi

e +Qiv, i = 1, 2, ..., nb (1)

C(qi, t) = 0

The system generalised coordinates qi are given by

qi =

[Ri

θi

](2)

where Ri is the position vector and θi is the attitude representation of body i. In this work, quaternions areused to represent orientations. The derivative of the generalised coordinates with respect to time t is denotedq. To account for the fact that, in general, the generalised coordinates q are not independent, constraintsC are introduced using the method of Lagrange multipliers. These multipliers are denoted λ. Cq is theconstraint Jacobian matrix with respect to the system generalised coordinates. Assuming the case in whichthe body reference is attached to the c.g. of the system, the mass matrix M i becomes

M i =

[mi

RR 00 mi

θθ

](3)

where

miRR =

mi 0 00 mi 00 0 mi

(4)

andmi

θθ = GiT

IθθGi

(5)

where the inertia tensor of a rigid body defined about its centre of gravity is denoted Iθθ. Gi

is a trans-formation matrix that depends on the attitude representation used. Forces external to the system are

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expressed via the external force vector Qie. This vector is defined as the sum of the external conservative

and non-conservative force vectors denoted by subscripts c and nc respectively,

Qie = Qi

nc +Qic (6)

Qiv is a term that results from differentiating the system kinetic energy with respect to time and is defined

as

Qiv =

[0T − 2ωi

T

Ii

θθGi]T

(7)

where ωi are the angular velocities about the local coordinate axes. This can be calculated by

ωi = Giθi

(8)

The rig is divided into three separate bodies denoted a, s and r; the aircraft, the arm and the compensatorrespectively. The coordinate systems chosen for each body are represented in Figure 2. The generalised

x

y

z

x

y

z

x

y

z

r

g

g

g

s

s

s

a

a

a

Figure 2. Rig equation coordinate systems

coordinates chosen are:

q =

Ra

θa

Rs

θs

Rr

(9)

The mass matrix then becomes

M =

ma

RR

maθθ

msRR

msθθ

mrRR

(10)

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Note that in this case the inertia of the compensator must be included in the msθθ term. The quadratic

velocity vector Qv is simply

Qv =

0[−2ωa

T

IaθθGa]T

0[−2ωs

T

IsθθGs]T

0

(11)

Given the external forces F aR, F aθ , F rR, F rθ, mag, msg and mrg, the external force vector becomes

Qe =

AaF aR + [0 0 mag]T

GaF aθ

[0 0 msg]T

GsF rθ

AsF rR + [0 0 mrg]T

(12)

whereA is a coordinate transformation matrix the definition of which can be found in Ref. 10. The constraintmatrix C contains 12 equations in this setup. These are: the two equations that fix the unit norm of thequaternions,

θa · θa = 1 (13)θs · θs = 1 (14)

the nine equations specifying the fact that the arm is connected to the pivot point, the aircraft and thecompensator and that the aircraft c.g. is not located at the pivot:

Ra = As[lf 0 0]T +Aa[lcgax 0 lcgaz ]T (15)

Rs = As[lcgsx 0 lcgsz ]T (16)

Ra = As[−lr 0 0]T (17)

and the one equation that represents the yaw and pitch gimbal of the aircraft:

AsTAa(3, 2) = 0 (18)

Equation (18) sets the roll angle of the aircraft about the x-axis of the strut to zero.In the pilot rig described in the following section, it has been found that friction is not negligible. To

account for this the mathematical model of friction of Ref. 11 is used. This model is simple to implementand captures most experimentally observed friction phenomena.12 The model takes the following form:

dZjdt

= qj −σ0|qj |S(qj)

Z (19)

S(qj) = Fc + (Fs − Fc) e−αf |qj | (20)

τfi = σ1e−

“qjvd

”2 dZjdt

+ σ0z + σ2qj (21)

The pre-sliding displacement friction phenomenon is captured by using the bristle deflection theory of friction.In this model, the variable Z is used to track average bristle deflections. In this pre-sliding mode the frictionalforce acts like a spring. One can imagine this behaviour as the interface between two brushes. On applyinga force the bristles deflect until some critical value when sliding occurs. Before this point, if the force wasremoved the two brushes would return to their original positions. The average bristle stiffness is denotedσ0 and bristle damping σ1. The breakaway force, more properly known as the stiction force, is representedby Fs. This force is combined with the Coulomb force Fc, together with αf into a scalar function, S, whichmodels the Stribeck effect. This effect is the observed dip in friction force at low velocities. The coefficientαf controls the shape of the transition region between pre-sliding and sliding. Between lubricated surfaces

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at high interface velocities friction is largely determined by viscous forces in the lubricant. This componentof friction is denoted σ2. The final parameter, vd, permits the reduction of the influence of bristle deflectionfor higher rotational velocities and ensures that the necessary conditions for passivity of the model aresatisfied.13 The total friction force, which is simply the sum of all these components, is denoted τf . For eachof the five joints in the system a separate equation tracking the average bristle deflection is required. Thefriction model is introduced into equation (1) as

M iqi +CTqiλ = Qi

e +Qiv, i = 1, 2, ..., nb (22)

C(qi, t) = 0dZjdt

= qj −σ0|qj |S(qj)

Z, j = 1, 2, ..., nj

and by including Fθj = Fθj + τfj in the external force vector Qe. In this case nj is the number of joints inthe system.

The equation system represented by Equation (22) is an index 3 differential-algebraic equation (DAE). Itis an initial value problem that can be solved numerically. A multi-step backward difference formula (BDF) al-gorithm was chosen, in particular the Modified-Extended-Backward-Differentiation-Formula-Implicit (MEBDFI)code of T.J. Abdulla and J. Casha was used. The code uses the modified-extended multi-step BDF methodsof Cash as detailed in Ref. 14 and Ref. 15.

III. Experimental setup

A pilot rig for the examination of the manoeuvre rig concept has been developed. In Figure 3 the rig canbe seen as mounted in the University of Bristol’s 7p × 5p closed-section low-speed wind tunnel.

Figure 3. The pilot manoeuvre rig mounted in the University of Bristol’s 7p × 5p closed section wind tunnel.

The BAe Hawk aircraft model used is a 1/16th approximate scale model with a wing span of 612mm.It has 5 actuators on board providing actuation for a conventional set of control surfaces, ailerons, rudder,elevator. The elevator is powered by two individual actuators which allows the left and right surfaces to bedeflected differentially, although it has not been used in this way.

The compensator has a cruciform configuration (Figure 1). Its four wings have a symmetric aerofoilprofile (NACA0018) and an individually actuated full-length 25% chord flap. The deflection of each of theseflaps is directly measured with a magnetic encoder. The total span of the compensator is 700mm. It wassized to singularly provide enough rolling moment to produce a steady roll of the rig at a wind speed of15m/s.

aThis can be freely obtained from http://www.ma.ic.ac.uk/ jcash/IVP software/itest/mebdfi.f (Retrieved 1 February 2009).

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The arm of the rig is made from a 30mm diameter steel tube. This was selected after a detailed trade-offstudy was completed. More detail on this arm and the reasons for selecting it can be found in Ref. 16.

The manoeuvre rig’s two gimbals are manufactured from aluminium and all the allowable axes of motionare instrumented with potentiometers. Internal stops in the 3-DOF gimbal prevent the aircraft model fromhitting the tunnel walls. When the potential exists for large amplitude motions a wire restraining system isattached from behind the compensator to the tunnel walls for extra protection.

Telemetry from- and control signals to the model and compensation device are provided wirelessly. Thisis necessary to allow unrestricted motion in arm roll and model-yaw and to avoid spurious damping causedby cables running across joints. It also allows control and monitoring of the rig in real-time and in particularthe potential for real-time feedback control of the rig, for the prototyping of control laws.

On board the Hawk and compensator a PIC microprocessor controls the moving surfaces and gathersdata from the 2-DOF gimbal in the case of the Hawk, and the four encoders in the case of the compensator.Using a microprocessor gives much flexibility to the system. It enables the recording of the data read timesand also the time of control inputs. When it comes to the data reduction stage this means that one canconsider the data independently of the wireless transmission delay and other processing delays. However,any real-time control of the system will still suffer from these delays.

A DSPACE real-time data acquisition system is used to gather data from the model and gimbals. Forthe cases presented in this work the measurement frequency is 80 Hz (The short period frequency for theHawk is approximately 1.2 Hz).

IV. Friction model parameter estimation

To reduce the computational burden during aerodynamic parameter estimation, the parameters of thefriction model are estimated in a wind-off condition following the techniques proposed in Ref. 17. The tech-nique involves applying a low frequency sinusoidal torque to each joint separately, to obtain the relationshipbetween the friction force and the joint velocity. The model’s seven parameters can then be obtained byexamining a graphical representation of this relationship. Here, the low frequency torque was achieved usinga system of springs and masses and the installed potentiometers were used to obtain the necessary data. Thesetup is shown in Figure 4(a). Due to the somewhat low-tech method used to generate the input, accuratedetermination of these parameters directly from the graph was impossible. Instead a two-step procedurewas followed: basic estimates were first obtained from the experimental friction-velocity graph and thenan optimisation technique was used to refine the parameters further to obtain the correct angular positiongraph. This was performed using several different spring stiffnesses and inertia combinations to give moreconfidence to the parameters. The identified parameters for the model-pitch gimbal are presented in Ta-ble 2. This resulting model displays acceptable behaviour and, as seen in Figure 4(b), the dissipative effectsof friction have been captured.

Table 2. Identified friction model parameters for the model pitch gimbal

Fs Fc αf vd σ0 σ1 σ2

1.95× 10−3 1.89× 10−3 95.6 0.10 9.37 4.88× 10−5 4.33× 10−4

V. Longitudinal dynamics estimation

To verify the manoeuvre rig concept it must be shown that useful data can be collected from the rig.A suitable starting case is a programme to identify a longitudinal model of the 1

16 scale Hawk aircraft.To do this the rig is initially configured in a one degree-of-freedom model-pitch mode, followed by a twodegree-of-freedom model-pitch and rig-pitch mode.

Two test cases, one for each configuration, are presented. The input to the model elevator is a 3-2-1-1 step input with a period of 0.28s. This is one third of the expected short period frequency of 1.2Hz.The amplitude of the input was chosen so that the angle of attack half amplitude was in the region ofapproximately 4 degrees. For the two-degree of freedom case, a doublet with a period of 1s was used as aninput to the compensator horizontal surfaces.

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(a) Friction characterisationsetup.

0 2 4 6 8 10 12 14 16-40

-30

-20

-10

0

10

20

30

40

Pitc

h jo

int a

ngle

(de

gre

es)

Time (seconds)

SimulatedExperimental

(b) Friction characterisation test on the model-pitch gimbal. Thefitted model is also plotted.

Figure 4.

The aerodynamic model to be fitted is of the form:

CL = CL0 + CLα(α− α0) + CLqqc

2Vt+ CLδeδe

CD = CD0 +1

eARπC2L

CM = CM0 + CMα(α− α0) + CMααc

2Vt+ CMqq

c

2Vt+ CMδeδe (23)

For both the Hawk and compensator, their respective centres of gravity are the reference points for all theirassociated forces and moments.

The equations of motion of the two cases can be derived from the 5-DOF mathematical model by addingmore constraints. More specifically, all terms involving the y-direction: translation and roll and yaw rotationsare forced to zero. The independent coordinates of the test cases presented can easily be identified andtherefore it is a relatively simple matter to substitute the constraints into the equation system to return anordinary differential equation.

The time-domain based filter-error method9 was used to estimate the parameters in the aerodynamicmodel given by Equation (23). It was used because it takes into account process and measurement noiseso it should provide better estimates of the models in the presence of noise due to turbulence. It washoped that this would lead to an improvement on the previous experience of a similar rig with the outputerror method.8 The data are smoothed using a robust locally-weighted scatter-plot smoother with a spanof 40 before performing parameter estimation. Additionally, the wind tunnel velocity was measured forthese tests and used as an input. On the Hawk model the actual deflections of the control surfaces are notmeasured and it has been found that the response of the servos is nonlinear.8 The servos have previouslybeen characterised18 and a simple model of the servo response derived. This model, containing delay and ratelimiting, was applied as a pre-processing step on the input data before it was used to estimate aerodynamicparameters.

The experimental traces and simulated values obtained from the 1-DOF pitch-only test are shown inFigure 5. As one can see a reasonable agreement is achieved. The coefficients obtained in this case arecompared to the previous estimates in Table 3. The previous estimates were derived from tests performedwith the same model but in a different wind tunnel and on a different test rig. Another setup difference isthe new wireless system which removes the spurious damping from the previous wired system but increasesthe pitch inertia. Friction was also neglected in the previous estimates. Given the differences and reasonableagreement achieved, the new estimates shown in Table 3 provide better evaluation of the aerodynamic modelparameters.

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0 2 4 6 8 10 12

0

5

10

Model P

itch (

deg)

0 2 4 6 8 10 12-40

-20

0

20

40

Model P

itch R

ate

(deg/s

)

Sim Exp

0 2 4 6 8 10 12

-1

-0.5

0

0.5

1

Ele

vato

r (d

eg)

Time (seconds)

1-DOF Model Pitch

Figure 5. 1-DOF wind tunnel test at 20 m/s.

A feature of this rig is the ability to estimate the longitudinal dynamics of the aircraft model that involveheave and pitch. This test is the result of a 3-2-1-1 step input to the Hawk model and a doublet input tothe compensator model. The traces from this test are shown in Figure 6. Once again a linear aerodynamicmodel has been fitted to the data and the same model used for the compensator. In this case the fit is lesssatisfactory. It is noticeable that the CM terms for the 1-DOF test are different to those of the 2-DOF testand the results for CLq and CMq are larger in magnitude than expected. The reason for this is unclear atthis stage and is under investigation. As reported by other experimenters,6 the Hawk model has a largeresponse in heave and although the inertia of the arm tends to suppress this somewhat, it results in a largeangle-of-attack at the compensator. This will cause the lift force from the compensator to fall off due tostall. A more detailed modelling of this effect is planned. The resulting coefficients are tabulated in Table 4together with the data that was previously obtained for the Hawk.

Table 3. Identified aerodynamic parameters from 1-DOF pitch-only test. Only the dominant parametersare shown.

Parameter Old Estimate New EstimateCM0 0.05 0.037CMα −0.75 −0.52

CMq + CMα −6.78 −4.36CMδe −1.26 −1.31

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0 5 10 15 20 25 30 35 40

-10

0

10

Model Pitch

0 5 10 15 20 25 30 35 40

-10

0

10

Rig Pitch

Sim Exp

0 5 10 15 20 25 30 35 40-40-200204060

Model Pitch

Rate

0 5 10 15 20 25 30 35 40

-50

0

50

Rig Pitch

Rate

0 5 10 15 20 25 30 35 400.51

1.52

2.5

Elevator

0 5 10 15 20 25 30 35 40

-5

0

5

Compensator

Elevator

Time (seconds)

2-DOF Model Pitch and Rig Pitch

Figure 6. 2-DOF model-pitch and rig-pitch (heave) wind tunnel test at 20 m/s. In this figure angles are measured indegrees and rates in degrees per second.

VI. Velocity vector roll

A major capability advantage of the proposed rig over other rigs of this type is the ability to perform avelocity-vector roll. Tests illustrating this capability have been performed. The traces from a velocity vectorroll demonstration are presented in Figure 7. In this particular case the compensator control surfaces areused to push the aircraft into a rolling motion and then fixed at a slightly lower deflection. A continualrolling motion results. The fluctuations in roll rate are due to the mass imbalance in the rig itself. This is afunction of the current rig design and is to be eliminated in future. As the c.g. of the Hawk model is belowthe pivot point, the model is also not properly balanced. This means that the model pitches to its stopsunder high roll rates.

VII. 3-DOF roll, pitch and yaw

Figure 8 shows the traces from a 3-DOF test. The three axes used are roll, model-pitch and -yaw. Thetraces show the response of the Hawk model subject to a long slow elevator input. Previously documentedlimit cycles in the region of 5◦ and above 12◦ angle of attack are clearly visible. What is also noted is thelarge yaw departure at the onset of the upper limit cycle. Previous tests have identified a loss of yaw controlat this position most likely due to the fuselage wake blanking the vertical tail.8 It must be noted that dueto the mass distribution of the rig, this yaw departure may not occur in a freely flying aircraft as it may bepreceded by a roll departure.

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Table 4. Identified aerodynamic parameters from the 2-DOF model- andrig-pitch test. Note that the previous estimate of CL derivatives wasderived from static tests.

Parameter Hawk Model CompensatorOld Estimate New Estimate New Estimate

CL0 0.14 0.52 0CLα 3.14 4.85 6.61CLq – 14.38 75.34CLδe – 1.39 1.694CD0 0.13 0.041 0.043CM0 0.05 0.063 0CMα −0.75 −0.57 −2.16

CMq + CMα −6.78 −21.58 −26.74CMδe −1.26 −3.25 −0.77α0 0.022 0.064 0

VIII. 4-DOF sway, roll, yaw and pitch

To achieve a bank-to-bank snake-like motion the rig is configured in a 4-DOF mode containing, roll, sway,model-pitch and -yaw. This should allow the estimation of some of the lateral-directional derivatives and aninvestigation of the dutch roll mode. A characteristic of these tests is a large amplitude, multiple-DOF limitcycle. With all surfaces set to zero deflection the rig and model are stable at non-zero roll, sway, pitch, andyaw angles. With the exception of model-pitch, the fact that these angles are non-zero is most likely due tomodel and rig misalignment. From this steady position, shown in the first 5 seconds of Figure 9, with allother surfaces unchanged, a 3-2-1-1 input is applied to the Hawk’s rudder. This leads to oscillations thatgrow into a larger limit cycle oscillation. The onset of this limit-cycle can also be triggered by an input tothe compensator yaw surfaces (not shown). This limit cycle is thought to be due to a rig-model couplingwith the offset Hawk model c.g. and not due to any particular aerodynamic instability with the Hawk model.More investigation is, however, needed.

IX. Conclusion

The proposed 5-DOF manoeuvre rig allows a large set of aircraft manoeuvres to be performed in thecontrolled environment of a wind tunnel. The rig mechanism aims to reproduce free-flight aircraft motionsas closely as possible within the confines of a horizontal wind tunnel and provide remote radio control anddata acquisition for the aircraft model. A mathematical model describing the dynamics of the rig has beenpresented together with modifications to account for friction in the gimbals. A pilot rig has been fabricatedand experiments undertaken in a variety of different rig configurations. A linear model of the longitudinaldynamics of the rig has been fitted to data from 1- and 2-DOF tests. A reasonable fit is achieved in bothcases. The rig’s capability for manoeuvres using more degrees of freedom is also demonstrated, in particularcontinuous velocity vector roll at high rotation rates. The 3-DOF and 4-DOF case presented uncoveredpreviously unseen but not unsuspected departure dynamics and limit cycle oscillations of the Hawk modeland rig system, which will be investigated in the future.

Acknowledgements

This work was sponsored by the Commonwealth Scholarship Commission (ZACS-2006-356).

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0 5 10 15 20 25

-6000

-4000

-2000

Roll

angle

0 5 10 15 20 25-600-400-200

0200

Roll

Rate

0 5 10 15 20 25-20

-10

0

10

Model Pitch

0 5 10 15 20 2514

16

18

Aile

ron

0 5 10 15 20 25

-30-20-10

0

Com

pensato

raile

ron

Time (seconds)

2-DOF Roll and Model Pitch

Figure 7. 2-DOF Roll and model-pitch wind tunnel test at 20 m/s. Angles are measured in degrees and rates indegrees per second. The Hawk elevator is kept steady during this test.

0 50 100 150 200

-10

0

10

20

An

gle

of

Att

ac

k

0 50 100 150 200

-10

0

10

Ya

w

0 50 100 150 200

-4

-2

0

2

Ro

ll

0 50 100 150 200

-5

0

5

10

15

Time (seconds)

Ele

va

tor

Slow Elevator Ramp 25m/s

An

gle

(d

eg

ree

s)

Figure 8. 3-DOF test at a wind tunnel speed of 20m/s. Two previously documented pitchlimit cycles19 are clearly evident.

Large amplitude limit-cycle

Small amplitude limit-cycle

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0 10 20 30 40 50 60

-10

0

10

Roll angle

0 10 20 30 40 50 60

02468

Model Pitch

0 10 20 30 40 50 60

0

10

20

30

Model Yaw

0 10 20 30 40 50 60

-10

0

10

Rig Sway

0 10 20 30 40 50 60-5

0

5

Rudder

0 10 20 30 40 50 60-5

0

5

Compensator

Rudder

Time (seconds)

4-DOF Result

Angle (degrees)

Figure 9. 4-DOF wind tunnel test at 15 m/s.

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References

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