M I N I S T R Y O F A V I A T I O N

R. & M. No. 3257

A E R O N A U T I C A L R E S E A R C H C O U N C I L

R E P O R T S A N D M E M O R A N D A

Measurements of the Direct Pitching-Moment Derivatives for Two-Dimensional Flow at

Subsonic and Supersonic Speeds and for a Wing of Aspect Ratio 4

at Subsonic Speeds By J. B. B R A T T , B.A., B.Sc., W. G. RAYMER, B.Sc. (Eng.)

and J. E. G. TOWNSEND

OF THE AERODYNAMICS DIVISION, N.P.L.

L O N D O N : H E R M A J E S T Y ' S S T A T I O N E R Y O F F I C E

1962

PRICE £1 2s. 6d. NET

Measurements Derivatives

Subsonic for

of the Direct Pitching-Moment for Two-Dimensional Flow at and Supersonic Speeds and

a Wing of Aspect Ratio 4 at Subsonic Speeds

By J. B. B~ATT, B.A. , B .Sc . , W. G. RAYMER, B.Sc. (Eng. ) and J. E. G. TOWNSEND

OF THE A E R O D Y N A M I C S DIVISION, N.P.L .

Reports and Memoranda No. 3257 *

January, ±959

Summary. Apparatus based on a self-excitation technique has been developed for the measurement of direct pitching-moment derivatives at high speeds, and has proved to function satisfactorily.

Measurements have been made at subsonic speeds on a two-dimensional RAE 104 aerofoil, both with and without spoilers, and on a rectangular wing of aspect ratio 4 with the same section; and at supersonic speeds (M = 1.42 and 1.61) on two-dimensional biconvex aerofoils of 7½ per cent and 5 per cent thickness.

Comparisons with theory are made and discussed.

1. Introduction. An account is given in this paper of oscillatory tests carried out in the

N.P.I . 9½ in. square High Speed Tunnel. The main purpose in designing these tests was to obtain

further information on the direct two-dimensional pitching-moment derivatives, both at subsonic

and supersonic speeds; at the same time, however, the choice of a self-excitation technique enabled

valuable experience to be gained in the application of this method to the measurement of these

derivatives at high speeds. Earlier high-speed measurements I at N.P.L. had employed a decaying oscillation technique and

were limited to very low values of the frequency parameter (0. 016 at M = 0.9) as a result of the high wind speeds involved. The attainment of higher values of the frequency parameter requires

still greater frequencies of oscillation, which in practice leads to difficulty in the measurement of the stiffness derivative by the method of decaying oscillations, since the aerodynamic stiffness tends to become very small compared with the elastic stiffness. The resulting small change in frequency due to wind loading, from which the derivative is determined, cannot easily be measured with sufficient accuracy from the limited number of cycles frequently available. This difficulty is overcome by employing a self-excitation technique in which damping moments are automatically

Previously issued as A.R.C. 20,714. Published with the permission of the Director, National Physical

Laboratory.

balanced by an electromechanical feedback system to give a sustained oscillation which may be allowed to continue for a sufficient length of time to give the required accuracy. The damping moment

is equal and opposite to the applied moment, and this is most conveniently determined from a measurement of the electrical driving power. The technique possesses considerable advantages at

high speeds and high frequencies when compared with other steady oscillation methods designed to measure overall force, since these all require the balancing out of inertial reactions which are in general large compared with the aerodynamic reactions to be measured.

Although the apparatus described below was designed for two-dimensional tests, it was found possible to utilize one -half of the system for measurements with finite aspect-ratio models by

employing a half-model technique. Some results for a rectangular wing of aspect ratio 4 obtained b y this method are included in the Report.

2. Experimental Technique. 2.1. Mecham;cal System. The general arrangement of the apparatus

in relation to the tunnel working section is illustrated in Fig. 1. The tunnel is of the induced flow

type with a vertical working section 9~- in. square. Two identical vibratory systems were mounted on opposite sides of the tunnel with their axes coincident with that of the model. Each vibratory system consisted of a rigid steel spindle supported by two crossed-spring bearings which permitted axial rotation but were very stiff for lateral displacements. This provided built-in end conditions for the model, which was damped to the spindles, and thus maintained a high value of flexural stiffness. Torsion bars connected between the spindles and earthed brackets provided elastic stiffness and determined the natural frequency of the system, which could be varied in discrete steps over a range of frequency by means of a graded set of torsion bars.

Each vibratory system was housed in an airtight box to prevent air flowing into the working section through the holes in the tunnel walls where the model was clamped to the spindles. One of

the systems was mounted on a floating platform supported on vertical flexures in such a manner as to relieve axial stresses in the spring bearings due to distortion of the apparatus resulting from changes in temperature and internal pressure.

2.2. Electromechanical Excitation. The system was oscillated by means of four electromagnetic exciters of the pot-magnet type, the moving coils of which were fixed to the ends of rigid arms attached to the spindles. All four coils were connected in series in such a way that when current flowed in the circuit a pure pitching moment was imposed on the system. This applied also to

each spindle separately, and either could be used independently for making tests with a finite aspect-ratio model.

When the system was oscillated, the phase of the alternating driving current was set to be in quadrature with the displacement in order to give a moment in phase with the damping moment. By adjusting the magnitude and sign of the current the damping moment could be balanced and a sustained oscillation obtained.

2.3. Displacement Pickup. The function of the displacement pickup was twofold, first to give a measure of the amplitude of oscillation, and second to provide a signal for the driving amplifier which supplied current to the exciter coils, and thus to complete the feedback loop which enabled self-excitation to take place. A photocell type of pickup, illustrated in Fig. 1, was devised for this purpose. By means of a lamp and lens a parallel beam of light was formed and then brought to a focus on a white screen by a second lens. The image on the screen acted as a light source illuminating

a vacuum-type photocell which was screened from the direct light from the lamp. A metal vane fixed to one of the driving coil arms partially intercepted the parallel beam, and a suitably shaped

mask placed near the vane ensured a linear relation between image intensity and angular displacement of the system. The indirect illumination of the photocell obtained with this arrangement avoided non-linearities due to non-uniform sensitivity of the cathode surface.

In order to obtain a high pickup sensitivity a large load resistance (22 megohm) was employed in the photocell circuit, and to minimise the shunting effect of lead capacity on this load the cathode of the photocell was connected by a short length of wire to the grid of a cathode-follower. The small phase shift due to residual stray capacity was compensated at each working frequency by an adjustable RC coupling circuit connected to the cathode-follower output.

A rapid d.c. calibration of the pickup could be made by rotating a metal shutter into the path of the parallel beam of light. This removed a fraction of the light forming the image on the screen known to correspond to a certain angular displacement of the system. The sensitivity of the pickup could be varied by adjusting the lamp current, and it was normally set to give an output of 1 volt peak when the system was oscillating.

For the measurement of amplitude the signal from the pickup was first amplified by a highly

stable negative-feedback amplifier (intrinsic gain 3000, gain with feedback 67) and then rectified

with a full-wave instrument rectifier, the resulting direct current being measured with an accurate microammeter.

2.4. Control Unit. The maintenance of a stable amplitude of oscillation of a self-excited system

requires some form of automatic control mechanism which adjusts the driving power to compensate

for changes in damping on the system. Where the damping arises from air-flow in a wind tunnel

some more or less rapid fluctuation must be expected, and the control requires to be rapid in action. The method employed was completely electronic and is illustrated in Fig. 2. The signal from the

displacement pickup was first amplified to a level of approximately 60 volts peak and then fed to a trigger circuit of the Schmitt type which produced a constant amplitude square wave of the same

frequency and phase as the pickup signal. A differencing circuit followed in which a controllable fraction of the pickup signal was subtracted from the square wave, and finally an RC filter circuit removed harmonic components to leave a sinusoidal signal of amplitude equal to a constant less some fraction of the amplitude of the pickup output. By using this signal as the input to the driving amplifier a controlled system was obtained in which a change in amplitude of oscillation resulted in a change of opposite sign in the amplitude of the signal to the amplifier and thus altered the driving power in a direction tending to restore the amplitude of oscillation to its original value. The tightness of control could be varied by adjusting the fraction of the pickup signal subtracted from the square wave. --

The method of control described above is much more rapid in action than the conventional A.G.C. technique in which the pickup signal is amplified, rectified and smoothed to provide a d.c. potential for biasing back a variable gain valve in the driving amplifier. This greater rapidity

results from the nature of the filtering employed, which in the first case is required to remove harmonics and leave the fundamental, whereas in the conventional A.G.C. method it must remove

the whole of the alternating components and thus involves much larger time constants. It has been

found also that the lag in response with the latter method is prone to give relaxation oscillations of the combined electromechanical system.

3

(sas27) A 2

2.5. Driving Ampl i f ier and Phase Shifter. A conventional push-pull amplifier (Fig. 4) was

employed, with a maximum output of 17 watts. The driver stage consisted of a long-tailed pair

acting as a phase splitter, and a negative feedback signal was fed to the grid normally held at a fixed

potential. Variable current feedback was employed and was derived from a low-resistance potentio-

meter in series with the output-transformer secondary winding.

The first stage of the amplifier, which preceded the phase splitter, was designed to act as a phase

shifter and comprised a pair of valves coupled by a conventional see-saw arrangement to give equal

and opposite output signals at the cathodes. A fixed capacity and a variable resistance were connected

in series across the cathodes, and a high-resistance potentiometer connected their junction point to

one of the cathodes. At each working frequency the variable resistance was adjusted to have the

same impedance as the condenser, and with this condition the signal fed to the phase splitter from

the potentiometer, could be shifted in phase by amounts varying from 0 deg to 90 deg relative to the input by rotating the wiper away from the valve cathode. The variation of signal amplitude over this

range amountedto + 20 per cent on a mean value. The direction of the phase shift could be reversed by interchanging the capacity and variable resistance by means of a switch (not shown in Fig. 4).

2.6. Electronic Wattmeter . The measurement of driving power required a wattmeter with ranges extending from a few milliwatts up to several watts, and since it was also desirable, from considera- tions of accuracy, to use the wattmeter as a null indicator in setting the driving current in quadrature

with the displacement, a further requirement was a fairly high input impedance to avoid the introduction of phase shifts in the circuits supplying the relevant signals. Since at the time there

appeared to be no commercial instrument available which satisfied these requirements, an electronic wattmeter was developed based on the work of Mallett 2.

Basically the action of the wattmeter depends on a pair of valves with identical square-law

characteristics. If the sum v 1 + vz of two sinusoidal signals is applied to the grid of one of the

valves and the difference v 1 - v 2 to the grid of the other, a difference between the mean currents

in the valves is produced which is proportional to v 1 v., cos e, where e is the phase angle between

v 1 and v 2. This difference may be conveniently observed as a steady voltage across the valve anodes by including equal resistances in the anode circuits.

The essential features of the circuitry developed are illustrated in Fig. 5. The input stage consisted

of a double triode V1 forming a long-tailed pair, with a pentode Vz as the common cathode load.

A signal applied to the grid of V 2 appeared at the anodes of V 1 with the same sign and equal ampli-

tudes, whereas a signal fed into V 1 appeared with equal amplitudes but opposite sign. Thus the

signals applied to the square-law valves V a and V~ were of the form av 1 + bv 2 and av 1 - bv~, and the steady potential developed between the anodes was proportional to abvlv 2 cos e. The voltage changes at the anodes were applied to the grids of a balanced pair of cathode-followers g 5 and V G

via resistance networks incorporating two stages of RC filtering for removing the sinusoidal components, and the voltage change between the cathodes of Va and V 6 was measured directly with a sensitive d.c. instrument. Two stages of RC coupling between V 1 and the square-law valves were employed in order to minimise voltage surges due to switching operations.

A number of different valve types were tested in searching for square-law characteristics, and the

EF 54 was selected as the type maintaining the required characteristic over the greatest range of control grid voltage. Since it is virtually impossible to find two valves giving exactly equal outputs,

the valve with the greater output received its input from a potentiometer P r Likewise inequality

between the two halves of K 1 was balanced with the potentiometer P~ which adjusted the relative

amounts of negative feedback at the cathodes. The setting-up process for the instrument consisted f

in adjusting P1 and Pe to give a null reading on the meter with a signal supplied to each input in

turn. A d.c. balance was obtained by adjusting the bias of V4. In order to obtain an acceptable

zero stability it was found essential to use a well stabilized I-I.T. supply and to run the heaters of

Va - V6 from an accumulator. Bias for Va and V 4 was also obtained from a battery. Full-scale meter deflection was obtained with input signals of approximately 0.1 volt peak.

These signals were supplied via precision 10,000 ohm 3-decade voltage dividers, one of which was

connected across a resistance placed in series with the driving coils, with one side earthed, to give

a voltage proportional to current, the other being connected through a 100,000 ohm resistance to

the 'live' end of' the driving coil chain for measuring power, or to the output of the displacement

pickup for setting phase. The voltage dividers performed the function of 'range changing' switches, and enabled a very wide range of powers to be measured. For greatest accuracy the voltage dividers

were normally set to supply equal input voltages to the wattmeter, which was attained by adjusting each divider in turn to give the same meter reading when switched to supply both inputs.

2.7. Frequency Meter. The determination of the aerodynamic stiffness derivatives from measure- ments of the change in natural frequency of the oscillating system due to the wind loading required an accurate method of measuring small frequency changes, since it was estimated that in the worst

case these might not be more than 1 per cent. The method adopted was based on the Maxwell

bridge and is illustrated in Fig. 3. In this arrangement a high-speed relay (Siemens type) operated by the square wave from the control circuit through a driving valve was made to charge a condenser

C to a potential V and to discharge it through a resistance network including R and P once per cycle

of oscillation. The average current flowing through the galvanometer G was thus proportional to frequency and could be balanced to give a null reading by adjusting the voltage divider P. Unsteady

components of the current were removed by an RC filter. The voltage divider P was a three-decade precision instrument, and by limiting settings to values

not greater than 100 in order to minimise changes in the network impedance, a relation between frequency and setting was obtained which was linear to within ½ per cent. The calibration of the

bridge could be varied by means of the variable resistance S, and it was normally adjusted to give voltage-divider settings numerically equal to the frequency at balance. This adjustment could be rapidly checked by feeding in an accurate 100 cycle signal derived from the N.P.L. standard

frequency source, and setting the voltage divider to read 100. In measuring a frequency, voltage-divider settings differing by unity were found which gave

galvanometer deflections on opposite sides of zero. From measurements of these deflections it was then possible to interpolate between the voltage-divider readings. The overall stability ~ of the instrument was such that frequencies of 10 cycles per second could be read to 1 part in 10 ~. This enabled frequency changes as small as 1 per cent to be measured to an accuracy of 1 per cent plus a maximum calibration error of ½ per cent.

For some conditions still smaller changes in frequency were obtained, and these were measured by a beat method in which the signal from the displacement pickup was applied to the beam of an oscilloscope together with the N.P.L. standard 1,000 cycle signal. The time base of the oscilloscope was synchronised to a convenient sub-multiple of the standard frequency, and the beat frequency, observed as a periodicy-displacement of the 1,000 cycle trace at a fixed x-position, was measured

with a stopwatch.

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2.8. Electrical Loss Measurement. In deriving the damping for the oscillating system from the power measurement a correction was required for the power losses due to the ohmic resistance in the driving coil circuit, to eddy currents in neighbouring metallic parts of the structure and to

hysteresis effects in the magnetic material of the exciter magnets. These losses were determined directly from a single power measurement with the oscillating system clamped and a current flowing

in the driving coils equal in magnitude and frequency to the current with the system oscillating. The current was obtained by supplying the driving amplifier with an input signal from a variable-

frequency valve oscillator, and the system was clamped by a brake mechanism in which spring- loaded friction blocks were brought into contact with the spindles from both sides.

The brake mechanism also served as a safety device where negative aerodynamic damping was

encountered, and for this purpose it was tripped automatically when the amplitude of oscillation

reached a predetermined value. Normally the brake blocks were held clear of the spindles by means of toggles held in by electromagnets. The current in the latter was momentarily interrupted and the

toggles were released when a contact spring was deflected by an arm attached to one of the spindles.

2.9. Calibration of Wattmeter and Amplitude Meter. In order to determine the true power corresponding to a given electronic wattmeter reading use was made of a 20 ohm resistive load which could be switched into the driving amplifier in place of the driving coils. A signal of the required frequency was supplied to the amplifier from a valve oscillator, and the power in the load was measured by the electronic wattmeter and a substandard dynamometer wattmeter simultan- eously. The power was adjusted to give a large deflection on the dynamometer instrument, and the voltage dividers were set to give the required reading on the electronic wattmeter. A relation was thus obtained between power and voltage-divider settings for this reading. When the power required to oscillate the mechanical system was large enough to be measured with sufficient accuracy directly on the dynamometer wattmeter, the electronic instrument was not used.

The calibration relating to amplitude measurement was divided into two parts, first a d.c. calibration of the displacement pickup, and then an a.c. calibration of the associated amplitude

amplifier with rectifier and microammeter. The d.c. calibration was effected by measuring the

output voltage from the pickup obtained on operating the metal calibrating shutter (see Section 2.3). A null method was used in which the output voltage was balanced against an equal voltage derived

from an accumulator by means of a precision 4-decade voltage divider, the voltage across the

accumulator being measured with a sub-standard dynamometer voltmeter. For the a.c. calibration

the driving amplifier was employed with the resistive load and input from the valve oscillator. The voltage divider was connected across the load and used to supply a signal to the amplitude

amplifier, the a.c. voltage across the load being measured with the same dynamometer voltmeter as was used for the d.c. calibration. These two calibrations determined the a.c. input to the amplitude

amplifier corresponding to a given microammeter reading and the d.c. output from the displacement pickup for a given angular displacement and thus enabled a relation between microammeter reading and amplitude of oscillation to be obtained.

2.10. Eddy-Current Damper. Where negative aerodynamic damping was encountered greater in magnitude than the apparatus damping, the mechanical system was excited aerodynamically without the aid of the electrical drive. To deal with this case an eddy-current damper was brought into action consisting of two aluminium vanes one fixed to each spindle and arranged to oscillate in the field gaps of a pair of electromagnets (not shown in Fig. 1). By adjusting the current in the

magnet windings to make the total damping positive, measurements could be made in the usual way and

the aerodynamic damping obtained as a difference between measurements with and without the wind.

2.11. Test Procedure. Before commencing a series of measurements the calibration of the

frequency meter was adjusted and the amplitude meter was calibrated in the manner described in

Section 2.9. Frequent checks of these calibrations were made during the tests. Measurements of driving power, current and frequency were made with the system oscillating at the required

amplitude and the current set in quadratur 9 with the motion, both with the tunnel running and in still air; electrical loss measurements and electronic wattmeter calibrations followed immediately.

The still-air driving power, corrected for loss, represented the apparatus damping, which was assumed to be due to elastic hysteresis in the torsion bars and at their clamping surfaces. On this basis the driving power in still air with a given torsion bar would be proportional to frequency, and the damping derivative Mo was calculated from the formula

W - AW' = - 26.77f20o~Mo, (1)

where A is the ratio of frequency in the wind to that in still air and corrects the still-air power W' for the change in frequency due to wind loading. In this formula power is in watts and 00 in radians.

The stiffness derivative M o was calculated from the formula

3/io = ~o(f + fo) ( f - fo) / fo ~ (2)

involving the frequencies with and without the wind and the elastic stiffness e0 of the torsion bar assembly, which was obtained from a static test.

Effects of neglecting the still-air damping and inertia on the moving parts not exposed to the wind and on the model in the tunnel working section were examined by separate experiments and

are discussed in Appendix I.

3. Experimental Results. 3.1. Details of Models. All the models tested were constructed of solid steel with flat rectangular end lugs for clamping to the oscillating spindles. The subsonic

tests were made for one axis position only, which was positioned at the c.g. of the RAE 104 profile,

coinciding with the mid-point of the lug section. In the case of the supersonic tests four axis

positions were employed, namely, at the leading-edge, quarter-chord, half-chord and three-quarter

chord positions. These were obtained with separate models in which the biconvex profile was suitably positioned in relation to the end lugs. Since the profile was symmetrical about the mid-chord

position, a single model served for the quarter-chord and three-quarter chord tests. The object in employing separate models for tile different axis positions in this case was to maintain a maximum clamping efficiency at the spindles with a minimum inertia; and since a rapid technique for making two-dimensional models with simple profiles had been evolved, based on a tangent plane approach, duplication of models was regarded as acceptable. The models tested are listed in Table 1 below.

TABLE 1

Model

Profile Aspect ratio Thickness/chord (per cent) Axis position

(from L.E.) Chord (in.) Span (in.)

RAE 104 cO

10 0.445

2-0 9.5

RAE 104 4 10

0' 445

2.5 5

Biconvex oO

7"5 0

2.0 9"5

Biconvex oo

7-5 0-25 0"75 2.0 9'5

Biconvex cX3

7"5 0"5

2.0 9.5

Biconvex CO

5 0-5

2-0 9-5

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3.2. General Remarks on Tests. The procedure followed in making the subsonic tests was to

run through the Mach number range for each still-air natural frequency, determined by the torsion

bar stiffness, to obtain basic experimental curves of m o and m o plotted against M. Since only small

changes in frequency occurred with change in Mach number, the frequency parameter ,co varied

approximately inversely as the wind speed, and in general this variation has been plotted against

M in the basic-curve diagrams. Curves showing the variation of the derivative coefficients with co for constant M have been obtained by cross-plotting from the basic curves.

Measurements at mean incidences other than 0 deg were complicated at the lowest frequency by

change in incidence with Mach number due to twist of the torsion bars under the mean aerodynamic

moment. In these cases the model was set by trial and error to give the required mean incidence

approximately at transonic speeds, and curves showing the variation of mean incidence with Mach

number are included in the Figures. At the higher frequencies the torsion bars were sufficiently stiff to prevent any appreciable change in incidence.

The subsonic measurements were made with solid liners fitted to the tunnel, and required

correction for blockage effects. Corrections appropriate to static tests were used, since the dynamic

case has not yet been treated theoretically, and in the finite aspect-ratio tests the method of the

equivalent doublet and its images as described by Thompson '~ was applied. In estimating the wake

blockage, existing values of the drag coefficient were utilized. For the two-dimensional tests blockage corrections were calculated from tunnel wall pressures in the manner described by Mair and Gamble 4.

The basic curves for the supersonic tests, were formed by running through the amplitude range

for each still-air frequency, Mach number and axis position. By cross-plotting, curves of damping

and stiffness against co for fixed 00, M and h, and against h for fixed co, 00 and M were obtained.

The semi leading-edge angle for the 7½ per cent thick biconvex profile is 8.58 deg. This is the

deviation of the flow when the model is at 0 deg incidence, and for this condition the bow wave

is attached at both the test Mach numbers. At the lower Mach number (M = 1-42) the bow wave

becomes detached for deviations greater than 9.97 deg, thus in the oscillatory tests it may be assumed

that the bow wave was detached for part of the cycle for amplitudes greater than 1.39 deg. At the

higher Mach number (M = 1.61) the bow wave was attached for all conditions of the tests. Estimates of the position of shocks reflected from the tunnel walls indicate that for all conditions, both with

attached and detached bow waves, these are well clear of the model.

3.3. Two-dimensional subsonic Tests ~. The measurements were made on a model with a 10% RAE 104 profile oscillating about an axis at a distance 0.445c behind the leading edge (model 1,

Section 3.1). This particular axis coincided with the c.g. of the section and thus avoided the

possibility of errors due to coupled inertial forces arising from flexure of the model. The test still-air

frequencies ranged from 11.5 c.p.s, to 68 c.p.s, approximately and the Mach number from 0' 3 to

0.9. Tests were made at mean incidences of 0, 2 and 3 deg and amplitudes of oscillation of 1 and 2 deg. At the higher Mach numbers the frequency parameter ranged from 0.01 to 0.08, and a Reynolds number of 0.8 x 106 was attained.

The subsonic results given in this report supersede earlier measurements, some of which appear in Ref. 15. The earlier tests were subject to considerable unsteadiness in the tunnel flow and are thought to be unreliable on this account. Improved flow conditions were obtained by modification to the tunnel intake before the present tests were commenced2

3.3.1. Tests at 0 deg mean incidence. Experimental curves showing the variation of - m s and

- m o with Mach number at three still-air frequencies are shown in Figs. 7 and 8 for a mean

incidence of 0 deg. At low Mach numbers the damping curves are flat, but a rapid increase in

damping takes place as M is raised, a peak value being reached at approximately M = 0-81, which

is somewhat higher than the critical Mach number (pressure critical) '~. For further increase in

Mach number the damping falls rapidly. The effect of amplitude of oscillation is small, being

greatest near the peak value of the damping, which is greater for the small amplitude. The general

shape of the curves is unaffected by frequency over the range covered, particularly in relation to the position of the peak damping.

The numerical value of the stiffness - m o shows a similar but much smaller increase up to

M = 0.8 approximately. This is followed by a minimum and second maximum at M = 0.89.

As in the case of the damping, frequency has little effect on the shape of the curves, and amplitude effects are small.

Tests were also made with a turbulent boundary layer on the model in the case of the middle frequency. For this purpose a coating of carborundum powder was applied to the model with an

adhesive, extending from the leading edge for a distance of 0. lc. The grade of carborundum powder

and the density of distribution of the grains were selected to give a drag, as measured by a wake

traverse at zero incidence, equal to the known drag for this profile with a turbulent boundary layer. The wake traverse tests were made at M = 0.5 and 0.75.

Curves of damping and stiffness for the turbulent boundary-layer condition are included in Figs. 7

and 8. The only significant difference when compared with the curves for natural transition is the

disappearance of the second maximum in the stiffness curve. However, since this occurs at the

end of the Mach number range, it is probable that it now appears at a slightly higher value of M.

3.3.2. Tests at mean incidences o f 2 deg and 3 deg. The major effect of incidence on - m 6 and

- m o occurs at the higher Mach numbers in the region between M = 0.75 and 0.9. At a mean

incidence of 2 deg (Figs. 9 and 10) and the lowest frequency the fall-off in damping is more rapid,

and leads to a narrow region of negative damping in the neighbourhood of M = 0.87. With further

increase of Mach number a rapid rise to positive values takes place. At the higher frequencies the peak damping is reduced and occurs at a lower Mach number (M = 0.77 approx.). This is followed

by a minimum and a second smaller peak leading into a rapid fall-off, but the damping remains positive.

At a mean incidence of 3 deg (Figs. 11 to 13) and at the lowest frequency the region of negative damping has become very deep and large amplitude effects are present, the damping being more

negative for the smaller amplitude. At the higher frequencies the curves are similar to the 2 deg mean incidence case in general shape, the minimum, however, is more pronounced and the final fall-off leads to negative damping.

• " The stiffness curves for a mean incidence of 2 deg and 3 deg are affected very little by frequency,

and they differ from the 0 deg case in that the second numerical maximum has become smaller and at 3 deg has almost vanished. The first maximum occurs at a progressively lower Mach number as the mean incidence is raised.

Curves of damping and stiffness for the turbulent boundary-layer condition are included in

Figs. 12 and 13, and a comparison with corresponding curves for natural transition shows no significant difference.

9

It is difficult to offer a physical explanation for the sudden changes and loss in damping at the

higher Mach numbers observed in the tests discussed above without a detailed knowledge of the

pressure distributions and flow conditions. It seems likely, however, that the effects are associated

with flow separation due to the presence of shock waves on the surfaces of the model.

3.3.3. Comparison with theory. In comparing the experimental results with theory it is important

to consider the effect of tunnel wall interference, since it has been shown theoretically by W. P. Jones 6

that this would be large for two-dimensional flow in the case of the damping derivative at the small

values of frequency parameter attained and for the size of tunnel and axis of oscillation employed.

This is illustrated by the curves in Fig. 14, which are taken from a report by Jones 6 and show the

effect of the tunnel wall interference on flat-plate theory for M = 0 and 0" 7. Curves based on the

equivalent profile theory 6, together with the influence of tunnel interference, appropriate to the

conditions of the present tests, are also included. This introduces the effect of aerofoil thickness and

boundary layer into the theory. In view of the large tunnel interference effects predicted by theory, no attempt has been made to

correct the experimental results to free-stream conditions, and the comparisons with W. P. Jones given in Figs. 7 and 8 are with equivalent profile theory including the effects of the tunnel walls. Agreement is satisfactory up to the peak value of the damping (Fig. 7), but the theory predicts a fall-off at a considerably higher Mach number. Free-stream flat-plate theoretical damping curves are also included in Fig. 7, and it will be noticed that these approach the experimental curves more closely the higher the frequency. This agrees with the prediction (Fig. 14) that the tunnel wall effects will become smaller as the frequency parameter is raised. The theoretical result by W. P. Jones

and flat-plate theory both show good agreement with the stiffness curves (Fig. 8).

3.4. Subsonic Tests with Spoilers. For the purpose of these tests spoilers were fixed to the

two-dimensional RAE 104 model, and measurements were made at subsonic speeds at a nominal

still-air frequency of 36 c.p.s., with an amplitude of oscillation of 2 deg and mean incidences of

0 deg and 3 deg. Three different spoilers were used, two being uniform in the spanwise direction

and square in section with sides of 0.02 in. and 0.04 in. respectively. The third was a castellated

spoiler formed by cutting 0.035 in. square teeth on a uniform 0.045 in. square-section spoiler,

the pitch of the teeth being equal to twice their width. The spoilers were made of steel and were

attached with Araldite.

Most of the tests were made with the spoilers fixed at the trailing edge of the model, either on

one surface or on both surfaces together. Some of the tests with the 0.04 in. spoiler, however, were

made with the spoiler at the 0.65 chord position. For all these tests the model boundary layer was

made turbulent.

3.4.1. Spoiler tests at 0 deg mean incidence. The effects of the 0-02 in. and 0.04 in. square-

section spoilers on the damping derivative - m o when placed at the trailing edge are compared in Figs. 15a and 15b. A single small spoiler has little effect on the damping curve, but two spoilers, one on each surface, produce a rapid loss of damping above M = 0-8, with a region of negative damping between M = 0.84 and 0.90 approximately. In the case of the larger size, a single spoiler now leads to a considerable loss of damping above M = 0.76, zero damping being reached at M = 0.87. Two spoilers produce an extrenlely violent fall-off in damping above M = 0.8 with no sign of recovery up to the highest Mach number of the tests.

10

Positioning the 0.04 in. spoiler at 0.65c' from the leading edge has a markedly different effect from that of the trailing-edge pos!tion. A single spoiler produces a loss of damping between M = 0.76 and 0.88 with full recovery at the latter Mach number (Fig. 16a), whereas two spoilers give a curve

similar in shape to the case with no spoilers, but with smaller damping on the whole. In both cases the damping at the lower Mach numbers (M = 0.4) is much reduced.

During the tests with the square-section spoilers a suggestion was made that the rapid loss of

damping at the higher Mach numbers might be associated with the formation of two-dimensional vortices by the spoilers, and that a castellated spoiler might reduce this loss of damping by breaking

up the regular vortices to give a more random flow pattern. Damping curves for a castellated spoiler,

given in Fig. 17a, have a similar shape to the curves for the smaller square-section spoiler (Fig. 15c),

but show a somewhat greater fall-off in the higher 1V~ach number region. In fact, in this region the curves are intermediate between those for the two square-section spoilers, and since the mean height of the castellated spoiler is also intermediate (0. 028 in. as compared with 0.02 in. and 0.04 in.) it appears that this may be the effective parameter rather than the castellated profile.

The effect of the spoilers on aerodynamic stiffness is shown in Figs. 19 to 22, in which the negative

stiffness (mo instead of - too) is plotted, since this brings out a marked similarity with the effect on damping. The curves relating to the uniform spoilers at the trailing edge (Fig. 9) show a falling off and change in sign of m o almost identical with that for - m o (Fig. 15), the exception being the case of the smaller spoiler on one surface, which appears to have more effect on m o than on the damping. Similar correlation exists for the castellated spoiler (Fig. 21), the region of negative too, however, being somewhat narrower than for - m O. In the case of the spoilers at the 0.65c position (Fig. 20), the similarity with the damping curves is not so close. The initial drop with one spoiler occurs at approximately the same Mach number as for the damping (Fig. 16a), but is quickly followed by a rise and second maximum. With two spoilers the curve is closer in shape to that without spoilers, as in the case of the damping.

3.4.2. S p o i l e r tests a t 3 deg m e a n incidence. In general the effect of the spoilers is to accentuate

any falling-off in damping or m o in the high Mach number region. Damping curves for the uniform spoilers at the trailing-edge position (Figs. 16b and 18) show this as a lowering of the minimum

in the curve obtained without spoilers, with the formation of regions of negative damping. It is

clear from Fig. 18b that, on the whole, a spoiler on the upper (suction) surface has less effect than one on the lower surface. This also applies to m o as shown by Fig. 22b.

The effect of 0.04 in. square spoilers at the 0.65c position is very similar to that for 0 deg mean incidence. With two spoilers the trend is for the shape of the original curve to be maintained, both

in the case of damping (Fig. 16b) and m o (Fig. 20b); with a single spoiler on the lower surface, however, an earlier falling-off in both damping and m o is obtained.

In the case of the castellated spoilers the fall in the original damping and m o curves is now continued to negative values, and there is not a great deal of difference between the effect of a single spoiler on the lower surface and two spoilers, one on each surface (Figs. 17b and 21b).

3.5. F i n i t e A s p e c t R a t i o S u b s o n i c Tes ts . These tests were made on a rectangular model of aspect ratio 4 (tip-to-tip aspect ratio) using the half model technique (model 2, Section 3.1). The profile and axis were the same as for the two-dimensional subsonic model, and the still-air frequencies ranged from 11 c.p.s, to 57 c.p.s, approximately. Measurements were made at mean incidences of

11

0 deg and 3 deg, and amplitudes of oscillation of 1 deg and 2 deg, for Mach numbers ranging from 0.3 to 0.93. At the higher Mach numbers the frequency parameter varied from 0" 01 to 0" 1

approximately and a Reynolds number of 1.0 x 106 was obtained.

3.5.1. Exper imenta l results. Curves showing the variation of - mo and - m o with Mach number are given in Figs. 23 to 26, and these are cross-plotted in Fig. 27 to show the effect of frequency

parameter. The damping curves for 0 deg mean incidence (Fig. 23) show a rise with Mach number to a peak value on approaching M = 0.9, which occurs somewhat earlier at the lowest frequency.

At a mean incidence of 3 deg (Fig. 25) the peak is still fm'ther delayed and occurs approximately

at M = 0" 92. A srnall amplitude effect is also apparent on the rising part of the 3 deg mean-incidence

curves, the damping being greater for the larger amplitude of oscillation. The stiffness curves for both 0 deg and 3 deg mean incidence fall slightly with increasing Mach

number to give a minimum a little above M = 0.8 followed immediately by a maximum at

M = 0.9. The minimum is more pronounced in the curves for 3 deg incidence, whilst in the curves

for zero incidence a small amplitude effect appears above the minimum, the stiffness being numerically

smaller for the larger amplitude. Apart from the difference in magnitude, the major difference between the results for A = 4 and

A = oo (see Figs. 7 and 8, and 11 to 13) is that in the case of the finite aspect ratio any fall-off in

damping is considerably delayed and fluctuations in the stiffness at the higher Mach numbers are

reduced. The variation of both damping and stiffness with frequency parameter (Fig. 27) is not large over

the range covered, and amounts to a reduction in magnitude with increasing co up to ~o = 0.1 approximately, after which the curves tend to flatten out. A comparison between results for mean

incidence of 0 deg and 3 deg is also given in Fig. 27.

3.5.2. Comparison with theory. Available theoretical points for the aspect ratio and axis position

of the tests are included in Figs. 23 and 24. With the exception of the single point at M = 0. 846 these values correspond to the frequency parameter shown in the experimental curves; the points at

M = 0.3, however, are strictly theoretical values for M = 0. The point at M = 0.846 relates

too~ -+0. The three points due to Lehrian v derive from an 'equivalent' wing theory s using a vortex lattice

method of downwash calculation, the values at M = 0- 866 being corrected to an exact value of the

downwash at the centre of the wing, in accordance with calculations by Acum '~. These points define

damping curves which are somewhat flatter than the experimental curves, but the agreement is

reasonably good at the lower Mach number and near the peak value of the damping, especially in

the case of the tests at the lowest frequency. The single points at M = 0. 866 due to Acum are derived from an extension to general Mach

number and frequency parameter of the low frequency Multhopp-Garner theory 1°. For the low values of the frequency parameter under consideration these results differ little from the correspond-

ing values due to Lehrian. The points due to Acum and the low frequency Multhopp-Garner values

at M = 0. 846 are both in fair agreement with the damping measurements. All the theoretical values show satisfactory agreement with the stiffness curves.

Wind-tunnel interference effects have not been indicated here since numerical values are not

available. :~ '~ " : :

12

3'~6. T w o - D i m e n s i o n a l S u p e r s o n i c Tes t s . Most of these measurements were made w i t h a

7½ per cent biconvex aerofoil oscillating about axes at the leading edge, and at positions 0.25c,

0.50c and 0.75c behind the leading edge in turn. Th e tests were made at Mach numbers of 1.42

and 1.61 with torsion bars giving still-air frequencies ranging f r o m 11-5 c.p.s, to 58 c.p.s.

approximately, and the amplitude of oscillation was varied from 1 deg to 4 deg in steps of 1 deg.

T h e frequency parameter ranged from 0. 006 to 0. 046 at M = 1.42 and 0. 008 to 0. 043 at M = 1.61,

and a Reynolds number of approximately 0 .8 x 106 was obtained. All the measurements were made

at a mean incidence of 0 deg. A small number of tests were made with a 5 per cent biconvex model oscillating about the

mid-chord axis.

3.6.1. T e s t s a t M = 1.42. T h e variation of - m 6 and m o with amplitude of oscillation is

shown in Figs. 28 to 31 for the range of axis positions and frequency parameter values covered in the

tests. For the most forward axis position (Fig. 28c) the derivative - m 6 decreases (increase in

negative value) with increase in amplitude 0o, the decrease becoming smaUer at the lower values of

the frequency parameter. This decrease with increasing 00 tends to become smaller as the axis is

moved aft, and at the mid-chord position it has changed sign at the lowest f requency parameter to

become a small increase. Results for the 0.75c axis position show an increase with 00 for all values of 03.

In the case of m o a similar but smaller decrease with increasing 00 is observed for the forward

axis positions. Frequency parameter, however, appears to have little effect. Results for the rearmost

axis position show practically no amplitude effect at all.

Curves of - m 0 and m o relating to the 5 per cent biconvex modei plotted in Figs. 30a and 30b

all fall below the corresponding curves for the thicker section.

T h e effect of frequency parameter is shown in Fig. 32, which was obtained by cross-plotting

from Figs. 28 to 31 at 00 = 2 deg. Over the range covered it is clear-that effects of 03 are not large

for this amplitude of oscillation. o

T h e curves in Fig. 32 are further cross-plotted at c~ = 0-025 to give Fig. 33, which shows the

variation of - m o and - m 0 with axis position at a fixed value of 00 and 03. Comparison is made in

this Figure with earlier tests 1 made by the method of decaying oscillations for the mid-chord axis.

Good agreement is obtained with - t o o , but - m0 shows considerable discrepancy, the damping

for the earlier tests being considerably higher (see Section 3.6.2).

3.6.2. T e s t s a t M = 1.61. Amplkude effects for this Mach number are shown in Figs. 34 to 37.

Broadly speaking, the effect on - m 6 of increasing 00 is similar to that observed in the case of

M = 1.42; rearward movement of the axis, however, now produces a much smaller change in slope,

and at the 0.75c position - m 0 still decreases with increase in 00. T h e results for 03 - 0. 034 are

inconsistent with the remaining curves, and in the case of the forward axis show practically no

effect of amplitude. For the 0.50c axis a rising curve is indicated, whilst for the rearmost axis the

curve has become very non-linear with a marked dip in the neighbourhood of 00 = 1.5 deg. No

explanat ion can be offered for this anomalous behaviour.

T h e variation of m o with 00 is much the same as for M = 1.42, the curves becoming almost flat

for the rearmost axis position.

As for the lower Mach number, curves of - m6 and m o for the 5 per cent biconvex model, plotted

in Figs. 36a and 36b, fall below the curves for the thicker section.

t 13

Results for a turbulent boundary layer, obtained at 00 = 1.5 deg and 3 deg with co - 0.025,

fall below the laminar-boundary curves at the L.E. and 0.5c axis positions but show little change for

the 0.75c position. The turbulent boundary layer was created in this instance by forming on the

upper and lower surfaces of the model a band of Frigilene 0'05 in. wide and 0.003 in. thick at 0.05c from the leading edge.

For this Mach number the effect of frequency parameter on - m6, as shown by the cross-plot

in Fig. 38, is somewhat more marked than for M = 1.42, the value for the rearward axes increasing

with increase in co for the 7-~ - per cent profile. For the 5 per cent profile the damping decreases with

increasing co. The stiffness - m o shows little change with co, as before.

Fig. 39 presents the cross-plot at co = 0.025 showing the variation of - m 0 and - m o with axis

position, corresponding to Fig. 32 for the M = 1.42 case. Comparison is agai,a made with the earlier

decaying oscillation tests (see Section 3.6.1), and a similar discrepancy to that for the lower Mach

number is evident. It is difficult to assign a reason for the large difference in damping. The decaying

oscillation tests were made in a circular tunnel, and interference effects due to reflected shocks might be expected towards the ends of the model. These effects, however, appear to have been too small

to make any appreciable difference in - m o. A more likely explanation is effects due to model flexure. It can be seen from Figs. 48 to 51, relating to flexure in the present tests, that the effect has the correct sign for explaining the discrepancy in the damping, and that the effect on stiffness would be very small. It seems most likely that the earlier rig was considerably weaker in flexure than the

present apparatus, since not only was the span greater (12 in.), but the end bearings were single and the built-in effect of the present double bearings was lacking. On the other hand, some attempt was made to increase flexural stiffness in the earlier apparatus by applying axial tension.

3.6.3. C o m p a r i s o n w i t h t heory . The experimental results given in Figs. 33 and 39 are compared

in the same Figures with theoretical values based on flat-plate theory 11 and a theory due to Van

Dyke 1~ which introduces the effect of profile thickness. Goo d agreement with Van Dyke is obtained for rearward axis positions at both Mach numbers in the case of the damping -too, but on approaching the leading-edge axis the experimental curves turn up less rapidly than the theoretical.

Agreement between experimental and flat-plate curves is poor over the whole range of axis position, a point of particular interest being the mid-chord position near which flat-plate theory predicts a

change in sign of the damping for M = 1.42. The measured damping changes sign at an axis position

considerably further forward (h -- 0.344), whilst Van Dyke predicts h = 0.442 for this point.

The experiments disagree completely with a theory due to Wylly 13, which predicts positive damping

( - mo) for all axis positions and Mach numbers if the thickness is greater than 4.5 per cent%

Measured values of the damping for the 5 per cent thick section (h = 0.5) are smaller than for the 7½ per cent model, which is in agreement with Van Dyke, the difference being considerably greater than given by theory, however.

The curves of stiffness derivative ( - too) against axis position are nearly linear and have the same slope as the theoretical curves for the forward axis positions. Towards the rearward axis the slope becomes slightly less numerically, and the curves approach the theoretical values of Van Dyke.

Agreement is closest with Van Dyke's theory, but the aerodynamic centre ( - m o = 0) lies further forward than predicted.

Since shown to be incorrect (see A.R.C. 15,864).

14

As in the case of the damping, measured values of the stiffness for the 5 per cent thick section (h = 0-5) are numerically smaller than for the thicker section, which again agrees with Van Dyke.

It has been suggested that departure from theory at the forward axis positions may be due to separation of flow towards the trailing edge. This would reduce the contribution of the pressures in this region, and its effect on the derivatives would be most marked for the forward axes. A

comparison is made in Fig. 40 with curves based on a theory due to W. P. Jones 14. This is a

numerical theory and, since results are available only for M = 1.5, comparison has been made with

the mean of the experimental curves which corresponds to approximately the same Mach number.

In one of the theoretical curves the integration of the pressures has been terminated at 70 per cent

of the chord from the leading edge to simulate the effect of separation. In the case of the damping

this gives much better agreement for the forward axes but is not quke so satisfactory at h = 0.75.

The effect on the stiffness is to displace the theoretical curve to the opposite side of the experimental

curve, which suggests that still better agreement on the whole might have been obtained by extending the integration somewhat further towards the trailing edge.

To test the influence of pressures near the trailing edge a few measurements were made at

M =: 1.42 with the axis at the 'leading edge, in which tile chord was progressively reduced by

cutting off squarely near the trailing edge. The results are given in Figs. 41 and 42. Unfortunately

there is a considerable amount of scatter in the damping results, but it is fairly clear that for small

amplitudes and the lowest frequency parameter removal of up to 20 per cent of the chord has little

effect. The stiffness is unaffected by removal of up to 10 per cent of the chord. These results support

the view that separation is making the rear part of the profile relatively ineffective.

The theoretical relationship between pitching damping and axis position h is parabolic in h and takes the form

m o = mo ' + h ( l o ' - m s ' ) - h 2 ~ ' (3)

where the dashed symbols relate to the leading-edge axis, and m o is the damping coefficient for an

axis at h. Since measurements were made at four axis positions, including the leading edge, it is possible to derive from (3) three independent linear relationships between the unknowns l 0' - m~'

and l~'. These are plotted in Fig. 43 for each Mach number, and from the error triangles formed by the intersection of the curves values for l 0, - m s' and 1 o' which agree best with the experimental observations have been determined. By making use of the relations

l 0 - m~ = lo' - m ~ ' - 2hl~' (4)

l , = l ; (5)

the variation of l 6 - m~ and L with h has been determined and is compared with theory (Van Dyke)

in Fig. 44. The slopes of the experimental curves for l 0 - m~ are considerably less than indicated

by theory, large differences in actual value occuring at the forward and rearward axis positions.

In the case of M = 1.42 the experimental and theoretical values of l0 - m~ for the leading-edge

axis are of opposite sign. It is noteworthy, however, that almost exact agreement with theory is

obtained for both Math numbers at the mid-chord position. The experimental values of I s are numerically considerably less than the theoretical values, agreement being closest for M = 1.61.

A similar relation to (3) exists for the stiffness derivatives, but since the corresponding curves (Figs. 33 and 39) are nearly linear, it may be assumed that l~' is zero, and values for l o' - m~ ' can

then be obtained from the mean slopes of the stiffness curves. The variation of l o - m z witl~ h is

5

given by the corresponding relation to (4), which is independent of h when /~' = 0. Values of l o - m ~ are included in Fig. 44 and compared with theory, the agreement being satisfactory for both

Mach numbers. It should be pointed out that the theoretical estimates of the supersonic derivatives assume an

attached bow wave, whereas in the tests at the lower Mach number the bow wave was detached

for part of the cycle of oscillation at amplitudes above 1.39 deg. Since, however, the amplitude

effect for M = 1.42 is very similar to that for M = 1.61 where the bow wave was always attached, it is argued that the effect of the bow wave becoming detached is small, and values of the derivatives for an amplitude of 2 deg were selected for comparison with theory, since these were regarded as

being somewhat more accurate than the smaller amplitude results.

4 . C o n c l u s i o n s . (1) The experience gained with the self-excitation technique showed that this

was satisfactory at high speeds, and the equipment enabled measurements to be made at higher

values of the frequency parameter than had been attained with earlier decaying-oscillation apparatus.

(2) The two-dimensional subsonic tests on the RAE 104 model at 0 deg mean incidence show a

peak value of the damping at about M = 0- 81, which is somewhat higher than the critical Mach

number, followed by a rapid fall-off. At mean incidences of 2 deg and 3 deg more complex changes

occur, and at the lowest frequency a narrow range of Mach number exists around M = 0.87 in

which the damping is negative. At the higher frequencies this region of negative damping is

suppressed, but in the case of the highest mean incidence the damping falls off to negative values

as the upper limit of the Mach number range is approached. The stiffness derivative shows corresponding, but less violent, changes with Mach number,

and the effect of frequency is small. Comparison of the damping measurements with two-dimensional flat-plate theory for free-stream

conditions illustrates the large effect of the tunnel walls predicted by theory for low values of the

frequency parameter, and shows how agreement is improved as the frequency is raised. Satisfactory

agreement is obtained up to the peak value of the damping when comparison is made with a theory which includes tunnel wall effects and also the effects of aerofoil thickness and boundary layer introduced by an equivalent profile treatment. The fall-off in damping predicted by the theory is

delayed, however. The measured stiffness derivatives show good agreement with free-stream

flat-plate theory up to M = 0-8.

(3) In general, spoilers fixed at the trailing edge of the model in the two-dimensional subsonic

tests produce loss of damping at the higher Mach numbers, this loss being greater with a larger spoiler, or with two spoilers, one on each surface. Spoilers placed at the 0.65c position have a different effect, a single spoiler producing a loss of damping over a range of Mach number with a subsequent recovery, whilst a spoiler on each surface tends to maintain the original shape of the damping curve. A roughly similar effect to that produced by the spoilers in the case of the damping

curves is apparent also in the m o curves.

(4) Measurements on a rectangular wing of aspect ratio 4 at subsonic speeds with the same

profile and axis as in the two-dimensional subsonic tests give curves similar to those obtained in the

latter tests at 0 deg incidence, but with smaller numerical values for the derivatives. The peak in the damping curves appears at a somewhat higher Mach number, however, and the fluctuations in

stiffness at the higher Mach numbers are reduced. The effect of frequency parameter over the range covered is small.

16

Comparison with theory shows satisfactory agreement with the stiffness curves and reasonable

agreement with the damping for the range of Mach number covered by theory.

(5) Two-dimensional supersonic tests on a 7½ per cent biconvex model show definite amplitude

effects over the range covered (1 deg to 4 deg). These effects depend on axis position, and to some extent on frequency parameter, and may be summarised roughly by stating that both - m o and

m o decrease with increasing amplitude for the L.E. axis position, this decrease becoming less as the

axis is moved aft. The effect is greater for - m o and for the smaller Mach number of the tests, in

which case an actual increase in - m o with increasing amplitude is obtained for the rearmost axis.

Broadly speaking the effect of raising the frequency parameter is to increase the tendency of - m0

to fall with increasing amplitude. In the case of m o the effect is negligible.

For a fixed amplitude (2 deg) the effect of frequehcy parameter on - mo and - mo is not large.

Comparison with theoretical curves showing the variation of - rn0 with axis position shows closest

agreement when the theory includes thickness effects (Van Dyke or W. P. Jones). Agreement is

most satisfactory for rearward axes, considerable discrepancy existing for the forward positions.

This is thought to be due in part to flow separations towards the T.E. reducing the contribution

of pressures in this region, somewhat better agreement being obtained if the theoretical estimate

ignores the integral of the pressures over the rear 30 pe~ cent of the chord.

The curves of - m o against axis position are almost parallel to the theoretical curves and also

agree more closely when the theory contains thickness effects. The bodily displacement of the

experimental curve, however, indicates an aerodynamic centre somewhat ahead of the position

predicted by theory. When the rear 30 per cent of the chord is ignored in the theory, the effect is

not so satisfactory as in tile case of -- rno, the theoretical curve being displaced to the opposite

side of the experimental curve. Values of - m o and rn o obtained from tests with a 5 per cent biconvex model for the mid-chord

axis position fall below the corresponding values for the 7½ per cent biconvex model, a result which

is in agreement with theoretical predictions. The measurement of the direct pitching derivatives for several axis positions in the case of the

7½ per cent biconvex model enabled values for l 0 - m ~ , - l ~ and l o - m ~ to be deduced. In comparison with theory (Van Dyke), l o - m~ shows good agreement, whereas the measured value

of - l~ is high. Curves of I o - m~ against axis position have considerably smaller slopes than indicated

by theory, but agreement is exact at the mid-chord position.

17

(83827) B

a ,b

C

f

h

l

InO~ m 0

8mo, 8too

P

P,

gSj) gl 2

Z

g

lo

azo

L~ + jpL~ =

Lo + jpLo =

M~ + jpM~ =

Mo +.ipMo =

8Mo + jpSMo =

M

P

LIST OF'. SYMBOLS

Gains associated with 1o;ag-tailed pair

Chord of wing

Pitching frequency

Distance of axis downstream of L.E. as fraction of c

Span of wing

Coefficients in non-dimensional lift due to downward displacement = l z + jcoL

Coefficients in non-dimensional lift due to pitching displacement = l o + jcol 0

Coefficients in non-dimensional pitching moment d u e to downward displacement = m, + jcome

Coefficients in non-dimensional pitching moment due to pitching displace- ment = m o + flom~i

Errors in m o and m 6 due to flexure

Circular pitching frequency

Circular flexural frequency

Input voltages to long-tailed pair

Downward displacement of reference section

Inertia related to z-displacement

Pitching inertia

Error in I o due to flexure

p V~l,%( l~ + jwl,)

p V~cl~l(m~ +jcom~)

p VZcZl(mo +jwmo)

p V2cg"l(Sm o +jco 8too)

Mach number

Driving moment

Product of inertia

P/pcal

18

p t

V

W

W'

O~

£

Oo

K 1

K 2

P

P

%

(7 0

co

L I S T OF SYMBOLS--continued

Still-air damping power for model

Wind speed

Driving power in wind

Driving power in still air

Mean incidence

Phase angle between v 1 and v~

Spanwise distance as fraction of span

Amplitude of pitching motion

Kinematic viscosity

Air density

Flexural stiffness

~/pV2l

Pitching stiffness

pc/V frequency parameter

Dashed non-dimensional derivative coefficients relate to the leading-edge axis.

The suffix 0 denotes still-air frequency or a value of co based on still-air frequency.

No. Author

1 J.B. Bratt and A. Chinneck

2 E. Mallet . . . .

3 J. S. Thompson .,

(ass27)

REFERENCES Title, etc.

Measurements of mid-chord pitching moment deriva- tives at high speeds.

A.R.C.R. & M. 2680. June, 1947.

A valve wattmeter. J.LE.E. Vol. 73. p. 295. 1933.

Present methods of applying blockage corrections in a closed rectangular high speed wind tunnel.

Unpublished M.o.A. Report.

19 B i

No,

4

Author

W. A. Mair and H. E. Gamble

5 E . W . E . Rogers, C. J. Berry and R. F. Cash

6 W.P .Jones

7 D.E . Lehrian

REFERENCES--continued

8 W.P. Jones . .

Title, etc.

9 W . E . A . Acum .. t

The effect of model size on measurements in the high speed tunnel. ,Drag of two-dimensional symmetrical aerofoils at zero incidence.

A.R.C.R. & M. 2527. December, 1944.

Miss D. E. Lehrian ..

Tests at high subsonic speeds on a 10 per cent thick pressure-plotting aerofoil of RAE 104 section.

A.R.C.R. & M. 2863. April, 1951.

Wind-tunnel wall interference effects aerofoils in subsonic flow.

A.R.C.R. & M. 2943. December, 1953.

on oscillating

Calculated derivatives for rectangular wings oscillating in compressible subsonic flow.

A.R.C.R. & M. 3068. July, 1956.

10 H. C. Garner

Oscillating wings in compressible subsonic flow. A.R.C.R. & M. 2855. October, 1951.

11 I . T . Minhinnick and D. L. Woodcock

Theory of lifting surfaces oscillating at general fre- quencies in a stream of high subsonic Mach number.

A.R.C. 17,824. August, 1955. Appendix.

Multhopp's subsonic lifting surface theory of wings in slow pitching oscillations.

A.R.C.R. & M. 2885. July, 1952.

12 Milton D. Van Dyke

Tables of aerodynamic flutter derivatives for thin wings and control surfaces in two-dimensional supersonic flow.

A.R.C.C.P. 382. October, 1957.

13 A. Wylly

Supersonic flow past oscillating airfoils including non- linear thickness effects.

N.A.C.A. Report 1183. 1954.

A second-order solution for an oscillating two-dimen- sional supersonic airfoil.

J. Ae. Sci., Vol. 19, No. 10, pp.685 to 696. 704. October, 1952.

14 W. P. Jones and S. W. Skan Aerodynamic forces on biconvex aerofoils oscillating in a supersonic airstream.

A.R.C.R. & M. 2749. August, 1951.

15 A. Chinneck, D. W. Holder and C. J. Berry Observations of the flow round a two-dimensional aerofoil oscillating in a high-speed airstream.

A.R.C.R. & M. 2931. August, 1952.

20

APPENDIX I

Still-Air Damping and Inertia

The assumption made in Section 2.11 that the apparatus damping was due to elastic hysteresis

alone is strictly incorrect since, during the power measurement without the tunnel running, still-air damping forces were present both on the oscillating structure in the side boxes and also on the model in the tunnel working section. The correct still-air power datum should relate to the sanle air density and frequency of oscillation as when the tunnel is running, and should not include still-air

damping on the model. Two sets of experi'ments were made to determine the magnitude of the errors due to ignoring

the still-air damping. In the first of these the model was removed, and each side box was sealed by a plate covering the entry to the tunnel working section. The power to oscillate the system in

each box was then determined with the box evacuated to various values of pressure, and the results

were plotted in the form illustrated in Fig. 45. The relationship between power and pressure is

linear, and extrapolation to zero pressure gives the hysteresis power. At any pressure the power due

to still-air damping is equal to the total power less that due to hysteresis, and it will be seen that

this is small compared with the latter. Corrections to - m0 based on these results were in general

small, but in the case of M = 1.61, where the pressure change is greatest, corrections amounting

to 20 per cent were obtained at the highest frequency and amplitude and rearmost axis position.

The still-air damping on the two-dimensional models might have been determined from the

measurements described above and an additional set of similar measurements with the model

present and the tunnel working section sealed off and made airtight. Since this would have involved

a considerable amount of constructional work, the method finally adopted was to measure the

apparatus damping power with a set of wooden models with chords ranging from 2 in. to 4.472 in.

These models were in the form of flat plates with sharpened leading and trailing edges and were

so designed that they could be fitted to the apparatus without disturbing the clamping of the

torsion bars, since this had been found to alter the hysteresis damping in the system. The

measurements were made at the lowest frequency of the test range (11-5 c.p.s.) in order to

minimize inertial distortion effects, and measurements of power for a given amplitude of oscillation and axis position were plotted against c ~. The resulting curves, an example of which is given i n Fig. 46, are linear, and the still-air power P' on the model is equal to the total power for a given

chord less the value obtained by extrapolating to c = 0. It may be shown by dimensional theory that the still-air power for the model takes the form

and since it has been shown experimentally that the relation between P' and c 4 is linear for a f ixedf

and p, it follows that the function $ is constant for the range of c~f/v covered. This is the same as

the range covered by variation o f f with a fixed chord of 2 in. in the derivative tests, and thus the

value of P' for the frequency range of these tests can be determined with the aid of (6) from the

measurements at f = 11.5 described above. Small changes in frequency occurred during the tests

with the wooden models due to their varying size, but it was estimated that the effect on the power

measurements could be ignored. Corrections to - m O derived from these measurements turned out

to be negligible.

21

No special tests were made to determine the still-air damping on the finite aspect ratio model,

gut application of the two-dimensional results discussed above indicated that the corrections were again negligible.

During the tests with the sealed side boxes, attempts were made to measure the frequency

change resulting from change in pressure, with the object of evaluating corrections to - m o to

allow for the difference in still-air inertia effects in the side boxes due to difference in air density

with and without the tunnel running. However, these changes were found to be less than the

normal scatter in frequency measurements, and reliable values could not be obtained. Similar

considerations apply to the model, the correct datum for the stiffness measurements requiring

still-air inertia effects on the model appropriate to the air density with the tunnel running. Values

for the still-air inertia on the models were calculated by means of flat-plate theory, and corrections to - m o based on these results amounted to not more than ½- per cent in the worst case.

22

APPENDIX II

Error Due to Flexuredr Model

It has been assumed that in the derivative measurements described in the present paper the

model has behaved as a rigidbody performing pure pitching oscillations. This is only approximately

true, since in general there is bound to be some flexure of the model due to aerodynamic lift forces

and out-of-balance inertial reactions. These flexural ~tisplacements will give rise by coupling to

aerodynamic and inertial pitching moments which will be included in the measurements and thus

constitute errors in the measured derivatives. In order to obtain some idea of the magnitude of these errors, calculations were made on the

basis of the following analysis of the system used in making the measurements. It is assumed that the system possesses two degreeS'of freedom, pure pitching about the axis

of oscillation and pure bending of the model, there being no elastic coupling between the two displacements. Also it is assumed that apparatus and still-air damping forces may be ignored. Only the two-dimensional case has been considered, the reference section being taken at the mid-span position. It is clear, however, that the analysis applies equally well to the finite aspect ratio case if

the appropriate mode of flexure is used. The determinantal equation for the system may be written

~ -p~ I~ + L~ + jpL~, - p ~ P + Lo + jpLo -~ = 0 , (7)

- p"P - M~ - jpM~, a o - p~I o - Mo - j p ( M o - 341)

where conventional derivative notation is used, and M 1 is the driving moment applied to the system. Aerodynamic inertias are assumed included in the structural inertias I~, -To, P. If no flexure were present, M 1 could be equated to M o. Due to the presence of flexure, however, an error S M o exists

in the damping measurement, and M 1 must satisfy the relation

M~ = Mo + 3Mo. (8)

Basically the stiffness derivative Mo is calculated from the relation

a o - p 2 I o - Mo = 0, (9)

where the pitching stiffness a0 is known from static measurements and the pitching inertia I o is

determined by a still-air test assumed to be made at the same air density as the wind-on test. Since

in general flexure is present in both cases, errors exist in the determination of both M o and Io;

thus relation (9) must be written

~o - P2(Io+ 31o) - ( M o + ~Mo) = 0 . (10)

An angular frequency p, relating to flexure may be" defined by

- po% = 0 , (11)

and substitution of (8), (10), (11) in (7) then gives

% 1 - - p y + L . + j p Z ~ , - p z P + Z o + j p L o = o . (12)

-p P - M . -jpm , + + j p , M o

23

In still air at the same density (12) becomes

o p 2 ] ' - P ~oP = o , (13)

- poeP, po2Mo

where the suffix 0 denotes still-air values. Elimination of M o between (12) and (13) then leads to the following expression for the errors in the derivatives

8Mo + jpSMo = { - pap _ (M.~+jpMe)}{_ p2p + (Lo+jpLo)}

% ( 1 - ~ ) + ( L , + j p L e )

p~po2P ~. % (1 _ p o ~ . (14)

This may be put in the non-dimensional form

~m o + joJSmo - ( - °J~ff - Kl(m" +jo)m~)}{ - J P + ~:1(lo +joJ/0) }

(.02(.O02/~2 (15)

(1 - 09z2 ]

where/~ = P/pc~l, ez = %/P VzI and K1, I¢ 2 are integra!s of the modal function f(~7) given by

fl fl ~q = f (~)d~, K s = f2(~)d~. (16) 0 0

In the application of (15) to the measuring system f(~7) has been taken as the static flexural mode as determined on the derivative rig for uniform spanwise loading. The same test gave %, and a value

for p~ was obtained by exciting flexural oscillations by means of an electromagnetic proximity exciter set up over the pitching axis. Pitching oscillations were prevented by applying the safety brakes. The product of inertia P was calculated from the known distribution of mass and the mode of distortion. From this point onwards the treatment was purely theoretical. Values for p and M were first assigned, and corresponding theoretical values for p and the derivative coefficients then obtained (Van Dyke's theory for supersonic speeds and flat plate for subsonic).

The calculation of ~ M o requires a value for P0, and this was obtained by first determining I o from (10) and (12). For still air (10) becomes

g0 - po (z0+ Io) = 0 , (17)

which gives a value for poZMo in terms Ofpo z since I o is now known. Substitution in (13) then gives a quadratic in po z-

Percentage errors relating to the subsonic tests are plotted in Fig. 47 against co and M. The effects are small over the range covered, largely due to ~he fact that the model was mass-balanced about the

axis of oscillation (P = 0), and it is reasonable to assume that the actual measurements will not be in very great error from this cause. In the case of the supersonic tests, however, considerable effects are indicated as shown in Figs. 48 to 51, which present curves of - (mo + 3mo) and - (m o + 8m0)

24

against axis position for a number of values of co. These curves are compared with curves of - m o

and - mo corresponding to the case of zero flexure. In general the errors are a min imum for an axis

near the mid-chord position, again due to the mass-balanced condition, and the departure f rom the

- m o and - m0 curves is considerably greater for the 5 per cent thick section than for the 7½ per cent

at the higher values of oJ. This might be expected in view of the smaller value of % (2/5 of the

value for the 7½ per cent section) giving rise to greater flexure. T h e tests for the 5 per cent section

were limited to the mid-chord axis position where the error is relatively small. In all cases the flexural

f requency was never less than three times the pitching frequency, and thus had only a small effect

on the calculated error, since it appears only in the terms a z ( 1 - p ~ / p ~ ) and %(1 - p o 2 / p z 2 ) .

25

l - ~ r ~ . . ! T u n n e l wall

I Y//////////////A ~ =

E~rbhed - ' - . sp~iog I I I I " " - 'x br-~cke¢ ( ~ I bearing I I I I ~ (C')P~ Drivina

~'~, ; II II '-Y-" _ _ - / wh,~ . ~ J " n ' - ® l l ~////,//////////~ b-'O-ti:O:-kll ~ t L - V - ' ' I I I Ir :=' / "--11 ILampII "'v--u--v"O/ Ph0~0 ~,--- I+o .... I, L I J un~,-5 ! L I__I L, Ix

Pickup FIG. 1. Diagram of apparatus.

Amplifier

t Differenc~r

unib

FIG. 2.

J ' ~ , Trigge r

K

~ _ 1 Filger"

Amplitude control.

R~I~

FIG. 3. Frequency meter.

26--

Phase shi f ter Driving ampiiffer 200 v H.T.

Inpu~_.j

FIC-. 4. Driving amplifier and phase shifter.

ing S

550 V tq.T.

',11

k_ - _ q

q

FIG. 5. Electronic wattmeter.

" 20E) V H : T .

Amplibude meber

Frequencg me~er

O~cillak-in 9 aerofoil 5Ljsbem Vad%

Driving coils Pick up

JX/X.

1 W&bbm~b~r

. . . . . I

I IJv i

i > I 1

. . . . J

FIG. 6.

J-U-L

Congrol unib

Pha~e ~hifl~r

Exciting and measuring circuits.

Jv% Driving &mplifier

JV%

28

4-'0

3 '0

- r~ b

~ ' 0

I'0

~'0

-r~

2.C

/

--+-- Oo =l° f-o=H'slc'R~" / / "*X' l ._0_. 0=20 . . ~ r ~ ,

_--oc- e~: 2 °, Turk B.L. t /~

-"~-- [=lal;plaM,,,~4 t ~.," -.-0-- .. ,,," •

I ~ , ~ ~ 0'04

o

_ ~ ~ . ~ ~ 0.o~

3.o 1 _1=v = sT-gs c.p.5

j J

2"0 I

0 0"~

F I G . 7 .

0

,o. ,-o.. --o \

i z ~ 0"2 LO

~O'J

M - criI-.(pre~ JrB~

n t 0

0-4 0"5 0% 0"7 0"5 0'9 M

D e p e n d e n c e of -- m 6 on M and co for = = 0 d e g - - R A E 104 profile, A = oo.

o ~ + t e o = I ° I

~ - - 9o=2 ° FlaP- pla~e (~heory)

- - d - - eo=2 ° Turb. 5.L7

-0"4 I 1

-0.B

-I.2 o

-0-4

W. P. Jones ( M = 0"2')

• -O"B ~

-1"2 0

I ~:_ = H.s~ c,p,5 g

~ll '~ • O.OA

t/

I fo = 3~ 7s ~P2

0 I

fo = 57. $5 C.lO.5__

-0"4 ~Y. P. Jones (M = 0'~')

-~0 ~ - ' - ' = = = ~ = = - " ~ " - - - - o - . ~

- 0 : 5

w

FIG. 8.

-/ '2 0'3

~ 0'2

/,0

B4-~ri~ .(pr¢~ ~ re ) 'o'

0"4. 0"6 0,6 ~ 0"7 0"8 0"9

D e p e n d e n c e of - m e on M and o3 for ~ = 0 d e g - - R A E 104 profile, A = oo.

4"0 I

fo = u.4~ c.p..s.

3"0

2"0

I'0

-I'0 5"0

2.C

I

- - + - - 0 o = I o .

(~o =20

. J

I Nom a. =1-3 °

0

3"D

2'0

FO

0"3 0'4-

F i g . 9.

j 00 + , t

i O

]

, ¢ %

l Lz2_ _

0"l

0

I

fo = ~8-oJ ~.R.__~.

• ~ w I

~ - - ~ ~---M--cri~ (upper 5urfac

0 I i Nom,a = f.3" ~o= II-4~ C.p.s.

-O.A-

-r~ e

-0 ,8

- I . 2 0

-0.4

-0",

- I -

_l

~- - - -q '==~ :~"=========== c~,=,,_.__

I + - - Oo: 16 ~o : ~,5,7-~ C. >.5.

0 - - 0 o : 2 °

O'

- o

f - M o

D'2 -O-9

LO

-')o - I . 2 0"5 0-6 0-7 0"8 0"9

M

D e p e n d e n c e o f - m 0 o n M a n d co for o~ = 2 d e g - - R A E 104 prof i le , d = oo.

I ~'o = GB .Ol - C.pS.

71@

0,3 0,4- 0 - 5 0 . 6 M

O -

M- c ~ . (up~,¢,.- ,s,.--r,~ e) ,-,_ 0. " / 0.8 o,g

F r o . 10. D e p e n d e n c e o f - - m e o n M a n d oJ for c~ = 2 d e g - - R A E 104 prof i le , A = oo.

8'D

5"0

4'0

2'D

0

-~'0

i

Nom. 6 = 2 0

_ - - + - 0 o = i °

---c----- ~ o = 2 °

c~

, I

fo = Jl.s3 c.R. 2

.I t

-4"q

1

4

C~

-~'0 0

M-cH~. (uppsr ( , , u r f a c e )

0"3 0"4 0"5 O'g M 0'7 0.8 0,~

FIO. 11. Dependence of - m ~ on M and co for = 3 d e g - - R A E 104 profile, A -- oo.

)'04-

/.0

0"02

3'0

2-0

I-0

-I '0 3'0

2"0

I'0

1 i

-- 'V 8o = }o fo = 35"75 C.R._.55.

~o =2° - - - q - - B o = 2 o , Turb. 13.k.

0

-I.D 0-3 0-4- 0.,5

I

fo =GB.05 c.p..~.

j!o,o ~o

0'05

/

/t~ --+

J o,, f5 f •

VJ

, - - I o o-o M o.7 o-~ o.~

FIG. 12. D e p e n d e n c e of - too on oJ and M for o~ = 3 d e g - - R A E 104 profile, A = oo.

O,} tO

-D'4

-0"B

°J,2

-I)-4-

-0":

-1"2

No, , . ~ : ~° ~=o : I ; - ~ ~.p.~ ~ , . . . ~ - 4 - ~ ~)

, 4

_ _ _t - - + - - 00 = I ~ fO "='~5 " 75" Cl ~ 'S~

- - -o- - - Bo:2 ° - - q - - Oo ~ 2 ° , Turb. B.L. .

fo = 88.05 c.p.s..._ a

-0'4-

-0 "~

- l ' 2 1 I)'3 {1"4-" 1)'5 Q'G M

I)'2

(o

{) 0"7 0"8 0"9

FIG. 13. Dependence of - m o on M and co for = 3 deg--RAE 104 profile, d = oo.

I Theore~c~l Tree sLrea, m With l:unnel fnl-er'-Fer-ence

------0.---- Wfkh ~.unnel inl:erFerence, khlck'ne~s, ~l, nct viscous

• - 0 e ~ T e c ~ . - - - - - ~ Ma~n ¢ x p e r ~ n n e n ~ l

(0 ° incidence).

2-5 r--,- i

" ' ' ' " - ~ ' ~ FILk. .~p~pl~ke (M=O-7)

~........~....~ - ...

Flat p~eCM:o@ ~ , ~

M=o-s ~ . . . .

E~ui~lenk lorof]le (M=O@

0 0"04 O-OS ~ 0-12 OdE, O' 20

FIG. 14. Theoretical tunnel-interference, thickness and viscous effects. (Two-dimensional theory)

4-"

0

- 4

- 8

O'4- 0.5 0'6 M 0"7 0-8 0'9 O'4 0'5

4

0

- 4

- 8

-12

0"6 M 0'7 0.0 0"9

FIG. 15a and b. Effect of square-section trailing-edge spoilers on - t o o for a = 0 d e g - - RAE 104 profile, A = co.

-2

. . . . . . . . . . . No . spoi ler O 0 "04 S~. spoiler cz~ 0.65c on one 6ur(~.ce

0.0~ Sq, 6po~ler ~F. O.GGc o~ ~ suFFa.ce

ck=O ° (,&)

o.4 0,5 0'5 0"6 M 0"/ 0-8 0'~1 0"6 "M o,'/ o,8

. . . . . . No spoiler" ~ O - - 0 . 0 4 ~ f~q. s p o i l e r ~ 0.6~Jc on lower ~ ~urfe.ce

• u - •

~ A - - O . O 4 Sq..spo~ler &k 0.65¢ on e&ch sur~&ce

t M = cr,l:. ( p r e s s u r - e )

0 , 9 ' 0 - 4

FIG. 16a and b. Effect of square-section spoilers at 0' 65c on - m O for a -- 0 deg and 3 d e g - - RAE 104 profile, d = oo.

33

(83827] C

. . . . . . No spoi ier - - 0 - - 0 . 0 4 5 " @~. ca."olell~.Eed spoi ler o n one surface. • - - , ' , - - O - O d - 5 " s g . Ca.sl 'etla.t.ed s p 0 i l e r o n e a c h s u r f a c e

. . . . . . iqo sp6~(er - - o ~ O.04.S'sop. ca,%}~ellaLed spoiler- or~ lower- sur4Ca, ce

,h.--0.0~'r$"$oF.e~R,5~'.eII6~Lec{ $po{ler or", e~.c~ 5ur-~6Lee

4

I

4

_ _ _ ~ -= 5_...._ ~

/ l

5 (pr~ $~re>

, , / 0-4 O'S 0.6 0-7 M 0"~ 0.~ 0.4 o.5 O.& M 0'7 O'S

FIG. 17a and b. Effect of castellated trailing-edge spoilers on - m 0 for c~ = 0 deg and 3 d e g - - RAE 104 profile, A = oo.

/ ! , ,', 2 i = *

/ * i

1 h/o Spoiler V

l ~ I 0.04'50.,Spo~er on ~ - - O.0~'s~. spoiler on

"~0.6 9 0.4 O.B

No spo{ler b - - D ~ O,OZ s~,$poller or% lower sur~:o-ce

- - ~ l O.OZ"Sq, spoileFon each %ur~Cace

I 0"5 o.g N 0,7 O'B

t o.~ 0.6 M 0.7

Fie . 18a and b. Effect of square-section trailing-edge spoilers on - m 0 for c~ = 3 d e g - - RAE 104 profile, A = co.

34

0'&

0'4 0"4

re 0

0

-0 '4 '0'¢

-O'& D'&

i 0

. . . . . . ~ ~ U ~ U ' ~ U' ~ 0"B M 0"7 0l ~ 0 ' 9

FIC. 1% and b. Effect of square-sect ion trai l ing-edge spoilers on m o for ~ = 0 d e g - - R A E 104 profile, A = oo.

0"8

0"4

• (a)

0

-0"4

( Z = 0 °

- - -o - - - Ho spoiler

o,o*" sq. spoiler aB 0.65 c on one surface o'od'sq, spoiler a.h~ 065c on each su/face

0'4 0"5

FIe-. 20a and b.

I

C~=~ °

N o spoiler- - - o - - - o'o¢"sq, spoiler aB O-GSc on lower 5urf&ce

O ' { } * " S q I S p O i l e l " &B O'~SC. or) each 5urfa.ce

0"~,

M J I - '.~urface)

O'G M 0"7 0"~ 0"9 0'~ 0"5 0"G M 0"7 o'& 0"9

Effect of square-sect ion spoilers at 0 .65c on m 0 for o~ = 0 deg and 3 d e g - - RAE 104 profile, A = oo.

35

0-6

0'4-

Ta

i (a)

°b!g4;'P'iqerP°i'er °n °he 5ur face L 0.045" sc ...spoiler on each 5urf&c~

-0'4 ~ I M-otis (ores~ure). ' I 3,

0 . ¢ ~ 0.5 0.6 hl 0.7 0.~

F m . 21a and b.

/~ (b) \ 0.8

x _ _ 9'4

C~ =3 ° ~ No 5poller

---o--- o,04¢ sq. spoiler on lower surface 0'045" ~q. 5poller on each surface

0"9 0'4 D'5

a-crib @pper 5urfa@

O.G k,I o,7 o-~

-0'4-

u'9

Effect of castellated trai l ing-edge spoilers on m o for eL = 0 deg and 3 d e g - - R A E ' 1 0 4 profile, A = co.

0'~

0"4

° \ No spoiler o' 07" sq. spoiJer on [~er surface o.o2" 5q. spoiler on each surface

"O'~

O'4 0"5

I I M- crik"~l* P P e r l 1 surface)

O'G ~'1 O'7 O'a

FIG. 22a and b.

u'9 0"4

~ ~,. (b) o.~

0'4-

D

No ~poiler 0-04:' ~q. spoiler on upper surface O" 04" sq. ~poiJer on lower surface 0.04; ~q. ~poiler on each 5urfac, e

-D'4.

M-crib. (.upper surface)

I ~ I -o. 0"5 0"& M 0"7 0'B 0'~

Effect of square-section trail ing-edge spoilers on m o for e~ = 3 d e g - - R A E 104 profile, A = oo.

36

l,C

o.~

I [ ~ I o I ~ o ~ o o = z ° ~ = . . 3 z c . p s . I ~ I m " + ~ O o = l ° - - - - I J~ I ---O--- Lehrie.m ~ I ~ r" ~ I o.o~

-"~d / I ¢ 15 a~

s< / / / o

o 0

I / / O /

I'C

---..<___

..~= s'7.Sl, c.p.~.

/ /

..¢,/ / -

o

-3 0,~- 0,5

td

o.2

o o -6 M 0 . 7 0.~, 0 . 9

FIo. 23. D e p e n d e n c e of - - m 0 on M and o) for ~ = 0 d e g - - R A E 104 profile, A = 4" 0.

° I i C'I ~ " 6

~--Oo= I Lehr~n ~ - 1 Acutn heo~ '

O. 2

c M=0 o.N ~ .

- 0 . 6

- 0 - 2

-o.~'

-O-G

I ~:o= 35-40 C.p.~;.

o . z o

o

- 0 - 2

-o-Z

- 0 - £

0"3

Fro. 24.

/ M = o

I

0 , 2 -

=

0

0 ' 4 0 - 5 O-g 0 - 7 O ' g 0 - 9 M

D e p e n d e n c e of - m o on M a n d o) for c~ = 0 d e g - - R A E 104 profile, A = 4"0.

O0

1.0

0-5

I ~ O o = z ° - - - - - 4 - - - 0 o = I °

I

4:0 = %5'41 ,F.p,~.

I

"~" 0"1

~o -'~ 57.4,G c.p.s.

1.0 ~ 0.4"

o-s " " - _ . . . ~ / , ' ~ o.z

- tie r

o ~ r I o o.~ 0.4 0.5 o,e o.7 0-8 o.9

M

FIG. 25. D e p e n d e n c e o f - m 3 o n M a n d co fo r c~ = 3 d e g - -

R A E 104 prof i le , A = 4 . O.

-0"~

-0.~

-01

-m e

e o = Z ° --+--?o = IO

J f o = ' ~ 5 " 4 1 c..p..2 : O'X

.,....,.03 0"(-

~ , ~ . - - - . ~ . ~

0

-0~

-0'4

--....,.

- m 0

fo = 5 7.45 o p.~_

j o O

-0.~ rA -cr; t {.upp~rlOurfa~a)

I

0"5 0"4 0-5 0'~0

0"4-

O.Z

0-7 0"8 0'8

FIG. 26. D e p e n d e n c e o f - m 0 o n M a n d co fo r o~ = 3 d e g - -

R A E 104- p rof i l e , A = 4-.0.

1.5

1.0

-rno

13

- 0 - 4

- m 0

-0"8

~ ×

-'-----o-.

-X" M= O.B7

M = O - 8

= 0 - 7

~ ~ = 0 °

_ 5 °

~ , ~ , M-0'3

40..__

0 0 . 0 5 0' 10 co 0"f~ O. 2 0

FIG. 27. Dependence of --m0 and - m e on oJ--RAE 104 profile, A = 4.0.

39

- 0 . 1 B

- o . ~ , 8

- o.15

- m 6

- o . ~ , 5

- 0.1t5

- m e

- 0 . 5 £

- o.18

- m e

- 0.88

- o . 1 8

-ra{)

-- 0.58

(a.)

( ~ -

a) = 0"0|15

I Z O o (deg) .5 4-

co = 0 ,0198

-.--.-.____.___

I i z oo (de9) 5 4

. ~ ~ [ ~ = O.OZ,91

I 2.. e o (de. 9) ~ 4

~0 = 0.0573

Z. O o (dag) .5 4

co = 0-0461

FIo~ 28a a n d b.

z Oo (d~ ) ~ 4

-- 0.60

m e

- -0"80

--0"60

m e

- -0"80

-O,coO

n] 0

-- 0 '80

-- 0 ' 6 0

m o

- 0'80

- 0.60

m e

-- 0.8O

(b)

z eo (de 9) 5

aJ = 0.0113

z % (deg)

z&

A

-- O'OIS8

4

4-

u.., = 0,0 2 . 9 1

z e o (deg)

~o = 0.0~7~

D e p e n d e n c e o f - m 6 a n d m o o n 0 o a n d o ) -77 ½ p e r c e n t b i c o n v e x prof i le , M = 1 . 4 2 .

4

~d = 0'0461

I Z 8 o ( d e @ ) 4

Axis at l e a d i n g edge .

J

4x

0

--m 6

--O'ZO

0

- - m e

-- O ' Z O

0

- r r l ~

- - O ' Z 0

0

- - rn 6

- - O ' Z O

0

- - m e

,..,3 = 0 ' 0 1 0 0

x ~ p

×

z e o (de9) a 4

~ = 0 ' 0 1 ~ 1 1

i Z e o (de9) 4-

co = 0 . 0 2 . 8 5

z Oo(da 9) s 4

- 0 ' 2 0

Z % (dag)

eo(deg) 2_

FIG. 29a a n d b.

CO = 0 ' 0 5 7 1

- - 0 . | 0

m e

- 0.50

- - O ' l O

m e

i

- O ~ O i

i

- - 0 " I 0

m e

- - 0-,50

--0'[0

rn 0

- 0 " ~ 0

. ~ = 0 . 0 4 5 6

- 0-I0

wl

- 0 - 3 0

z eo (deg) 3

~- -~ O ' O l O 0

--..- ~

2. O o ( deg ) 5

= 0 ' 0 1 9 1

co = 0 . 0 7 - 8 5

Z e o (deg) 5 4

+ " - b ~

cO = 0 ' 0 5 " 7 1

4

1

4 z % (deg)

D e p e n d e n c e o f - t o o a n d m 0 o n 0o a n d co--7½ p e r c e n t b i c o n v e x prof i le , M = 1 .42 . Axis at 0"25c.

z e o(decJ) 5 4 "

oo = 0 " 0 4 5 6 !

4~ I'O

O.ZO

0

(a)

: 0 ' 0 0 8 1 7 I

I z e o (deN) 5 4-

O'Z5

- m e

O-OS

2__ I z e o [deg)

~_= o .o i a4

-lkClo ~

s°lo L_ 0

4

O.ZO ' J - - ~ , . . . ~ Lo = O-OZ.'/8 ~_

i : i - me - ' . . . . . . . . ,___ 5% I- 0 4 I Z {3 o ( d e £ ) -~

tO : 0 . 0 3 6 9 025

- r % - ~ L _ - ' ~ - ~ - ~ . ! -°--- ! O'OS

i Z e o (de9) 5

0-25

- .% ~e-'----~

O'OS'

~o = 0 . 0 4 5 4

0'35

0 . 1 5 I

(b)

uO = O . O O B I 7

Z Oo(deg)

7a°lo

0"30

= 0.011~4

I o -~ lzi<> )"

,_ 5 %

OI0 I ~ Oo(deg) 3

I L 0 o (decj) 5 4

0"50

7kL°Io

9 3 0 I Z eo (deg) ,:3

I i 4

0%0

;00

= 0 ' 0 3 6 ~

z e o ( d ~ ) s ~- z 0o (de~) s

co = 0 - 0 4 5 4

F i e . 30a a n d b . D e p e n d e n c e o f - m 0 a n d m o o n 00 a n d co- -7½ p e r c e n t a n d 5 p e r c e n t b i c o n v e x p ro f i l e s , M = 1 - 4 2 .

A x i s a t 0 . 5 c .

4 x

0"75

- z ; z b

0"55

0"75

- r n b

0.55

0-70

-m~

0"5C

O" 7C

-#'zO

0'50

0"75

- ~ e i

0"55

®

j / x / / - x w : 0-0067

t

A

Oo @~9)

I

I .

I

aJ = O' 0175

/

w = 0"0271

t j

+ . . , ~ _ _ . _ _ + ~ / ' ~ # cu= 0 ' 0 3 6 4

2 0o (deg) 3 4-

/ co = 0' 04..47

FIC. 31a and b.

0"80

77Z O

uJ = 0"0062

0-60

0"75

~O

0'55

0,75

~o

0"55

0"75

8

0"55

, Oo @eg) 4

13,

A

I

co = 0 . 0 1 7 5

03 = 0.0271

8o @~9) 3 4

co = 0"0364

2 eo @~9) 3 4 0"7~

~Tz o

0"5~ =

co = 0 " 0 4 4 7

Dependence of - m 0 and m o on 00 and co--71 per cent biconvex profile, M = 1.42. Axis at 0.75c.

-N

0.8

0"4

-0 4

0"6

0"3

0

-r/z 0

-0"3

~- ~ --(9-

I

O" 750 aXiS

)9.5c axis ;

~- . . . . . . . . . ~___ 5%

t 0"25c axis

L.E axis x

÷.

-0"6

0-01

LEa~--~"

f 0.250 ax~s , t &

/ / 0-5c~xis

o

5%

,I

#PO" 75c axis

0"02 co 0'03 0-04

' - - - - - - v -

FIe. 32. D e p e n d e n c e o f - m 0 and - m 0 on co for 7{- per cent and 5 per cent b i c o n v e x p r o f i l e s - - 0 o -- 2 deg,

M = 1-42 .

0"8

0"4

.~Tz@

-0

-0"4

----o---- E×perlmenb --7"~ % bhick

.... ×--.- Theory ~Yan D.~ke)--7~ % [ hick

. . . . . . . Theory (flaL pla~e) u ExperimenL -5% chiok ] # ~ " / /

z / / A Theor~ ~,~n D9~-5 % chick @ Earlier Lesbs

./ ' l/z//

ii ¢/'*

-72Z e

-0"3

0-6 ~ ~\ \

o J %

-0"5

h E

, "\,

FIc. 33.

\ , ~\x

, • \x

\ \ \

\ \

0"25c 0-5c -7~0 Axis posibion

Dependence of - m ~ and - m o on axis

"I.E.

p o s i t i o n - - 0 o = 2 deg, w = 0 - 0 2 5 , M = 1"42.

0 0.5

- m . 6 0

- 0 " 0 5

@) o

""~ ~ . . , . . . . . , . . . . . . . ~ . ~ co = 0'0183

(b)

• -0 '53

~ " , , - ~ " co = O. 0183 0 " ~ 7ZL 8

-0'63 I 2 0o (deq~ 3 4- \ u /

0'05

0 - 7 ~ . ~

-0'05

0,10

- zrc~

0

Turb. B.L.

u3 = 0 '0262

, 2 0o @09) ~

- - " 4" + +

m=0'0348

-0-55 ~-= ]

- -~ ~ co = 0,0262

-0"65

I 2 e o ( d e g ) 3 4 I

Turb. B.L.

- 0 - 5 4 ...%_ [

- 0 " 6 4

m = 0"0348

7 Oo (de9) 3

o.05 "" ~ ' ~ " - - - 2 ' -os6

nz 8

70"05 '1 2 8 . ( ,deg) 3 4

- 0 ' 6 6

uJ = 0"0425

I

FIG. 34a and b. Dependence of - t o o and m o on 0 o and co--7½ per cent biconvex profile, M = 1.61. Axis at leading edge.

0 ,[0 ~J-" 0.0090

4-

-O- I~ '

7n 0

(6)

0;=0.0090

i 2 O~(deg)

0%

0.07

-~

0

- 0 .05

0 . 0 5

0

-0.05

J ~=0.0176

4

I~=0,0263

~=~04Zg

:-n o

- 0 . 2 4

-O'L4

-D-Z4

.o...._. CO = 0"0 I"~b

z Oo (~e9) 5

--..< tJ= 0.0265

2 8 o (oJecJ~ 3 4.

0 . 0 6

0

- 0 , 0 7 5

- O . I g

~0

- 0 , 2 0 4

~O =O-O425

FIG. 35a and b. Dependence of - m 0 and m o on 00 and o~--7½ per cent biconvex profile, M = 1.61. Axis at 0-25c.

4

"-3

O'lg

- ~ 6

0-05

Oq5

0.05

0.165

-'m(~

O" 0'55

"--"X'------------,Y.-- ~ J ~?o

~ " .0-25

i i

~n o i

3 4

61 "- 0.017l I

>--____.~& . . . . . . . . . s% -I~---

_ ~ ~ *Turb.B L.(? ~ )

'~ ~ ~ _ [ ~ =o.o26"7 .

, . . . . . . . . . . . . . S%

7 I z Oo (a~9) s

0"15 o. o 340 - -

->'~ . . . . . . . . . . ~}

D-05 I 2 O~ ~ - "~

X C6 = 0.0418

C]

0.05

b ~

0.1"5

O.Zo

t O'I0

O.Zl

"m, e

0'11

O.zo

®

[0= 0-O07B

7~O/o

I ~ = 0.0171

' -~ ........ "~ .... T /Y

. . . . . . ~ . . . . . . . . . . . . . . . . . . . . . . . . . . . _~)_ ?__

I A t , r~rb.S.t (7{%5 -- 0 . 0 2 5 7

e/. I i z Oo (a*9) s ÷

O, I0 {

O.2~ m

"z " ~ - - a _ .

__] i ~:.._a-o,~o | + ~ +._.._____ • . _ _ _ _ _ _ _ . _ _ ~ _

7"~%

I ........... i " ~Caeg) , +

~=0,0418

i 0. x4

Dependence of - - m O and m o on 00 and ~o--7½ per cent and 5 per cent b iconvex profile, M = 1" 61. FIG. 36a and b. Axis at 0' 5c.

0 " 4 0

0'50

®

o ~

G)=0.0165 0.60

0,~0 i

(b)

03 = 0 "0165

0"43

--2#

0"53

0,41

0.5[

T • ~.-Tu ," b. 5. L.

'~ 03 = 0 - 0 2 4 "

ft.. ~ = O. o 54.._,_~1

0 . ~

0 " 4 ~

0.60

%

0.60

~ T u r ~ 6.L. co : 0 . 0 Z 4 -

+

(0=o-o"%41

z I Z 0o (cte @ 3 4

0 '44

0,34

C~ = 0 . 0 4 2 0

I 2

FIC. 37a and b. D e p e n d e n c e of - m 0 and m 0 on 0 o and co--7{- pe r cent b iconvex profile, M = 1" 61.

0.69 / aJ=O.O4ZO

-ta0 D l - -

/ 0"49

I Z O o ( d e ~ ) S 4

Axis at 0 ' 75c .

- N

0"4

0-3

-rag

0.9.

-0 " I

0.6

0

-0-6

+

7¢/.---a. _~_~_...~ _

A

. . . . . -*0- . . . . ~- - - - o - - -

0-5c &xi5

0.'/5c &x[5

O.Sc a.xi~ 7zOlo

. . . . .

5% O.2£ ~xi~. L ~ /~xis

L.E. ~. xis

O.ZSc ~Xi5 / ,

L_

x - '17 .

v~X ,,. ~.._.

. . . . . o__~_~o

O.Of 0.~2 03 0-03 0"04

Fro. 38. D e p e n d e n c e o f - m 0 and - m o on ~o for 7 } per cent and 5 per cent b iconvex prof i l e s - -

0 o = 2 d e g , M = 1"61.

0.4.

0"3

D.7.

0 ' l

-Off

- + E;,..perirr~nl:," - 7~ O/o ~hick I/i//~ - -

- - ~ - - - ~ o ~ s (wDsk~)-7*°/oLhi~ /~//0," .... . . . . . Theory (~,~ pl~) / / ' / :

/ / m gxperimenb- 5 % thick / ? /

. . / '

•

; ~ Earl ier ~ S s ~ # '__/ / / I / I I

\"~\\\ / / 1 1 II ; " \ / / ' , / \ \ I I .',

,, \\ ~i" ~"x " ' ' - / , - ' / / ~ "

0"6 " L ~ \ \ \

0"6 .. "%

-~r~ o \ \ \

o ",%.-

-0"3

-0 '6

\ \

L.E

FIG. 39.

\% " \

005c 0.5c 0.75c T.E Axis posiSion

D e p e n d e n c e of - m 0 and - m 0 on axis pos i t ion- -O 0 = 2 deg, ~o = 0 .025 , 3 I = 1 .61 .

0"75

0'50

- ~z 0

0'25

-0'25

I I

Esbimabed exFerimenLal curve Theorebical curve ('/0% chord') . . . . . "). / lnd order Theoreblcal curve ~ull chord ) . . . . . . . . . . . . ~ f /--" bheor~.

/, I / /

/ 0.7~//// //0"g0

/ , / / , /

0'8

0.4 \ ,

-~o \ \

0

-0.4

t

/ / /

0'75 ~(9~e posi~',on)

0"75 I'0

\x'X , ,

x£ ,

rio. 40. Comparison with theory for 70 per cent chord--7½ per cent biconvex profile,

00 = 2dog, co = 0"025, M = 1"5.

50

tart

--0"1

- - 0 " 2

--0"3

--0"4

-0"1

--0"2

- r i Z ~

-0"~

- 0 ' 4

" ~ /00 ~

Oo @~.) 4

• tu - o ' o4--/t

Oo (d~9.) 3 4

FIo. 41. Effect on - m O of cutting off trailing edge- - 7½ per cent biconvex profile, M = 1.42. Axis at

leading edge.

0.$

0'3

0-3

0"9

2 8o@e~" )

"zd=0"296

I00 °,4 c

90 ~ o

8 0 % c

70%c L

0.'7

-7/i

0,!

0 . 3

• " ~ L ~ ~ :3"" ~ ~

Z 0 = 0 ' 0 4 7 1

,__2O2/o~_

90 % c

8 0 % e

70%c

0o@~9)

FIG. 42. Effect on -- m o of cutting off trailing edge- - 7½ per cent biconvex profile, M = 1.42. Axis at

leading edge.

2"0

..> /-

1"2

1.0 e1=1.47

. . 5 / / - °° J 0-8"

°8 oo - 0 , ~ J ~ °~ 4 -0 .3 ~o-~ - "~ o

/ j / J / / / '

/ / , /

/A'"

/

. J

J / /

~ 0.8 /

I 0.2 o-a 0.4 0-5 0.6 0.7

Fro. 43. Error triangles for determination of I~ and l O -me--7{- per cent b iconvex profile, 0 o = 2 deg, co = 0' 025.

52

~ o

I ' 0

-I'£

&

- ~ i - 2 <

-3"(

N " x

" \ \ \ \ N \"~7%z,~ \ x o/

~ N " " x

M=I '61

/ , I

• \ N

'"¢2 \N \ " \

\ \

Expenrnenb Th o j (va

M=1.42 d L~

t M=I'61

\ *~-'d'.~/,N@\ \ \

\ \ \ \

T _

M ~ t . 4 7 / /

1.5 ~ - . . . . . . . . . ~ . . . . . . . . . . . . . . . . . 7 . . . . . . . . . . . . . . .

t ' 0

M=[.61

0"5 t

h (axis pOSlCion) I L 0'25 0-~ 0,75 ~. 0J

FIG. 44. Dependence of ( l 0 - m~), - l~ and ( l o - m~) on axis position--7½ per cent biconvex profile, 0 o = 2

deg, co = 0-025.

0"05!

oo~o

voz~ o

Oo = ~o

...---.o

i I

H~ ". bere~i 5

I i i I

~ b l l l a i r

FIG. 45.

I 0 20 5O Pce~,suce,,, {iqch¢5 of 149}

Apparatus damping in side box No. 1.

D00;

pi (wab~a

0"001

I 0 o = z °

.., c l = 11'5" c / ~

= 0 5

5id ¢., b o ~ e . 5

I 0~) I00 ZOO

c 4 ( , n c h unib.5)

/ I

F m . 46.

5OO 400

Two-dimensional still-air damping.

- - 0 ' 1 0

[o ~ rl"or"

into O

- - O ' Z O - -

- - 0 "50

\

- 0 '40

0 O'OZ 0 '04- 0 . 0 6 b3 O' 08

I ' Z 0

°/o ~rrer in m~

-0"8

-0 ,4

j ~ ~ J

J J

f f

J

O- IO O' I?.

M = 0 " 5

~ = 0 - !

o.oz o,o4 o,o6 ~ 0.o8 o./o o.lz

FIG. 47. Percentage error due to f lexure--RAE 104 model, h = 0.445.

54.

t.,n

['4= 1.42, ! I -nq 6

x-- ~ = O O Z

0"~ - - -------o----- ~o = 0 0 4 ~t ,..~--' ~ =006

-----o co =0 OB

0"4

0-2 - (% + Sr%l ,/'/'/~" - 0 " ; ~

0"6 M = 1-61

- m 6

0""~ ~ ~ = 0 - 0 ? _

o3 =0-04 ua =0 06 ~ =098

O'Z

-0"1

L.E.

FIG. 48.

O ZSc O'5o O'75 c~ Ax15 ~o~itAon

Error in --m~ due to flexure--7½ per cent biconvex model.

1"0

0-5

-OS

~.o

06

- 0 . 5

I

-('rn e ~- S m e)

I M = 1 '4z

me

OJ =0 02

o to = 0"0.¢

u.;, =0'06

LE. O-ZSc O 50~ 0"750 A×Js po~ bioq

FIG. 49. Error in --m e due to flexure--7½ per cent biconvex model.

I

M=I"GI

.m 0

x 00=0"02

~-- co = 0"04

a 00=0'06.

{Jo

Ox

I'C

= 1.4Z

o.~ - ~e

x w = O'OZ

i ~ - - - - Io = 0 0 8 O'E

0 4

0 '2

- O.i~

- 0 . 4

0 '8

0"6

0 "4

M = I 'GI

. . . . . . . n31~

u) = O.OZ

u0 =008

- O - Z

/ / / "

f

/

0

FIC. 50.

O ' Z S o 0 . 5 ~ 0 ' 7 5 o A x i s p o s i a i o n

E r r o r in - m 0 d u e to f l e x u r e - - 5 p e r c e n t b i c o n v e x m o d e l .

1.0

0"5

~0"5

1.0

0"5

- (ma+Srn e )

- O ' 5

L.E.

Fic. 51.

O ZSc

I M = 1 ' 4 Z

rn 8

× co = 0 -02 .

co = 0 - 0 8

M = 1"61

• . . . . . . rn A

c o = O.OZ

m----- Lo = O-Og

1-5c 0.75c

A x i ~ po~ ic , i o n

E r r o r in - m 0 d u e to f l e x u r e - - 5 p e r c e n t b i c o n v e x m o d e l .

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