The determination in flight of aircraft dynamic characteristics

THE DETERMINATION IN FLIGHT OF AIRCRAFT DYNAMIC CHARACTERISTICS By H. C. Lyster

Flight Research Section

National Aeronautical Establishment

Since the earliest days of powered flight, an increasing understanding of the dynamic behaviour (stability and controllability) of the aeroplane has

characterized the good designer. The success of the Wright brothers was to a large extent due to their insight into control requirements developed during their gliding experiments. The importance of aircraft dynamics in the determination of aircraft structural loads was brought out in an article (Ref. 1) appearing in the last issue of this Bulletin. It is intended here to describe briefly the position we haye now reached in our ability to Measure these dynamic characteristics in flight.

During the thirty-five years between the Wrights I first powered flight and the beginning of World War II, a number of important developments occurred. The wing-warping system ofthe Wrights was replaced by the aileron system of Bell, Baldwin and Curtiss (Ref. 2). Lanchester described and named phugoid motion (Ref. 4). During World War I, spinning was sufficiently _common that investigations were made which led to an understanding of its dynamics (Ref. 5).-During the twenties, a good deal of testing was carried out in the effort to under-stand and prevent flutter (Ref. 6). hi 1937 Bryant and Gates attempted to under- standard-ize a nomenclature for stability coefficients (Ref. 7).

During the thirties and World War II, the autopilot developed until it became standard equipment on all but the smallest aircraft. Autopilot design, as

a branch of servo-mechanism theory, was indebted to the work of Nyquist (Ref. 8) and of Bode (Ref. 9).

Since World War II, the tremendous expansion of the performance envelope of airborne vehicles from ground effect machines to supersonic aircraft has resulted in intense activity in the field of stability and control. Milliken, surveying the situation in 1951 (Ref. 10) listed 127 references, of which only 20 were written before 1946. Much of the advance has been made possible by applica-tion of new tools - analogue and digital computers. An indicaapplica-tion of the rate of advance and the change of emphasis may be seen by comparing textbooks written in 1950 and in 1958 (Ref. 11 and 12).

One of the most promising of the late developments is the adaptive auto-pilot, a device designed to remove the dependence of aircraft response character-istics on dynamic pressure, altitude or Mach number (Ref. 13).

Following this brief history, let us examine the underlying theory common to all airborne vehicles leaving aside any detailed discussion of the complex problems

of non-linearity, compressibility and structural deformation.

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DME/NAE 1961 (1) 2 -DIVERGENCE SUBSIDENCE TIME 2

ATIME

V

_s CL UNDAMPED OSCILLATION

(NEUTRAL DYNAMIC STABILITY)

-DAMPED OSCILLATION ( DYNAMIC STABILITY) NEGATIVE DAMPING (DYNAMIC INSTABILITY) FIG. 2

- 3

THEORY OF DYNAMIC STABILITY AND RESPONSE CHARACTERISTICS A system is stable if, following a disturbance, it tends to return to its previous state. It is statically unstable if it tends away from that state. If there is no oscillation the stable mode may be called a subsidence, the unstable mode a divergence. If, during the return to equilibrium, there is an oscillation, the amplitude of which decreases with time, the system is said to be dynamically stable and the rate of decay becomes the damping of the oscillatory mode. If the amplitude of the oscillation remains constant, the motion is undamped. If the amplitude in-creases with time, the damping is negative and the system dynamically unstable. The frequency of the oscillation and its rate of decay are dynamic stability

characteristics. The parameters describing the dynamic behaviour of the system in response to control movement (in aircraft, usually elevator, rudder, aileron) are dynamic response or dynamic control characteristics.

When an aircraft can be considered as a rigid body, six equations (the equations of motion), one for each degree of freedom of translation and of rotation, are sufficient VS specify its motion. The assumption of rigidity includes any specified movement of a control. If, however, instead of control movement, control forces and hinge moments are to be investigated, an additional equation relating forces - inertia, aerodynamic and input - must be written for each control, raising the total number of equations of motion to nine.

Finally, if there is significant distortion of the aircraft under load an additional set of equations may be required to specify aeroelastic effects. If the distortion is not too great, it may be possible to treat the actual aircraft behaviour as if it were that of a rigid aircraft modified by elastic factors which vary With dynamic pressure and with Mach number (Ref. 19 and 20).

Let us return to the six non-linear force and moment equations. It is common practice to linearize and non-dimensionalize them, but one rarely finds all six non-dimensional, linearized equations given by one author identical with those given by another. Table I contains a comparison of the normal force equa-tion as written by one author with the same equaequa-tion as written by another. It will serve to illustrate some of the several factors which cause differences.

First, linearization by the method of small disturbances consists essentially of expanding a function in a Taylor series of which only the first few terms are retained. One author may retain more terms than another.

Second, there are several different axes systems in use, the most common being stability axes, which are fixed in the aircraft but for a single manoeuvre. In the stability axes system, the X-axis is defined to be through the centre of gravity of the aircraft parallel to the direction of the flight path before movement of the control. Each change in angle of attack or in centre of gravity of the undisturbed aircraft (whether due to altitude, Mach number or all-up-weight) results in a change in the system of axes. On the other hand, many of the sensing elements, accelerometers, etc., remain fixed in the aircraft, necessitating a

4

-TABLE I

ILLUSTRATION OF DIFFERENCES IN THE NON-DIMENSIONAL EQUATIONS OF MOTION AS WRITTEN BY TWO_AUTHORS

EXAMPLE: THE NORMAL FORCE EQUATION

According to Seckel (Ref. 35) 6 C

M L

c + u + + D0 - C 6

L 2 -am z z L e

a 6.

According to Etkin (Ref. 12)

(2CLo C + (24D - C D Cz ) a - [(2k + CZq ) D -CLo tang a - C n = o

To compare, Seckelts equation doubled and re-ordered becomes

a cL (2cL + m u + (2D + CL) a - 2D0 + C, 6_ = 0 a _L'6e A cz. cz tan00 CZ 6e Comments

Undisturbed aircraft lift coefficient.

Compressibility effect on lift coefficient. Z is downward (M = Mach number)

Ratio disturbance longitudinal velocity /undisturbed velocity

1 d Both DT s indicate -1: cyt

but Etkin,s r = ₂₁₁₀ SeckePs T

-pSuo

Derivative of normal force or lift of aircraft with respect to elevator displacement

Elevator displacement Equivalent terms Etkin Seckel CL acL czu am Etkin s IL = 2m pSU

Second terms in expansions Omitted by Seckel

resolution into components. Next in popularity is the system which uses the principal axes of the aircraft - so that terms containing products of inertia do

indeed vanish. Unfortunately, the principal axes vary slightly with load distribu-tion and with fuel as it is burned or moved from tank to tank.

The third major reason for diversity in the equations of motion lies in the choice of non-dimensionalizing parameters, especially r, the aerodynamic

second. Some authors use m/p SV, where m is the mass of the aircraft (slugs), p is the density of the air (slugs/cu. ft. ), S is the plan-form area of the wing

(sq. ft. ) and V is the steady-state forward velocity (ft./sec. ). Others use where V retains its meaning and I is a characteristic length, e. g., semi-span

of the wing.

Finally, there is a lack of standardization in the symbols themselves. Here there are two words of caution, the first to authors, the second to readers. The writer should define every symbol he uses. The reader should consult the list of symbols the first time he reads a symbol, even if it appears familiar, because, as in the case of T just mentioned, the same symbol may be used to

mean different things.

In 1904 Bryan and Williams (Ref. 3) pointed out that the six equations

could be separated into two sets: the first set, longitudinal or symmetric - the lift, drag and pitching moment equations; the second set, lateral or asymmetric -side force, rolling and yawing moment equations. Separation is possible because an ordinary aircraft has a plane of symmetry (containing tae longitudinal and normal axes) and deflection of the elevator (the symmetric control surface) does not produce any significant asymmetric motion. There is, of course, a gyroscopic

yawing moment when the propulsive system contains a significant rotating component (c.f. modern jets, World War I rotary engines).

Separation is not complete, because deflection of an asymmetric control (ailerons or rudder) produces effects in the longitudinal variables - loss of speed, dropping of the nose, loss of height - as well as in the desired lateral variables

-bank or turn. Provided such coupling is small it is convenient to treat it as non-existent and the sets of equations as independent. Unfortunately, there are important cases where separation is not possible. First to be noted is the case where large angles of bank are involved. Second, in many modern aircraft the mass is concen-trated in a long, slender fuselage, with consequent large ratios of pitching and yawing moments of inertia to rolling moment of inertia, and large inertially coupled moments. (Ref. 14).

With the control surfaces fixed, the fourth order differential equations specify the characteristic modes in which the aircraft can move. The modes are not mutually exclusive, but may occur in combinations. An aircraft normally has two oscillatory longitudinal modes - the short period mode and the phugoid. The short period mode is characterized by vertical translation and by rotation in pitch, but by negligible change in forward speed. It is often well damped. On the other hand, the phugoid mode, with a very mnch-longer period, and characterized by an

DME/NAE 1961 (I) 2.0 10 (0)/conl 41(02 W2 I ZERO FREQUENCY GAIN UNDAMPED NATURAL FREQUENCY = DAMPING RATIOS C .5 FREQUENCY 1.0 RATIO 2.0 w/wn 4.0 1/2 FIG. 3

GRAPHICAL FREQUENCY RESPONSE DETERMINATION

+

BODE PLOTS SECOND ORDER DENOMINATORS

FRCP

=TANI 2 W/CLI tr.AI (C0/041

7.

Kai

.5 1.0 20

40

FREQUENCY RATIO WI wn 9

J

C.D <I 13

7

-oscillatory change of energy between potential and kinetic (oscillations of altitude and air speed), may be very poorly damped.

-A conventional aircraft has three typical lateral modes. The first two are not oscillatory: a roll subsidence with a Very short time constant, and a spiral mode, usually a divergence, with a very long time constant. It may be noted that the roll subsidence mode is neutrally stable with regard to bank angle; the sub-sidence has to do with roll rate. It causes no trouble in practice. The second, spiral divergence, implies that an aircraft with fixed ailerons and rudders does not continue to fly straight and level, but slowly rolls into a spiral, and must be kept straight by a pilot or an autopilot. The third lateral mode, the Dutch roll, often poorly damped, is characterized by oscillations in roll and yaw.

TRANSFER FUNCTIONS AND FREQUENCY RESPONSES

Two sets of three equations of motion have been mentioned. As normally written, each equation of each set contains three output variables, their derivatives and one or two input variables. With the help of operational calculus, transfer functions connecting output with input can be written directly. The transfer function is the quotient obtained when the transform of the output is divided by the transform of the input. Either a Laplace or a Fourier transform may be used In either case both input and output (functions of time) must begin at zero and, in the case of the Fourier transform, they must also end at zero in a finite time. In that case, the transform variables s and iw are interchangeable. Numerical evaluation of the Fourier form of the transfer function yields the frequency response which can be plotted as amplitude ratio and phase angle as functions of frequency. On log-log paper such plots are called Bode plots (Ref. 9).

The denominator of a transfer function may be factored into linear or first order and quadratic or second order factors. Non-dimensional Bode plots of second order denominator transfer functions have been plotted for various damping ratios (Fig. 3) and allow graphical determination of three parameters: zero

frequency gain, undamped natural frequency and damping ratio.

TEST METHODS

Test methods can be classified according to the type of input and according to the processes used in coefficient extraction.

TYPES OF INPUT

Whether the disturbance should be an autopilot or a manual input depends on at least three factors:

Has the aircraft an autopilot capable of being adapted to produce the desired control movement?

What data reduction equipment is available? What is the purpose of the test?

A 1.0 DME/NAE 1961 (I) 8 INTEGRAL SINGLE DOUBLET SECOND

a-10 15

FREQUENCY ca RADIANS /SEC.

20 SINGLE DOUBLET DOUBLE. INTEGRAL STEP TIME SECONDS TIME SECONDS FIG. 4

PULSED

INPUTS - THEIR

FOURIER TRANSFORMS

9

-To illustrate, if a peak in- afrequency response curve requires checking, a steady continuous sinusoidal movement of the control at the required frequency by a pilot may be carried out in a short time, and the amplitude ratio and phase angle measured with a minimum of effort directly from oscillograph recordings.

To obtain a reasonable band-width of input, a continuous oscillation, but with a slowly changing frequency; can be used (Ref. 15). Either constant or slowly changing frequency oscillations might be produced by an autopilot.

The control may be pulsed manually - as a step, a single pulse or a doublet pulse (Fig. 4). Which should be chosen depends on the purpose of the test and the data handling technique proposed. If primary interest is in low frequencies, approaching zero, the step function is most suitable. If a Fourier transform method is to be used, requiring all variables to return to their initial values, the doublet pulse is usually best If a matrix method is to be used (e. g. , Donegan-Pearson), the doublet pulse is not suitable; because the values of the variables and their integrals do not increase throughout the integration period (Fig. 4), the determinant of the coefficients of the matrix is small and the transfer function coefficients are likely to be badly in error.

In general; pulsed' inputs and their resultant transient outputs are preferred to steady oscillations for two reasons: (a) The conditions of the test (c .g. and weight of aircraft for example) can change significantly during a pro-longed oscillation. There is also a greater chance of invalid data resulting from a gust encountered during a prolonged run. - (b) The flying time required is much less than for an adequate number of steady oscillation tests.

METHODS OF COEFFICIENT EXTRACTION

Several authors have attempted to list, classify, and to some extent evaluate the many methods of coefficient extraction (Ref. 16, 17, 18, 22, 23). Let us examine a few representative cases.

Practically all methods could be termed curve fitting techniques in either the time or frequency domain. In the time domain the lightly damped free oscillation continuing after the input disturbance has ceased can be fitted with one or more exponential functions together with sinusoidal oscillations. Such decaying oscillations can be readily presented in time-vector diagrams (Ref. 21, 22, 33 and

Fig. 5).

In the time domain again, the curve fitting may extend from the beginning of an arbitrary control movement. When the form of the transfer function is known, or can be assumed, there are two types of equipment which can be used - annlogue and digital. Analogue computers have been used extensively for verifying transfer function coefficients (Ref. 24), and can be used directly providing not more than three coefficients are to be sought (Ref. 25). Since the settings (representing the coefficients) are not independent of each other, the direct procedure can be quite

DMEMAE 1961 (1)

(a) TIME VECTOR (PHASE PLOT) AND TIME HISTORY OF A DECAYING

Xt

OSCILLATION e.g. X = A 0 e COS. cat 10

DISPLACEMENT X

ACCELERATION

VELOCITY

MAGNITUDES CIjgiCJr TO BE FOUND

(DIRECTION KNOWN)

Cfp ESTIMATED

(b) TIME VECTOR SOLUTION OF THE ROLLING MOMENT EQUATION

Ix IS Ixz C ip Cip Cir 0

qsb r qsb r

FIG. 5

tedious and one may have difficulty in obtaining the best fit values. In general, a chart follower will be required to pick up the oscillograph record of the input, or a tracing of it.

Digital techniques are practically all dependent on the least squares principle. Ideally, the fitting should be of the variables and their derivatives as they appear in the chosen differential equation. In fact, many of the derivatives are not measured. Numerical differentiation is possible, but inaccurate, so that instead the equations themselves are integrated, yielding integral equations. The fitting then consists in minimizing the square of the difference between the observed value and the calculated value based on the integral equation (Ref. 26). Just as an analogue computer is often set up to simulate the six-degree-of-freedom motion of the aircraft, so it is theoretically possible, when several functionally related outputs have been measured independently, to extend the least squares technique to deter-mine the best value of each independent parameter. In practice such a direct synthesis may be/difficult to achieve and one may well treat his system of transfer functions in sequence rather than simultaneously (Ref. 27).

In the frequency domain, a mechanical harmonic analyzer may be used with an oscillograph recording to determine the Fourier integral of the function (Ref. 28). Alternatively, regularly sampled input and output data may be Fourier transformed numerically either with a digital computer, or with a desk calculator and a set of tables (Ref. 29). No assumption of the form of the functional relation-ship is required to obtain the frequency response of output to input.

Unfortunately there are at least two good reasons why spurious results may be obtained. First, if the frequency content of the input at the natural

frequency of the aircraft is low, the relative error in the transform is large, and therefore the amplitude ratio may have a spurious peak (or valley). Second, if there is an appreciable frequency content at frequencies for which the period is less than twice the sampling rate, aliases will appear. (Ref. 30 and 31).

From frequency response curves, it is usually required that transfer function parameters be obtained - undamped natural frequency or frequencies, damping ratio, zero frequency gain, etc. Two methods are in general use: the first, a graphical method, is based on fitting by eye the best of a series of trans-parent templates (Ref. 32); the other is a least squares digital technique. Analysis

in the frequency domain has one noteworthy advantage over the time domain. In the event that any instrument used in recording data has a variation in amplitude ratio over the frequency range under analysis, corrections can be made.

Finally, it has been suggested that power spectra might be calculated. Such computation is faster than Fourier transform calculations but discards any information on phase relationships. This technique is not yet common, and is not even mentioned in most of the summaries of techniques.

STABILITY DERIVATIVES FROM TRANSFER FUNCTION COEFFICIENTS Transfer function coefficients are easily calculated from stability

Expanding A a [CLa 2 12 -TABLE II

DERIVATION OF TWO APPROXIMATE TRANSFER FUNCTIONS

Shert duration manoeuvres can often be made without a perceptible change in airspeed. In this case the longitudinal equations of motion can be simplified by omitting the drag equation completely, together With the forward velocity terms in the lift and pitching moment equations, leaving

rc + C

a.Da

J L-mDE1'

- DO - C ma e

Then, providing a, Da, DO and oe are all initially zero, Laplace transformations allow the

A

transfer functions and 1-119 to be written directly.

ue be CL6 a _{+ s} + D I a -C C - hs m -mDA CLa _{+ S - 1} 2 Cm

+ CaS C

mD ha a 111DO 2 CL (C + C

)

+ (C +C _{- 2a h} s - hs2 m 2 m a a 0 mDa mDe C La CL C

+C

)m 2 e 2 m _a

(c

a + 2 a Cm CL

C)+

e hs mDO 2 + (Cm Da Lae -DO = - _ae -DO C +C s _{- Cm} taDa ma _Se CLa _{+ s}

_-1

2 Cma

+C

s _Cm - hs mD a DO

Note that the denominators define the natural frequency and damping of the short period

longitudinal mode. CL

(

2 m S e

a h) s - hs

-CL6 e 2

- 13

derivatives (e .g. Table II), but the inverse problem is much harder. Most important, from the practical point of view, is thefact that there are significant errors due to atmospheric turbulence and instrument errors, and to digitizing (Ref. 23). One can state as a general qualitative rule that "the accuracy with which one Can obtain each of the stability derivatives is associated with the importance of that stability deriva-tive in influencing the motion of the aircraft during the test, the highest accuracy being obtained for the more important stability derivatives "- (Ref. 34).

The lateral stability derivatives may be obtained in a reasonably straight-forward manner, although experimenters, using the time-vector method (where the directions of the vectors are all known but magnitudes are to be found), must

usually assume (from wind tunnel data, etc. ) one vector magnitude in order to close the polygon, i.e. , one can find only two unknown magnitudes in a single plot.

Two longitudinal derivatives can be found (CLa and Ciao ), but the remaining four cannot be completely separated (if data is

restrictea

to a, n,

). The damping derivatives, in particular, occur as an inseparable pair Cm& + C, (equivalent to Cmpa + Crape or mAT + mg in other notations). To separate them, one may measure tail loads, or make some supplemental assumption: e.g. ,

20ma = Cmq (Ref. 27), or use a strongly coupled motion, e. g. , a rolling pull-out (Ref. 18).

Even a rocket pulse in the Z direction has been suggested (Ref. 18). dE

Probably general practice will be to take Cma. =

da -mg,

where has been obtained from wind tunnel tests.

DATA COLLECTION, TRANSMISSION_ AND. RECORDING

There is a sound general principle that the parameters an experimenter wishes to investigate should be measured as directly as possible. For instance, when the damping of the short period longitudinal mode of a conventional aircraft is heavy, the usual instrumentation measuring n, a, Ei, and 6 is less than

adequate for the determination of the damping derivatives Cm. + Cm as a sum,_a and separation is impossible. Since most of the longitudinal damping is provided by the tailplane a significant improvement in accuracy can be made and the

-derivatives can be separated if the incremental load on the tailplane is measured, e.g. , with strain gauges. As a further desirable-by-product, the effect of elevator deflection on aircraft lift and hence elevator control effectiveness can also be accurately obtained (see Table III).

Following the choice of the variables to be measured there is an even wider choice of equipment to do the measuring. Two requirements must be satisfied if the measurements are to be of value. First, resolution and repeatability must be good for a minimum of two-figure accuracy (with three-figure accuracy highly desirable), and second, the undamped natural frequency and damping ratio must be

NOTES:

14

-TABLE Ill

COMPARISON OF METHODS OF EXTRACTION OF THE STABILITY DERIVATIVES OF A HEAVILY DAMPED MODE

Cm +

C

_in.

obtained by the Fourier Transform and Matrix Integral

q

a

methods pertain to the complete aircraft.

Cm + Cm

obtained by the Matrix direct when tail loads are

q &

measured are the contributions of the tailplane only.

Tailplane only, contribution, corrected for compressibility to M = 0, and for flexibility.

Independent of Mach number, but corrected for flexibility.

*ot P.E. = 0.6745

1E(obsetved average)2

_{n - 1}

- Probable Error

Derivatives + P (Average . E. Tail Loads Not Measured Tail Loads Measured

(per rad. ) Fourier Matrix Matrix

Transform Integral Direct

Method Method .Method

t

Cm + Cm q a -3.95 ± 0.75 (19%)

-2.76 ±0.59

(21%) -2.55 ± 6.25 (10%)

t

Cm q

Not separable Not separable -1.65 ± 0.12

(71%)

t

C

m_a Not separable Not separable -0.75 ± 0.10_(13-1/3%) *

0.373 + 0.036 Not done 0.390 ± 0.015

- 15

such that the response of the tansducer-galvanometer system is flat across the frequency range under investigation.

For ease of synchronizing data, a single large galvanometer recorder with enough channels to record all the data on a single roll of paper is desirable.

Either of two factors, space limitation or cost, may make it necessary to use two or more smaller recorders, in which case there is a requirement for synchronizing data from the various recorders. The sampling interval should not be greater than

0.1 second, but there are losses as well as gains when the interval is reduced, so that 0.1 and 0.05 second appear to be most generally used. A timer providing 100 c. p. s. oscillations can be connected to a galvanometer in each oscillograph recorder and can thereby ensure that the synchronization error does not exceed 0.005 second. Slowly varying quantities, altitude, airspeed, and temperature, may be recorded at a relatively low paper speed, if they have their own recorder. Still other items, e. g. fuel remaining, can best be written by hand.

Rotary potentiometers may be used to sense the rotation of control

surfaces. Duplication here, i.e.

, one on each aileron, is recommended since

it reduces the chance of abortive flights. Great care should be taken to keep the relation between oscillograph deflection and control movement linear. Backlash in particular must be prevented.

The need for minimizing bearing friction is the governing factor in the case of wind-vanes (sideslip and attack). One system uses an autosyn coupled directly to the shaft of the wind-vane, but driving a galvanometer rather than another auto syn.

Acceleration measurements are important. Fluid-damped, thermo-statically controlled, heated accelerometers of the appropriate range (e .g., 5g

maximum for the normal acceleration) should be mounted as close to the centre of gravity of the aircraft as possible. If tail lOads are being measured, another should be mounted close to the centre of gravity of the tailplane to allow the measured air loads to be corrected for the effect of inertia of the tailplane. Lateral and longitudinal accelerometers should be mounted in such a way as to minimize the undesirable effects of normal ac-Celeration.

Rate gyroscopes are suitable for measuring rates of roll, pitch and yaw. Attitude gyroscopes measuring angular positions in pitch and In bank are usually of secondary importance.

Strain gauges may be installed on tailplane spars to measure tail loads. The validity of their output will require ground testing, with jack-applied loads. Temperature effects on strain-gauge behaviour can be neglected since the incre-mental tail load is used in determining response characteristics rather than tail load itself.

Finally, if the pilot is to provide a manual input for the test, he should be presented with some control surface position indication. Preferably it should

have an easily adjusted datum which can be set just prior to the start of control movement and available as the target for the return of the control to its original position.

POTENTIOMETERS

(CONTROL POSITIONS)

ACCELEROMETER S _{RATE GYROSCOPES}

WIND VANES'

STRAIN

GAUGE

BRIDGES

ACCELEROME TER (TAIL PLANE) _{PRESSURE CAPSULES} THERMOCOUPLES

ATTITUDE GYROSCOPES 100 C. P S. TI MER CODER AND 2 C.P S. TIMER GALVANOMETER RECORDER GALVANOMETER

RECORDER _GENERAL PURPOSE _RECORDER

OSCIL LO

-GRAPH S

ALTIMETER _{MACH METER} % R P. M

FUEL GAUGES SENSING ,EQUIPMENT SYNCHRONIZING ELEMENTS RECORDING EQUIPMENT FIG. 6 AIRBORNE EQUIPMENT

DATA REDUCTION

It will be convenient to consider the data 'reduction process in the following stages:

First, there should be some sort of a "quick look", the purpose of which is twofold - to determine whether there was any obvious failure in technique

or equipment which might necessitate a re-test and/or repairs and to pick out the data to be reduced in order of its suitability for the purpose. Usually oscillograph

recordings are quite suitable for a "quick look" . If the data were obtained

directly as magnetic tape, a digital to analogue conversion with plotter output might be required.

Second, the analogue representation embodied in the oscillograms must be sampled and digitized, if a digital computer is used, or re-drawn to appropriate scales, if an analogue computer is to be employed. The sampling will depend on the type of test. A continuous sinusoidal input and output might require reading at peaks only, with careful measurement of the frequency and of the phase

relationship. A set of lightly damped output responses might be adequately defined by determination of the rate of decay in addition to frequency and phase. Finally, for heavily damped responses the Sampling rate might well be 1/20-second intervals.

In this last case, semi-automatic, tape-producing, oscillograph readers which punch tape at the touch of a button (when cross-hairs coincide with the

observed point) are invaluable. Without such equipment data can be punched on tape digit-by-digit from a keyboard.

Third - Normalizing of Data. Two operations which may well be carried out during the first rtm through the computer are the calculation of the deflection of the trace from its original value normalized with respect to the distance between reference lines and the correction of the instrument reading _for the" known influence of other readings. For example, the indicated angle of attack should be corrected for vane position in the field of flow and for the rate of rotation in pitch of the aircraft.

Fourth - Tail Load Calculation. _{Calculation of the aerodynamic force} on the tail involves the calculation of the tail load measured by the shear strain gauges corrected for the relieving effect of the normal acceleration of the hori-zontal tailplane. When considering increments' in tail load from the steady-state value before the input disturbance, slow zero drift has a negligible effect. If the total tail load were sought, close attention would have to be paid to temperature effects on strain gauge circuits.

When this data has been calculated, prior to coefficient determination or frequency response calculation, it is advisable to check against gross errors. The time required for machine replotting of data at this stage is well spent, as it prevents a great deal of trouble searching for the causes of incongruous results.

OSCILLOGRAPHS OBSERVER'S NOTE-BOOK READER TYPED READINGS TAPE DIGITAL COMPUTER TYPED INTERMEDIATE AND FINAL RESULTS FIG. 7

DATA HANDLING EQUIPMENT

TAPE 'PLOTTER GRAPHICAL PRESENTATION INTERMEDIATE RESULTS TAPE

.7 19

-If a plotter is not available a great deal of manual checking should be done here. .

At this stage, the process of coefficient determination, or of frequency response determination, may be carried out. Here the details vary, depending on the particular technique chosen.

Finally, the validity of the stability derivatives or transfer function coefficients should be checked. Either a digital or an analogue computer may be used to check the consistency of the output, calculated from input plus transfer function, with the actual measured output.

CONCLUDING REMARKS

Whether the flight measurement of the dynamic stability and response characteristics of an aircraft is undertaken to establish the dynamic behaviour of a new aircraft for airworthiness or development purposes, or to provide full-scale data for establishing the validity of theoretical estimates or model data, its success depends on careful planning. Selection of test instrumentation, of test techniques and of data reduction methods must be based on their individual

and collective suitability for extracting the required information.

The steady sinusoidal oscillation technique, attractive for its ease of frequency response determination, is costly in terms of flying time and should normally be used only if there is good reason to suspect that an output to input

ratio is seriously dependent on the amplitude of the input movement.

If the form of the transfer function is not known and cannot be assumed, the Fourier transform method .may be considered. Its use is limited by the

requirement that both input and output must return to their original values in a finite time.

If the stability derivatives are to be found (equivalent to assuming the form of the transfer function) the Fourier transform method is a detour to be avoided. The simplest of the transient oscillograph recordings to analyze is the decaying oscillation of a lightly damped mode (e.g. , Dutch roll or phugoid), where period and time to damp to half amplitude can be obtained directly. Of course such recordings contain no information concerning control derivatives. The heavily

(almost critically) damped motions require more complex treatment as well as additional instrumentation. As an example, analysis of a transient response to an elevator input should be extended back to the beginning of the elevator motion, and the tailplane should be strain-gauged to measure incremental tail loads. Numerical integration is inherently more accurate than numerical differentiation. The transfer function may therefore be determined by the least squares method involving an

input, an output and their first and second time integrals. Here information which can be obtained directly (without integration) should be sought first.

It is an unpleasant fact that the accuracy with which many of the dynamic stability derivatives can be found is quite poor. On the other hand, the over-all

20

-response of the aircraft, the matter of fundamental importance, may still be deter-mined with adequate precision, because the accuracy with which a stability derivative can be found is almost directly proportional to its importance in influencingthe motion of the aircraft during the test.

While the limited scope of this article has prevented any detailed discussion of the effects of Mach number, aeroelasticity or non-linearity, these effects are present and often of considerable magnitude in all modern high speed aircraft. In general the isolation of such effects requires the further elaboration of the experimental and analytical techniques described.

REFERENCES

1. Becze, E. J. Aircraft Structural Loads.

NRC Report No. DME/NAE 1960(4).

2. Parkin, J. H. A History of Aeronautical Research in Canada. NRC, June 1955.

3. Bryan, G. H. Longitudinal Stability of Aerial Gliders.

Williams, W. E. Proceedings of the Royal Society, Vol. 73, 1904, pp. 100-116.

4. Lanchester, F. W. Aerial Flight, Aerodonetics. Vol. II.

Archibald Constable and Co. Ltd., London, 1908.

. Glauert, H. Investigation of the Spin of an Aeroplane.

Advisory Committee for Aeronautics, R&M No. 618, June 1919.

6.

Frazer, R.A.

The Flutter of Aeroplane Wings.

Duncan, W. J. ARC R&M 1155, 1928.

7. Bryant, L. W. Nomenclature for Stability Coefficients.

Gates, S. B. -ARC R&M 1801, October 1937. 8. Nyquist, H. Regeneration Theory.

The Bell System Technical Journal, Vol. XI, January 1932.

9. Bode, H. W. Network Analysis and Feedback Amplifier Design. Van Nostrand, 1945.

10. Milliken, W. Dynamic Stability and Control Research.

Proceedings of the Third Anglo-American Aeronautical Conference, Brighton, 1951; pp. 447-524.

21

-11. Duncan, W. J. The Principles of the Control and Stability of Aircraft. Cambridge University Press, 1952.

12. Etkin, B. Dynamics of Flight: Stability and Control.

13. Gregory, P. C.

John Wiley & Sons, New York, 1959.,

Proceedings of the Self-Adaptive Flight Control Systems

(ed.) Symposium.

WADC Technical Report 59-49 March 1959.

14. Pinsker, W. J. G. Critical Flight Conditions and Loads Resulting from Inertia Cross-Coupling and Aerodynamic Stability Deficiencies.

RAE Tech. Note Aero 2502, March 1957.

15. Crane, H. L. A Manual Frequency Sweep Technique for the Measure-ment of Airplane Frequency Response.

NASA TN D-375, April 1960.

16. Eggleston, J. M. Application of Several Methods for Determinin.g Transfer Mathews, C. W. Functions and Frequency Response of Aircraft from

Flight Data.

NACA Report 1204, 1954.

17. Donegan, J. J. Comparison of Several Methods for Obtaining the Time Huss, C. R. Response of Linear Systems to Either a Unit Impulse or

Arbitrary Input from Frequency-Response Data. NACA TN 3701, Ju1y_1956.

18. Wilkie, L. E. Final Report on Extraction of Aircraft Stability Coefficients from Flight Test Data and Theoretical and Experimental Studies on Selected Problems of High Speed Aerodynamics and Dynamic Stability.

WADC Tech. Report 57-723, December 1957.

19. Huss, C. R. Effect of the Proximity of the Wing First-Bending Donegan, J. J. Frequency and the Short-Period Frequency on the

Airplane Dynamic-Response Factor. NASA TR R-12, 1959.

20. , Klawa.ns, B. B.

Johnson, H. I. Some Effects of Fuselage Flexibility on LongitudinalStability and Control. NACA TN 3543, April 1956.

21. Sternfield, L. A Vector Method Approach to the Analysis of the Dynamic Lateral Stability of Aircraft.

Wolowicz, C. H. Holleman, E. C. Zbrozek, J.K. Triplett, W. C. Brown, S. C. Smith, G. A. Milsum, J. H. Donegan, J. J. Pearson, H. A. Donegan, J. J. Seckel, E. Huss, C. R. Donegan, J. J. Shannon, C. E.

Press, H.

Tukey, J. W. Draper, C. S. McKay, W. Lees, S. 22

-Stability-Derivative Determina.tion from Flight Data. AGARD Report 224, 1958.

On the Extraction of Stability Derivatives from Full Scale Flight Data.

RAE Tech. Note Aero 2559, April 1958.

The Dynamic-Response Characteristics of a 350 Swept-Wing Airplane as Determined from Flight Measurements. NACA Report 1250, 1955.

Transfer-Function Discovery on the Pace Analogue Computer.

NRC Mech. Engrg. Report MK-2, February 1959.

Matrix Method of Determining the Longitudinal-Stability Coefficients and Frequency Response of an Aircraft from Transient Flight Data.

NACA Report 1070, 1952.

Matrix Methods for Determining the Longitudinal-Stability Derivatives of an Airplane from Transient Flight Data. NACA Report 1169, 1954.

Suggested Procedure for Using the Corradi Analyzer to Obtain Airplane Frequency Responses from Transient Data.

Cornell Aero. Lab. FRM No. 104, May 1950.

Tables for the Numerical Determination of the Fourier Transform of a Function of Time and the Inverse Fourier Transform of a Function of Frequency, with some Applica-tions to Operational Calculus Methods.

NACA TN 4073, October 1947.

Communication in the Presence of Noise. (The Sampling Theorem..)

Proceedings of the I. R. E. , Vol. 37, No. 1, January 1949, pp. 10-21.

Power Spectral Methods of Analysis and their Application to Problems in Airplane Dynamics.

AGARD Flight Test Manual Vol. IV, Part IVC, pp. IVC: 1-16. Methods for Associating Mathematical Solutions with

Common Forms.

23

-33. Doetsch, K. H The Time Vector Method for Stability Investigations. ARC R&M 2945, August 1953.

34. Breuhaus, W. O. Dynamic Response Techniques.

Segel, L. AGARD Flight Test Manual, Vol. II, Ch. 10. 35. Seckel, E. Airplane Motions.

24

-LIST OF SYMBOLS (See also Table I)

Symbol Definition

Wing span, ft.

CL Lift coefficient = qs

CL Aircraft lift curve slope dCL/da, per radian or per degree

a

CL6

Elevator lift slope dCL/doe, per radian or per degree

Rolling moment coefficient

Rolling moment due to sideslip 8 Waft, per radian 0

Rolling moment due to rolling Rolling moment due to yawing

Pitching moment coefficient

dC /da per radian

ma _C 1 m Trl T Da 1 a cm cmDO ao

Pitching moment due to elevator e

Mean chord, ft.

T dt- _{I pS}

Non-dimensional pitching moment of inertia - _m2.6

Rolling Moment of inertia, slug-ft? Product of inertia, slug-ft?

Symbol

LIST OF SYMBOLS (Contld) Definition

Characteristic length, ft.

Load factor = normal acceleration in g units Rate of roll, radians per second

Acceleration in roll dp/dt, radians per second2

2V

P 2

Dynamic pressure V2, _{pounds per ft.} Rate of yaw, radians per second

Acceleration in yaw dr/dt, radians per second2 rb/2V

Reference area - plan form of wing, ft2. Laplace variable

Time, seconds

u)

_o)

Forward velocity, ft./sec. V)

All-up-weight of aircraft, pounds

a

Angle of attack, radians

da/dt

Angle of sideslip, radians

e Elevator angle, trailing edge down positive, -radians Angle of downwash, radians

Angle of pitch, radians de/dt

26

-LIST OF SYMBOLS (Cont Id)

Symbol Definition

Density of air, slugs/ft3.

Air Sec., usually _{p SV}