NASA Technical Memorandum 4747

/' • _J

(.__SS .: ,._ . ,._...z_

Structural Dynamic Model

Obtained From Flight for Use

With Piloted Simulation and

Handling Qualities Analysis

Bruce G. Powers

June 1996

NASA Technical Memorandum 4747

Structural Dynamic Model

Obtained From Flight for Use

With Piloted Simulation and

Handling Qualities Analysis

Bruce G. Powers

Analytical Services & Materials, Inc.

Edwards, California

National Aeronautics and

Space Administration

Office of Management

Scientific and Technical

Information Program

1996

ABSTRACT

The ability to use flight data to determine an aircraft model with structural dynamic effects suitable for piloted

simulation and handling qualities analysis has been developed. This technique was demonstrated using SR-71

flight test data. For the SR-71 aircraft, the most significant structural response is the longitudinal first-bending

mode. This mode was modeled as a second-order system, and the other higher order modes were modeled as a time

delay. The distribution of the modal response at various fuselage locations was developed using a uniform beam

solution, which can be calibrated using flight data. This approach was compared to the mode shape obtained from

the ground vibration test, and the general form of the uniform beam solution was found to be a good representation

of the mode shape in the areas of interest. To calibrate the solution, pitch-rate and normal-acceleration instrumentation is required for at least two locations. With the resulting structural model incorporated into the simulation, a

good representation of the flight characteristics was provided for handling qualities analysis and piloted simulation.

INTRODUCTION

Performance requirements generally call for a structure to be as light as possible. For large cruise vehicles,

such as the high-speed civil transport and single-stage-to-orbit vehicles, maneuvering requirements are generally

low. The combination of low maneuvering requirements with a high vehicle gross weight often results in a relatively flexible structure where the frequencies of the structural modes approach those of the rigid-body modes. Therefore, it is highly desirable to include the structural modes in the analysis of the handling qualities and the piloted

simulation of the vehicle. A simulator study of a vehicle that included structural dynamic modes I concluded that

structural dynamics can have a significant effect on the handling qualities of the vehicle. However, most piloted

simulations do not include structural dynamic modes primarily because of the complexity of the structural modes.

For a full-envelope simulation, the structural mode calculations can be of a similar order of magnitude as those for

the rigid-body modes, which can often tax the capability of the simulation computers.

Another difficulty arises when providing simulation support for flight research programs. In this case, the

structural modes that are available from either a prediction or ground vibration test (GVT) often do not correspond

to the current configuration being tested. It is generally not feasible from a cost and time standpoint to update the

structural dynamic models to support handling qualities experiments. The goal of this study was to develop an easy

method to implement a structural model that can be calibrated with flight data. Because this model is being developed to support piloted simulation and handling qualities analysis, it was preferred to have comparatively fewer instrumentation and flight test maneuvers than for the handling qualities experiment itself. This report documents the

development and calibration of the structural models using actual flight test data.

In the SR-71 program, the flight data have shown a noticeable structural response. The structural modes have

generally not been modeled in the simulations or past handling qualities analyses. Many handling qualities criteria

today depend on frequency response characteristics that generally extend to frequencies that include some structural dynamic mode effects. As a result, it was necessary to expand the current SR-71 models to include structural dynamic effects. The first fuselage bending dynamics were easy to distinguish. Although the pilots were very

conscious of the structural dynamic motion, it was not a handling qualities problem because the motion was far

enough removed from the rigid-body frequencies. The mode was well enough defined and of sufficient magnitude

that it could be used to develop the modeling method for incorporating the structural dynamic effects. The SR-7 !

aircraft was used to validate the simulation and analysis capability. This modeling technique provides the capability to use flight data to establish structural models suitable for simulation and analytical studies that will be applicable to other large, flexible vehicles of the future.
NOMENCLATURE

A n

A1

A2

c.g.

cF 6

dz/dx

F8

FS

FS o

g

GVT

Kl

Kz

L

L a

L s

M

Mq

MeL

M 8

q

4

S

S

SAS

V

W

X

Z

6

Ax

normal acceleration, g

mode shape bias

mode shape angular scaling factor

center of gravity, percent

control surface input effectiveness coefficient, per deg

slope of normalized structural mode vertical deflection

control surface input effectiveness, in/deg

fuselage station, in.

reference fuselage station, in.

acceleration caused by gravity, ft/sec 2

ground vibration test

displacement constant

slope constant

characteristic length, fi

rigid-body lift caused by angle of attack, l/sec

rigid-body lift caused by elevator, l/sec

Mach number

rigid-body pitching acceleration caused by pitch rate, l/sec

rigid-body pitching acceleration caused by angle of attack, l/sec

rigid-body pitching acceleration caused by elevator, l/sec

pitch angular rate, deg/sec

dynamic pressure, lb/ft 2

pitch angular acceleration, deg/sec z

Laplace operator

wing area, f12

stability augmentation system

velocity, ft/sec

aircraft gross weight, lb

horizontal axis coordinate, fl

vertical displacement, in.

vertical velocity, in/sec

elevator deflection, deg

pilot stick input, in.

distance from center of gravity to sensor, ft

incremental elevator deflection from trim, deg

2
.q

0

"[an

(0

aft

d

fwd

r

s

0

damping ratio

structural modal displacement, in.

structural modal velocity, in/sec

structural modal acceleration, in/sec 2

pitch angle, deg

normal-acceleration time delay, sec

elevator time delay, sec

natural frequency, rad/sec

aft instrumentation location

delayed

forward instrumentation location

rigid value

structural value

bias term

Subscripts

STRUCTURAL DYNAMIC MODEL

The most significant structural dynamic response of the SR-71 aircraft is the longitudinal first-bending mode.

Figure 1 shows the flight-measured pitch rate and normal acceleration caused by a sharp, longitudinal pitch stick

input as well as the rigid-body motion obtained from the simulation response to the same input. The flight response

4

2

0

-2

q,

deg/sec

-4

1.8

1.2 An,

g

Figure 1. Comparison of flight response at forward-sensor location with rigid-body simulation response.
hasaconsiderableadditionalhigherfrequencycomponent becauseof thestructuralresponse. A structuralresponse

of thistypecanbedescribedbytheproductof thedynamicresponseof themodeandthespatialdistributionof the

structuralmode(i.e.,themodeshape):

Z = lq Kl(X )

The dynamic portion of the response, T1,is the response at one point along the fuselage, usually of maximum

amplitude. The mode shape, K l (x), is the distribution of the response for other locations along the fuselage. The

exact distribution of the deflection can be obtained from a structural analysis using the actual mass and structural

characteristics from a GVT or from flight data. In general, the distribution would be a function of loading and flight

condition. In the following section, an approximate distribution will be developed based on the solution for a uniform beam. It will be shown that this distribution provides a reasonable representation of the structural mode for

the purpose of performing handling qualities analysis. Flight data can be used to calibrate the approximate solution

with a minimum of instrumentation to provide the best match of the actual vehicle as tested.

Mode Shape Model

The displacement constant, K 1, is the distribution of the response for locations along the fuselage. For a uniformly loaded beam, the displacement constant is a sinusoidal function:

K 1 = sin(p x/L+¢)+A 1

The nodes of the first-bending mode are located at x = 0.224L and at x = 0.776L, where L is the beam

length. 2 The other constants are p = 1.51t, ¢ = 0.757t, and A I = 0.267. The primary variable is the nondimensional beam length x/L. Figure 2 illustrates the resulting mode shape.

1.0

.5

K1 0

-.S

- 1.o I I I I I

0 .2 .4 .6 .8 1.0

x I L _o2M

Figure 2. Mode shape for uniformly loaded beam.
Theuniformbeamsolutionhasthreevariables that can be used to fit a particular mode shape: L, p, and A 1.

The form of the solution can be rearranged slightly to provide more meaningful parameters. Because the structural

coordinate system of the uniform beam solution is not necessarily coincident with the aircraft coordinate system, a

bias between the two coordinate systems is required so that x = (FS - FSo)/12, where FS is the fuselage station

of the aircraft reference system in inches. For easier interpretation of the variables, it is useful to change the phase

parameter so that FS o corresponds to the center of the beam where the slope is zero. The resulting expression for

the deflection constant is as follows:

( FS - FSo_I

Kl=-C°S_ l'5n 12L J+A,

To test the validity of the assumption that a uniformly loaded beam solution can reasonably represent an actual

aircraft mode shape, this form of solution was used to fit the mode shape obtained from a GVT. The GVT data obtained from the YF-12 aircraft, 3 which is geometrically similar to the current SR-71 test aircraft, provide a definition of the shape of the first-bending mode. A curve fit of the data was obtained using the sinusoidal form of the

uniform beam solution. Figure 3 shows the GVT data and the curve fit. The fit of the GVT data results in an effective length of 139 ft (compared to the actual fuselage length of 107 r) and an inflection point location of FS 0 798.

The bias, A 1, was 0.72 rather than the uniform beam solution of 0.267. The uniform beam form of solution provides a good fit of the actual aircraft deflection curve from the GVT data, except near the aft end of the airplane.

For handling qualities analysis, the range of interest is from the nose to the center of gravity (c.g.) location

(FS 900), which encompasses the pilot location, control system, and instrumentation sensors.

1.o

.5

Normalized

deflection

-.5

O

/

O GVT data

Curve fit

I I I I I I 1

200 400 600 800 1000 1200 1400

Fuselage station, in. _02s9

Figure 3. Fit of GVT data using form of uniform beam solution.

5
Dynamic Response Model

The dynamic portion of the response, rl, is modeled as a second-order system with the elevator as the forcing

function:

fi+2 _ o) _1+c02 TI = 032 FsA5

The term F_ is the effectiveness of the elevator which, in conjunction with the normalized deflection K 1,

scales the structural deflection response zs in inches when the elevator input is in degrees. The pitch angular deflection of the structure is the slope of the deflection curve with respect to the longitudinal distance and is given by

the following:

dz/dx = -K 2 11

The slope dz/dx is approximately the angle of the fuselage relative to the rigid-body x-axis.* The slope constant, K 2, is given by the following:

-A 2 ( FS - FSo_

J

The scale factor, A 2, provides the proper scaling for the pitch angle equations. The structural dynamic displacements and angles are perturbations from the rigid-body values. The structural dynamic components of the

various response parameters are as follows:

Displacements: zs = K 1 11

Rates: Zs = KI ¢i

Accelerations:

(in.) 0 s = K 2 11 (deg)

(irdsec) qs = K2 1_ (deg/sec)

(g) qs = K2 i_ (deg/sec 2)

For the nonlinear simulation, the elevator input to the structural mode, A& is the incremental elevator deflection from trim. The effects of static structural deflections are included in the basic aerodynamic model, which contains the static aeroelastic corrections. The complete equations for the two types of measured parameters that were

available on the test aircraft, normal acceleration and pitch rate, are as follows:

An, = 1-2"ggL-C°stl'5= 12L J+ A1 's2+ 2_l_s+o 2 F_ At3

F-A ( (FS- FS0)_']F s co2 1

The structural dynamics are then added to the rigid-body motion to obtain the total vehicle response. To account for the effects of higher order structural modes that have not been modeled, a time delay was introduced between the elevator deflection and the input to both the rigid-body aerodynamics and the structural modes. This

*Because of the sign convention ofx (+ x towardthe rear) and z (+ z up), a positive slope correspondsto a negativepitch angle.
procedureissimilartothatusedin handlingqualitiesanalysistodeterminelowerorderequivalentsystemswhere

thehigherordercontrol-systemdynamicsarerepresentedasatimedelay.4Theplausibilityof suchadelaycanbe

visualizedbyconsideringanaircrafthathasawingwithtorsionalflexibility.Theinitialresultof asurfacedeflectionwill bea wingtwisting.Afterthewinghastwisted,aerodynamicforceswill begenerated,resultingin rigidbodypitchaccelerationandlongitudinalfirst-bendingmodeacceleration. Thisphenomenonwasrepresentedasa

delaybetweenthesurfacepositionasproducedby theactuatorandthesurfacepositionthatisusedtodriveboth

therigidandflexibleresponses. Anadditionaldelaywasallowedfortheaccelerationresponseasshownin thefollowingtransferfunctions:

cSd(s) -_ss

-- e

CS(S)

q(s) [qr (s) qs (s)]

= [ 7;5÷

An(S) -Ancg(s)

_Sa(s) _ia(s)

Ax ¢Jr(s) Ans(s)" ] -x%s

_+ +

g 8d(S )

For the current SR-71 flight test program, there was no instrumentation for the control-surface positions. The

control-surface responses were calculated using the simulation models for the actuators and the pilot control input,

which was recorded. As a result, it was not possible to discriminate between delays caused by structural effects and

delays caused by errors in the actuator models.

CALIBRATION TECHNIQUES USING TEST DATA

The unknown variables required to define the structural model are o s, ks, FSo, L, A 1, A 2, FS, "Ca , and 'csFlight data were available for normal acceleration and pitch rate at two fuselage stations: one at FS 23_', near the

nose, and the other at FS 683, near the c.g. The best method to solve the unknown parameters is a parameterestimation technique such as the pEst software, 5 which readily identify the unknown structural characteristics and

the generally unknown rigid-body aircraft characteristics. The time delay can be estimated by shifting the input

time history various amounts 6 or solved directly by using the nonlinear estimation capability of the pEst program. 5

For the SR-71 aircraft, the structural mode identification can also be accomplished with a simplified analysis because the structural mode frequency is sufficiently removed from the short-period frequency and the stability augmentation system is not significantly coupled with the structural mode because of the sensor location. In general,

these conditions will not exist. The motivating factor for including the structural modes in the handling qualities

analysis is that the structural frequency is near enough to the short-period frequency to cause a problem for pilot

control. This also makes a simplified analysis difficult and generally can only be analyzed by using parameterestimation techniques. The following section will develop the method used for the simplified analysis to provide

some insight into the results.

Simplified Analysis Technique

The four mode shape parameters--FS 0 , L, A], and A2--can be determined from the ratios of the magnitudes

of the two pitch-rate and two normal-acceleration responses. These ratios are An/q at the forward and aft locations, Anfwd/Anaft and qfwd/qafl. When these ratios are obtained from the free-oscillation part of the response following a sharp excitation input, the response is primarily caused by the structure. From the previous expressions

7
for Ans and qs, the ratios can be evaluated at the structural frequency cos. The phase information can be used to

determine the location of the motion along the x-axis. When the positive Ans peak leads the positive qs peak by 90 °,

the ratio A,_,nsis considered positive. The ratios Ansf*C and qsfwd are considered positive when both responses

qs Ans_ t i qs_t

are on the same side of the node. Figure 4 provides an example of the free-oscillation portion of a time history. In

this example, the amplitude ratios can be read directly from the time history. However, in many cases, it was more

difficult to directly measure the amplitudes because of the noise on the measurements. As a result, a curve fit of

each parameter was made and the amplitudes were determined from the fitted curves.

An,

g

1,2 B

1.1

1.0_

o9

.8

r-I 0

D[] []D [] Measurements

I:_ O& D_ Forward Aft

.7 I I I I

0 .5 1.0 1.5 2.0

Time, leo

0

--4

_6

-8

2.5

96o290

Figure 4. Free-oscillation portion of response to sharp pitch control input, M 0.8.

-2

q,

down©

For the example time history, the measured value of the structural mode frequency was 14.6 rad/sec. The amplitude ratios from the measurements at the forward location (FS 234.5) and the aft location (FS 683) resulted in

the following equations:

• Ratio of acceleration to pitch rate at the two fuselage stations

(.. (234.5 - FSo)" _

cos[l.3_ 12L J-A]

sin( 1.5_ (234"5 - FSo) ]

i- c j

14.6 = 0.058 g/deg/sec

An, L

qs laet = 12gA2

(,. (683 - FSo)'_

cos_,.,n 12L _ -AI

sin(1.5_ (683 - FSo)I

12L j

14.6 = -0.034 g/deg/sec
• Ratio of accelerations at forward and aft fuselage stations

cos(l.5r_ (234"512L- FSO)I) -A1

tr (683 - FS0)" ]

cosL1.5n 12L ) -A1

= -5.72

• Ratio of pitch rates at the forward and aft fuselage stations

( (234.5 - FSo) )

sinL1.5_ ]'2"L" J= 3.40

sin(1.5/t (683 - FSo) _

12L )

These equations were then solved for the four unknowns L, FS O, A 1, and A 2 . The solution required an iterative procedure, and a spreadsheet with an equation solver was used for this process. Good starting values for the solution are L = fuselage length and FS o = L/Z Figure 5 shows the solution. It can be seen that the assumption of

the sinusoidal form for the deflection distribution provides a rather nonlinear solution obtained from a minimum

number of measurements. The next step was to determine the structural gain F 8 and the time delays. This was accomplished by putting the mode shape data into the simulation and then adjusting the gain and delays to obtain the

best fit to the flight data. The delays were adjusted only to the nearest simulation frame (0.020 sec). The delay in

the elevator control input, determined from the pitch-rate response, was found to be 3 frames or 0.060 sec. A

4-frame delay (0.080 sec) had to be added to the normal-acceleration responses. This is likely caused by delays in

the accelerometer instrumentation itself rather than any structural phenomena.

.1

An/q' 0

g/_eg/sec

-.1

--_ \ 0 Flight data

I I I I

Amplitude

ratiio

8 m

4-

0

--4 _

-8

0

i

, q

A' i i ' i t i i

200 400 600 800 1000 1200 1400

Fuselage station, In. 960291

Figure 5. Pitch-rate and normal-acceleration amplitude ratios and calculated fit.

9
Parameter-Estimation Techniques

Two parameter-estimation techniques were used: the pEst software 5 and the spreadsheet identification method,

which uses the equation solving capability of a spreadsheet on a personal computer. These techniques use the longitudinal short-period rigid-body equations of motion and structural equations of motion. The input to the system

of equations was the elevator deflection. Because the elevator deflection was not measured in flight, a calculated

value was obtained by using the measured stick position and the simulation to obtain the elevator position. The elevator position was the average of the inboard and outboard elevator positions. The simulation actuator models

were second order, with the inboard elevator driven by the stick command and the outboard elevator driven by the

inboard elevator position. The elevator input to the aerodynamic and structural models was delayed by x8 so that

the input at time t was 8(t - xS). The following equations were used to model the dynamic responses:

ilr(t ) = Mq qr(t) + M a O_r(t) + M s 8(t - XS) + M 0

(_r(t) = qr(t) -- L a Or(t ) - L_ 8(t - X8) + L0

i_(t) = -2 _CO _l(t)-to 2rl(t)+to 2F 8_5(t-xS)+F 0

The output equations were as follows:

v ax. 1 r ( (rs- rSo)

An(t)= g(qr( t- "[an)-_rlt-'[anl)+ "g'qrlt-'[an) + l_gL-c°s[ 1"5_ 12L )+ A13i_/t-"[an) + Ano

q(t) = qr(t) + sin 1.5_ 12L +q0

where FS and Ax were evaluated for the forward and aft instrument locations.

The cost function was the square of the difference between the calculated outputs and the flight-measured values of the four measurements. The unknown variables used to minimize the cost function were the structural coefficients, short-period rigid-body coefficients, and various bias terms: to, _, F 8, L, FS O, A 1, A 2, x8, Xa, Mq, Mix,

MS, Lcx, LS, V, LO, MO, FO, Anofwa, Ano , qot_ d, qo_"

For the spreadsheet solution, the time delay was determined by evaluating the cost function for various time

delays, which were incremented as multiple sample intervals. 6 This was done for only one flight condition, and the

results were used for the other conditions. For the pEst technique, the time delays were implemented as interpolated values of tables (i.e., 5 as a function of time) and the time delay was estimated directly for each time history.

With both techniques, the parameters of the nonlinear mode shape could be solved for directly. Other parameterestimation techniques that require linear equations could be used by estimating K 1 and K 2 for the forward and aft

location. The mode shape parameters L, FS o, A 1, A 2 could then be calculated from K 1 and K 2 .

Figure 6 provides a comparison of the mode shapes obtained from the simplified analysis results and the

parameter-estimation technique results. All of the results show good agreement in the area of interest from the nose

to the c.g. Some disagreement is seen at the aft end of the aircraft; however, if this area were of importance,

additional instrumentation could be added in this region. Figure 7 shows a typical example of the fit of the flight

data using the spreadsheet parameter-estimation technique for a flight condition at M 3.0. A reasonably good fit

was generally obtained for all the flight conditions.

10
Normalized

deflection

1.0 B

- .5 0

Simple analysis

Forward Instrumentstton ..... Spreadsheet estimation

_ - _ pEst estimation

_,_ Cockpit

Nose ,'_

I ;:\_ Aft Instrumentation /

: ; ,, , : Z

: ! _"_'_ : , 25% c.g. _'"

I I I '1 I I

200 400 600 800 1000 1200

Fuselage station, In.

Figure 6. Comparison of mode shapes resulting from three analysis techniques, M 0.8.

I

1_o

960292

SUMMARY OF PARAMETER-ESTIMATION RESULTS

The data used to analyze the structural characteristic data were collected on five flights of the SR-71 airplane.

The maneuvers consisted of sharp raps of the longitudinal control stick, which primarily excited the structural

mode. In one case, a frequency sweep maneuver was performed. A summary of the flight conditions and maneuvers is contained in table 1. The maneuvers at M 3.0 were performed with the stability augmentation on; the other

maneuvers were performed with the stability augmentation off. The following discussion will present the results

obtained from the five flights for the simplified analysis, spreadsheet parameter identification analysis, and pEst parameter identification analysis.

Table 1. Flight conditions for the test maneuvers.

Flight Maneuver

Dynamic

Mach pressure, Aircraft

number Altitude, ft lb/ft 2 weight, lb

15

15

18

18

18

19

19

Stick rap 1

Stick rap 2

Stick rap 1

Stick rap 2

Stick rap 3

Stick rap 1

Stick rap 2

0.785 20,200 440 72,900

0.800 19,800 440 72,400

0.825 18,000 490 120,000

0.825 18,400 490 119,900

0.825 18,700 490 119,800

0.850 22,800 400 115,900

0.850 23,200 445 115,800

20

20

20

21

21

Stick rap 1

Stick rap 2

Stick rap 3

Stick rap 1

Frequency sweep

0.800 20,600 426 11,750

0.820 21,000 440 117,100

0.830 21,200 447 116,700

3.000 75,300 460 80,300

3.000 75,300 460 80,000

11
Anfw d,

g

Anllft'

g

qfwd'

deg/sec

1.0

.9

..... Flight

I I I

ell f° °

i s i I

, • ,, ,

I I i I I i i

1.0

.9

" i | od II

. ,

I I I I I I I I I I

- 1.5

1 °5 B

B

1 I I i I I i I I I

1 m

-1 i I i I I I I I I I

1 o5 B

deg

8p,

In.

- 1.5

0

., ;Sp

¢ I

i i ""1 I I I I I I i

.5 1.0 1.$ 2.0 2.5 3.0 3.5 4.0 4.5 5.0

Time ,lec 960293

Figure 7. Time history of match between flight data and calculated time history using spreadsheet identification

technique, M 3.0.

12
Mode Shape

Figure 8 shows the results of all the points in terms of the mode shape parameters L, FS O,A 1, and A 2 . The

simplified analysis and the two parameter identification technique results showed good agreement, although the parameter identification results were usually more consistent. Good agreement was found between the spreadsheet

3OO

L, 200 ft

100

O Spreadsheet identification

[] Simple analysis

A pEst

A

O _ B

Uo

I I I I

A1

1.0 m

°8_

.6

____----UlID

I I I I

1000

FS0' 800--

in.

i

600 i

AOD

l I I I

150 -- A

A2 100

5O

.......

) I I I

60,000 80,000 100,000 120,000 140,000

Weight, Ib 96o294

Figure 8. Mode shape parameters L, A 1, FS O, and A 2 as function of aircraft gross weight, (open symbol, M 0.8;

solid symbol, M 3.0).

13
parameter-estimation technique and the pEst technique. The faired line shown in figure 8 is a least square fit of the

two sets of parameter identification data. The parameters FS 0 and A l were the most significant in determining the

mode shape. Figure 9 shows the effect of the parameter variations with weight on the mode shape for two weight

conditions. A fairly large shift in the forward node toward the aft is seen as the gross weight is increased.

Normalized

deflection

1.0 --

I

- "50 1400

960295

Forwsrd instrumentation Weight

: Aft Instrumentation Light

: Cockpit : Heavy

[ , SAS gyro

Nose :\_, : '

' 'i_" ' "

, * i% ,

, t • , ,

1 , l % _ '

• , : 25% c.g. //

t % ) i t

, t % , _ /

l o i • o , ¢ •

l l i • l I •

t l i • i i

l a • , l i

t I qlk_

t • i • ,

'_ u 4

i

o i n

• I

I I I ' I I I

200 400 600 800 1000 1200

Fueelage station, In.

Figure 9. Comparison of mode shapes for heavy-weight and lightweight aircraft using faired data from figure 8.

Control Effectiveness

Figure 10 shows the control effectiveness parameter. The control effectiveness was modeled as Fs = _-_WCFs" It

would be expected that the control effectiveness for the structural mode would be similar to the control effectiveness for the rigid-body motion. A curve proportional to pitching moment caused by elevator is shown in figure 10.

The flight data are in general agreement with this trend although data as a function of Mach number are very

CF 8'

per dog

.12

.10

.06

.06

.O4

.O2

[]

I I I I

0 Spreadsheet Identification

r'l Simple Intlyeta

pEst

Proportional to 1RA

.... Proportional to pitching

moment coefficient

.5 1.0 1.5 2.0

Mach number

Figure 10. Structural control effectiveness parameter.

2.5

14
limited.A simpleapproximationto thisfunctionof Machnumberis a functioninverselyproportionalto Mach

numberwithanupperlimit,asshownin figure10.

Frequency and Damping

Figure 11 shows the structural frequency and damping. The frequency was a function of vehicle gross weight

but was not significantly affected by flight condition. The damping ratio was independent of airplane gross weight

but was noticeably reduced at the M 3.0 flight condition. Figure 12 shows the decrease in damping with Mach

number. A function similar to the one used for the control effectiveness is also shown as an interpolation over the

recUsec

20--

15--

10

.15 --

.10 --

.0S --

..... _----JL_

O Spreadsheet Identification

D Simple analysis

A pEst

12

.......

I I I I

8

A

O

60,000 100,000

Weight, Ib

_8

A D

80,000 120,000 140,000

960297

Figure 11. Structural frequency and damping characteristics as function of airplane weight (open symbol, M 0.8;

solid symbol, M 3.0).

.16

.10

.05

O

0 Spreadsheet Identification

[3 Simple analysis

pEst

Proportional to IlM

I I I I } I

.5 1.0 1.5 2.0 2.5 ;:.0

Math number _029e

Figure 12. Structural damping characteristics as function of Mach number.

15
Machnumberange;however,there are only marginal data to support this trend. No effect on damping was seen

because of dynamic pressure over the limited range tested.

Time Delay

The time delay was estimated with the spreadsheet parameter-estimation technique at one flight condition.

Several fits were obtained using a range of time delays for both the elevator and the normal acceleration. Table 2

lists the cost function results from the matches. An interpolation of these data produced an estimate for the elevator

delay of "_8 = 0.064 sec and Xa_ = 0.077 sec. It can be seen that the simplified analysis estimate of a 3-frame delay (0.060 sec) for the elevator and a 4-frame delay (0.080 sec) for the normal acceleration agrees well with the

spreadsheet parameter-estimation results at the same flight condition. An easier estimation of the time delay was

obtained from the pEst program, which had the capability to directly estimate the time delays for each maneuver.

Figure 13 shows the three sets of results. Considerably more scatter is seen when all of the pEst points are included. The pEst results indicated an average elevator time delay of "_8 = 0.036 sec and an average normalacceleration time delay of Xa, = 0.100 sec.

Table 2. Cost function from the spreadsheet parameter estimation as a function

of pitch-rate and normal-acceleration time delays.

1;_, see

0.04

0.06

0.08

0.06 0.08 0.10

18.5771 16.2385 20.1536

12.2156 10.696 13.9351

14.7924 13.6397 16.7074

x8,

line

.15 --

.10 --

.05 --

0

60,000

• Spreadsheet Identification

• Simpleanalysis

A pEst

---- Average

I I

A

1

80,000 100,000 120,000

Weight, Ib

(a) Elevator.

Figure 13. Elevator and normal-acceleration time delays.

I

140,000

96o299

16
.10

"_An,

sac

.15 --

.05 --

• Spreadsheet identification

• Simple analysis

Zl pEst

--- Average

A

A

A

A

A

A

i

o I I I I

60,000 80,000 100,000 120,000 140,000

Weight, Ib 9eo3oo

(b) Normal acceleration.

Figure 13. Concluded.

SIMULATION RESULTS

Simulation Model

A simulation model of the longitudinal first-bending mode was created from the parameter estimation results.

The structural mode equations were added to the rigid-body states. A low-pass filter was used to approximate the

steady-state trim elevator deflection. The incremental elevator deflection, which is the input to the structural mode

model, is the difference between the current and the filtered elevator positions. The transfer functions for the structural motion and the combination with the rigid-body motion are shown in the following equations as a function of

the fuselage station:

0.1

Ali(s) =(1 s+_.l.)l_(s)

qs (s)

ZXS(S) F-A 2 ( FS - FSo'_'IF s 0) 2 ]

A,,_:s)_pA"o,,_;s)Ax'il=)A,,<s)'l-'°?-'_=

L 8(s) +g li(s)+A--'_J e e

q(s) _ Iqr(S) qs(S)7 -xss

L8-Tzr+ e

17
ThecontroleffectivenesswasmodeledasafunctionofMachnumberanddynamicpressureasfollows:

CF s = 0.06/M-0.01

CF 6 < 0.065

Fa = W CFs

The structural mode shape parameters were modeled as a function of airplane gross weight as follows:

L = 208-0.000105 W

FS o = 752+0.00063W

A 1 = 0.803 + 0.00000049 W

A 2 - 113-0.000214 W

As shown below, the structural frequency was modeled as a function of airplane gross weight, and the structural

damping ratio was varied as a function of Mach number:

cos = 23.6-0.000079 W

ks = 0.075/M-0.007

ks < 0.087

The time delays were modeled as pure time delays with the following values:

x8 = 0.036sec

Xa_ = 0.100 sec

Comparison of Flight/Simulation Frequency Response

The frequency response characteristics are of particular interest for handling qualities analysis. A frequency

sweep maneuver was performed in flight at M 3.0, and this was used to compare the results from the simulation

model that was developed with the flight data. The stick input from the flight maneuver was used to produce a simulation time history of the frequency sweep maneuver. Fast-Fourier transforms were used on both the flight and

simulated data to find the pitch rate and normal acceleration to pilot input transfer functions. Figure 14 shows the

results for the forward instrumentation location, and figure 15 shows the results for the aft instrumentation location. The simulation model shows reasonably good agreement with the flight data at the M 3.0 flight condition. The

simulation model also allows the extrapolation to other fuselage locations. Figure 16 shows the frequency response

characteristics for the pilot location, which are of interest for handling qualities analysis. Figure 16 also shows the

original rigid-body responses from the simulation. Significant improvements in the model fidelity are shown in the

higher frequency region, especially for the pitch-rate response. The results indicate that the desired objective, to

provide a model suitable for handling qualities analysis which includes the structural dynamic effects, has been

achieved.

18
-10 --

- 20 -- "

- 40 -- Flight _ t'_ "

..... Simulation _' ........

-so _ l lllll I !_!1 I IIlil I I I i illli

100--

O--

Phase,

deg - 100 --

- 200 --

-300

.1

I I I IIIIII I I I I LIIII

1 10

Frequency, red/sac

I I I I 11111

lo0

96O3O1

(a) Normal acceleration for pilot stick input.

10

0

Megnhude,_lO

dB

-20

- 30

-- -- Flight ,_

..... Simulation _ /7

/ '.

I I I I IIIII I i i I I IIll I I

I I 11111

100

0

Phase,

deg - 100

- 200 --

I I I tilt

- 300

.1 100

i I i I llill i I I I i llil I I

1 10

Frequency, red/sac 9603O2

(b) Pitch rate for pilot stick input.

Figure 14. Comparison of normal-acceleration and pitch-rate transfer functions from flight and simulation for

forward-sensor location at M 3.0.

19
-10

- 20

Magnitude, _ ao

dB --

-40

-SO

200

100

B

i #,

i "_ s. . ,t

..... Simulation ' |I _1_ "

I I I I lllll [ I t t ;tlt_ t t t t llttt

!

O-- Phase,

(leg

-100 --

- 200

- 300

.1 1

Frequency, rad/sec

i

III I "'1-

10

I I IIIII

IO0

96o3o3

(a) Normal acceleration for pilot stick input.

0

Magnitude, _ 10

dB

-20

- 30

10-

Flight

..... Simulation

I I I I l llll I I I I IIIII I I

Phase,

dig

100_

0--

-100 B

- 200 --

- 300

.1

I I I llll I I I I I llll

1 10

Frequency, rad/sec

I I IIIII

I I I I I IIIIII

100

96O3O4

(b) Pitch rate for pilot stick input.

Figure 15. Comparison of normal-acceleration and pitch-rate transfer functions from flight and simulation for aftsensor location at M 3.0.

20
11I - 20

Magnitude,

dB

- 50_-- ..... Rigid

-so| I I I i iltll I I _ l lllll I I I I l llll

I I I IIIIII I I I li1111

1 10

Frequency, rad/sec

\

II IIIII

100

960305

(a) Normal acceleration for pilot stick input.

2O

10

0

Magnitude,

dB - 10

- 20

- 30

100--

O--

Phase,

deg - 100 --

- 200 --

- 300

.1

-- With structural mode

I I i I IIIII I I I I I IIll I"-I I II litl

I I I I IIIII I I i I IIIII I I IIIII

1 10 100

Frequency, raci/sec

(b) Pitch rate for pilot stick input.

Figure 16. Simulation results for normal-acceleration and pitch-rate transfer functions for pilot location at M 3.0.

21
CONCLUDING REMARKS

The ability to use flight data to determine an aircraft model with structural dynamic effects suitable for piloted

simulation and the analysis of handling qualities and flight control system has been developed. This technique was

demonstrated using SR-71 flight test data. For the SR-71 aircraft, the most significant structural response is the longitudinal first-bending mode. This mode was modeled as a second-order system. The effect of other higher order

modes was modeled as a time delay to both the rigid-body and structural motion. The distribution of the modal response at various fuselage locations was developed using a uniform beam solution, which can be calibrated using

flight data. This approach was compared with the mode shape obtained from GVT, and the general form of the uniform beam solution was found to be a good representation of the mode shape in the areas of interest. A simplified

analysis calibration technique was developed and is applicable when structural dynamic motion is not significantly

altered by the control system and is separated significantly from the rigid-body motion. For the more difficult case,

where the control-system or short-period interactions prevent a simple analysis, a parameter-estimation technique

was demonstrated. This analysis can be accomplished by using either standard parameter-estimation programs or a

personal computer spreadsheet analysis with equation solving capability. To calibrate the uniform beam solution,

pitch-rate and normal-acceleration instrumentation is required for at least two locations. This technique provided

structural mode shape information that is comparable to a GVT with a minimum of instrumentation, flight time,

and analysis. With the resulting structural model incorporated into the simulation, a good representation of the

flight characteristics was provided for handling qualities and control-system analyses.

REFERENCES

IWaszak, Martin R., John B. Davidson, and David K. Schmidt, A Simulation Study of the Flight Dynamics of

Elastic Aircraft, NASA CR 4102, 1987.

2Freeberg, C.R. and E.N. Kemler, Elements of Mechanical Vibration, New York: Wiley & Sons, 1960.

3Smith, John W. and Donald T. Berry, Analysis of Longitudinal Pilot-Induced Oscillation Tendencies of YF-12

Aircraft, NASA TN D-7900, 1975.

4Hodgkinson, J. and W.J. LaManna, Equivalent System Approaches to Handling Qualities Analysis and Design Problems of Augmented Aircraft, AIAA 77-1122, Aug. 1977.

5Murray, James E. and Richard E. Maine, pEst version 2.1 User's Manual, NASA TM-88280, 1987.

6Sharer, M.F., Low-Order Equivalent Models of Highly Augmented Aircraft Determined from Flight Data, J.

Guidance, Control, and Dynamics, AIAA 80-1627R, Sept.-Oct. 1982.

22

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June 1996 Technical Memorandum

4.TITLE AND BUBTn'LE S. FUNDING NUMBERS

Structural Dynamic Model Obtained From Flight for Use With Piloted

Simulation and Handling Qualities Analysis

6. AUTHOR(S)

Bruce G. Powers

7. PERFORMING ORGANIZATION NAME(S) AND ADDREBS(ES)

NASA Dryden Flight Research Center

P.O. Box 273

Edwards, California 93523-0273

9. SPONBORING/MONOTORING AGENCY NAME(S) AND ADDREBS{ES)

National Aeronautics and Space Administration

Washington, DC 20546-0001

WU 537-09-22

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REPORT NUMBER

H-2075

i

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NASA TM-4747

11. BURPCEMENTkqY NOTEB

Bruce G. Powers, Analytical Services and Material, Inc., Edwards, California

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13. ABSTRACT (Maximum 200 WORIS)

The ability to use flight data to determine an aircraft model with structural dynamic effects suitable for piloted

simulation and handling qualities analysis has been developed. This technique was demonstrated using SR-71

flight test data. For the SR-71 aircraft, the most significant structural response is the longitudinal first-bending

mode. This mode was modeled as a second-order system, and the other higher order modes were modeled as a

time delay. The distribution of the modal response at various fuselage locations was developed using a uniform

beam solution, which can be calibrated using flight data. This approach was compared to the mode shape

obtained from the ground vibration test, and the general form of the uniform beam solution was found to be a

good representation of the mode shape in the areas of interest. To calibrate the solution, pitch-rate and normalacceleration instrumentation is required for at least two locations. With the resulting structural model

incorporated into the simulation, a good representation of the flight characteristics was provided for handling

qualities analysis and piloted simulation.

14. BUBJECTTERMB

Handling qualities; Parameter estimation; Simulation; Structural dynamics

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