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Masters Theses Graduate School

5-2003

State-Space Modeling of the Rigid-Body Dynamics of a Navion State-Space Modeling of the Rigid-Body Dynamics of a Navion

Airplane From Flight Data, Using Frequency-Domain Identification Airplane From Flight Data, Using Frequency-Domain Identification

Techniques. Techniques.

Robert Charles Catterall University of Tennessee - Knoxville

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Recommended Citation Recommended Citation Catterall, Robert Charles, "State-Space Modeling of the Rigid-Body Dynamics of a Navion Airplane From Flight Data, Using Frequency-Domain Identification Techniques.. " Master's Thesis, University of Tennessee, 2003. https://trace.tennessee.edu/utk_gradthes/1912

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To the Graduate Council:

I am submitting herewith a thesis written by Robert Charles Catterall entitled "State-Space

Modeling of the Rigid-Body Dynamics of a Navion Airplane From Flight Data, Using Frequency-

Domain Identification Techniques.." I have examined the final electronic copy of this thesis for

form and content and recommend that it be accepted in partial fulfillment of the requirements

for the degree of Master of Science, with a major in Aviation Systems.

Ralph Kimberlin, Major Professor

We have read this thesis and recommend its acceptance:

Peter Solies, Frank Collins

Accepted for the Council:

Carolyn R. Hodges

Vice Provost and Dean of the Graduate School

(Original signatures are on file with official student records.)

To the Graduate Council: I am submitting herewith a thesis written by Robert Charles Catterall entitled "State-

Space Modeling of the Rigid-Body Dynamics of a Navion Airplane From Flight

Data, Using Frequency-Domain Identification Techniques.” I have examined the final

electronic copy of this thesis for form and content and recommend that it be accepted

in partial fulfillment of the requirements for the degree of Master of Science, with a

major in Aviation Systems.

Ralph Kimberlin * Major Professor We have read this thesis and recommend its acceptance: Peter Solies * Frank Collins * Accepted for the Council: Anne Mayhew * Vice Provost and Dean of Graduate Studies

(Original signatures are on file with official student records.)

STATE-SPACE MODELING OF THE RIGID-BODY DYNAMICS OF A NAVION AIRPLANE FROM FLIGHT DATA, USING FREQUENCY-DOMAIN IDENTIFICATION TECHNIQUES

A Thesis Presented for the Master of Science

Degree The University of Tennessee, Knoxville

Robert Charles Catterall May 2003

ii

ABSTRACT

State-space models of the open-loop dynamics for the cruise flight condition (V = 90

KCAS) of a modified Navion were extracted from flight data using a frequency-

domain identification method. The identified longitudinal and lateral/directional

models closely match the aircraft’s dynamic response, in both magnitude and phase,

for dissimilar flight data. The identified lateral/directional dimensional stability and

control derivatives, the model parameters, compare well to values obtained from

previous wind tunnel testing. The identified longitudinal derivatives, however, differ

from the wind tunnel results, in some cases significantly. The results for the

longitudinal derivatives are attributed to insufficient excitation of the aircraft,

particularly at the low frequency end (from about 0.2 Hz to 0.6 Hz), and to

differences in the test conditions (most notability, the flight data was for the gear

down configuration, not gear up as in the wind tunnel study). Further investigation,

including additional flight data (preferably from a Navion without fixed gear), is

needed to fully resolve these issues, and necessarily identify the longitudinal

derivatives.

iii

PREFACE

This study is part of ongoing flight-testing research by the Aviation Systems and

Flight Research Department at the University of Tennessee Space Institute. It follows

a 1998 study by Randy Bolding of the “closed-loop” handling qualities of the

Navion’s (N66UT) conventional and fly-by-wire controls using frequency-domain

techniques, entitled: “Handling Qualities Evaluation of a Variable Stability Navion

Airplane (N66UT) Using Frequency-Domain Test Techniques”.

This study involved the planning and execution of a flight test to collect input and

output parameter data for the Navion (N66UT), and data analysis and model

development using CIFER® (Comprehensive Identification from FrEquency

Responses), a system identification and verification software facility. It is intended to

lay the ground work for future efforts aimed at developing a gain scheduling routine

that ties together derivatives identified at select airspeeds to form a [single], variable

coefficient, state-space model of the Navion (N66UT) that is applicable over a wide

airspeed range.

iv

TABLE OF CONTENTS

1. Introduction and Purpose ................................................................................... 1 2. Background .......................................................................................................... 3

2.1 Previous Research........................................................................................... 3 2.2 State-Space Modeling..................................................................................... 4 2.3 Rigid-Body Dynamics .................................................................................... 5 2.4 Identification................................................................................................... 6 2.5 Data Presentation............................................................................................ 8

3. Navion (N66UT) Research Aircraft ................................................................. 10 3.1 Aircraft Description...................................................................................... 10

3.1.1 Aircraft Modifications........................................................................... 10 3.2 Control System ............................................................................................. 14

3.2.1 Cockpit Controls ................................................................................... 14 3.2.2 Moment Controls................................................................................... 14 3.2.3 Normal Force Control ........................................................................... 14

4. Flight Test and Data Aquisition ....................................................................... 15 4.1 Flight Test..................................................................................................... 15

4.1.1 Flight Test Conditions........................................................................... 15 4.1.2 Flight Test Technique............................................................................ 16 4.1.3 Special Flight Test Precautions ............................................................. 18

4.2 Data Aquisition............................................................................................. 18 4.2.1 Cockpit Instrumentation........................................................................ 18 4.2.2 Flight Test Instrumentation ................................................................... 19 4.2.3 Data Acquisition System....................................................................... 19

5. Parameter Identification and Modeling........................................................... 22 5.1 Data Conditioning and Compatability.......................................................... 22 5.2 Spectral Analsis ............................................................................................ 24 5.3 Model Setup.................................................................................................. 26 5.4 Model Structure Determination .................................................................... 29 5.5 Time Domain Verification............................................................................ 30

6. Results ................................................................................................................. 31 6.1 General.......................................................................................................... 31 6.2 Longitudinal Dynamics ................................................................................ 31 6.3 Lateral/Directional Dynamics....................................................................... 37

7. Concluding Remarks ......................................................................................... 43 8. Recommendations .............................................................................................. 44 List of References ...................................................................................................... 45 Appendix .................................................................................................................... 50 Vita............................................................................................................................ 131

v

LIST OF FIGURES

Figure Page

Figure 2-1. Bode and Coherence Plots for Aileron Input (DAIL) to Roll Rate (P), Lateral Accel. (AY), and Roll Angle (PHI). .......................................................... 9

Figure 3-1. Photo of Ryan Navion (N66UT). ............................................................. 11 Figure 3-2. Three-View Drawing of a Navion............................................................ 11 Figure 4-1. Example Frequency Sweep Input (Aileron Control Deflection).............. 16 Figure 5-1. Frequency-Domain Identification/Modeling Methodology. .................... 23 Figure 6-1. Time Histories of the Identified Longitudinal Model for an Elevator

Doublet................................................................................................................. 33 Figure 6-2. Root Locus Diagram of the Identified Longitudinal Model..................... 36 Figure 6-3. Root Locus Diagram of the Comparison Model Based on the Wind

Tunnel Results. .................................................................................................... 36 Figure 6-4. Time Histories of the Identified Lateral/Directional Model for an Aileron

Doublet................................................................................................................. 38 Figure 6-5. Time Histories of the Identified Lateral/Directional Model for a Rudder

Doublet................................................................................................................. 39 Figure 6-6. Root Locus Diagram for the Identified Lateral/Directional Model.......... 42 Figure 6-7. Root Locus Diagram for the Comparison Model Based on the Wind

Tunnel Results. .................................................................................................... 42 Figure A-1. CIFER® Naming Convention for Plots................................................... 51 Figure A-2. Weight and Balance Sheet for Frequency Response Flight Test............. 52 Figure A-3. Photo of DaqBook Installation in N66UT’s Cabin. ................................ 53 Figure A-4. Example Time Histories and Frequency Responses for Two

Concatenated Aileron Frequency Sweeps ........................................................... 55 Figure A-5. LOE Transfer Function Model of Dutch Roll Dynamics. ..................... 106 Figure A-6. Frequency Responses for the Longitudinal Model................................ 111 Figure A-7. Frequency Responses for the Lateral/Directional Model. ..................... 113 Figure A-8. Bode and Coherence Plots of the Relevant Responses to Elevator Input.

............................................................................................................................ 117 Figure A-9. Bode and Coherence Plots of the Relevant Response Parameters to

Aileron Input...................................................................................................... 121 Figure A-10. Bode and Coherence Plots for the Relevant Responses to Elevator Input.

............................................................................................................................ 126

vi

LIST OF TABLES

Table Page

Table 3-1. Navion Specifications. ............................................................................... 12 Table 4-1. Test Matrix................................................................................................. 17 Table 4-2. Measured Input and Output Parameters..................................................... 20 Table 5-1. Parameters Removed During Initial Model Setup..................................... 27 Table 5-2. Frequency Ranges for Selected Frequency-Response Pairs. ..................... 28 Table 6-1. Identified Longitudinal Dimensional Derivatives. .................................... 34 Table 6-2. Longitudinal Dynamic Stability Modes for the Identified Longitudinal

Model. .................................................................................................................. 35 Table 6-3. Longitudinal Dynamic Stability Modes for the Comparison Model Based

On the Wind Tunnel Results................................................................................ 35 Table 6-4. Identified Lateral/Directional Dimensional Derivatives............................ 40 Table 6-5. Lateral/Directional Dynamic Stability Modes for the Identified Model. .. 41 Table 6-6. Lateral/Directional Dynamic Stability Modes for the Comparison Model

Based on the Wind Tunnel Results. ..................................................................... 41 Table A-1. Input and Output/Response Parameters .................................................... 54 Table A-2. Derivative Values and Associated Costs for the Minimally Parameterized

Longitudinal Model. .......................................................................................... 108 Table A-3. Derivative Values and Associated Costs for the Minimally Parameterized

Lateral/Directional Model.................................................................................. 109 Table A-4. Relevant Longitudinal Dimensional Derivatives.................................... 129 Table A-5. Relevant Lateral/Directional Dimensional Derivatives. ......................... 130

vii

NOMENCLATURE

Unit Description dB Decibels deg Degrees ft Feet lbs Pounds rad Radians sec Seconds Abbreviation Description a/c Aircraft C.G. Center of gravity CZT Chirp z-transform DFT Discrete Fourier transform FBW Fly-by-wire FFT Fast Fourier transform Hd Density altitude Hz Hertz KCAS Knots calibrate airspeed LOE Low-order equivalent MAC Mean aerodynamic chord MIMO Multi-input/multi-output OAT Outside air Temperature SISO Single-input/single-output UTSI University of Tennessee Space Institute Symbol Description Units ax,AX Longitudinal acceleration (positive out the nose of

the a/c) ft/sec2

ay, AY Lateral acceleration (positive out the right wing of the a/c)

ft/sec2

az, AZ Normal acceleration (positive downward) ft/sec2 f Frequency Hz g Acceleration due to gravity (at sea level) ft/sec2 Gxx Input autospectrum power-dB Gxy Cross-spectrum power-dB Gyy Output autospectrum power-dB H Frequency-response dB

NOMENCLATURE (CONT’D)

viii

Hd Density altitude ft l, L Rolling moment ft-lbs m, M Pitching moment ft-lbs n, N Yawing moment ft-lbs p, P Roll rate (positive right-wing down) deg/sec q, Q Pitch rate (positive nose up) deg/sec r, R Yaw rate (positive nose right) deg/sec s Laplace variable 1 S Wing plan form area ft2 u, U Longitudinal speed component ft/sec u Control input vector v, V Lateral speed component ft/sec w, W Normal speed component ft/sec x State vector y Measurement vector β Sideslip angle deg δa, DAIL Aileron surface deflection (positive for right-

aileron tailing edge up, which results in a positive roll rate)

deg

δe, DELE Elevator surface deflection (positive for elevator trailing edge down, which results in a negative pitch rate)

deg

δr, DRUD Rudder surface deflection (positive for rudder trailing edge left, which results in a negative yaw rate)

deg

φ, PHI Bank angle deg θ, THTA Pitch angle deg ψ Heading angle deg ω Frequency rad/sec

1

1. INTRODUCTION AND PURPOSE

Determining an aircraft’s dynamic characteristics from flight data is a primary

objective of most any flight test program. For flight dynamics investigations needing

only to describe an aircraft’s handling qualities, conventional flight-test and analysis

techniques which can only characterize the frequency and damping of the dynamic

stability modes, and some static parameters, are usually adequate. A reasonably

complete math model, however, is needed to develop control laws for modern flight

control systems, and for piloted simulation.1 Development of comprehensive analysis

tools, along with advances in computational capabilities, has substantially enhanced

the use of parameter identification techniques to fulfill this modeling need. In

particular, flight test organizations now have the option to use frequency-domain

identification techniques, which have a distinct advantage over time-domain

techniques in that they are well suited for unstable systems, and are unbiased for

process and measurement noise. Frequency-domain techniques are also advantageous

because they do not require comprehensive, high fidelity analytical modeling with its

inherent assumptions. That is to say, frequency-domain techniques can be used to

describe the dynamics of an aircraft a priori using frequency responses (non-

parametric describing functions) before any model structure is assumed. Where as,

time-domain methods such as the Maximum Likelihood Method require that a

[parametric] model structure be determined before any analysis can begin.

2

The Navion (N66UT) was selected for this investigation due to its availability as an

instrumented test aircraft, and because derivative estimates determined from previous

wind tunnel testing were available for comparison.

The purpose of this study is to create a 6-DOF state-space derivative model of the

rigid-body dynamics for the Navion (N66UT) from flight data, using frequency-

domain identification techniques.

3

2. BACKGROUND

2.1 PREVIOUS RESEARCH

Scientific interest in describing an aircraft’s dynamic behavior in terms of stability

and control derivates data dates back to reports published by the National Advisory

Committee on Aeronautics (NACA) in the early 1920’s.2 Research involving

frequency-response identification is not new either. In fact, the earliest reported

research of frequency-response identification of aircraft dynamics from flight data

was conducted in 1945 at the Cornell Aeronautical Laboratory.3 For this study, low-

order transfer-function models were derived for the B-25J aircraft using a least-

squares fit of frequency-responses extracted from steady-state sine-wave inputs.

Comprehensive surveys of identification research conducted during the 1950’s,

1960’s and early 1970’s can be found in Reference [3]. More recent frequency-

domain identification efforts are described in References [4], [5] and [6] in which

transfer-function and 6-DOF rigid-body models of various helicopters and tilt-rotors

are derived using CIFER®, the frequency-domain identification analysis software

used by this study.

The Navion has also been used for a number of research efforts. Of particular

relevance to this study are three (3) dynamics studies conducted at Princeton

University—Princeton was the previous owner of this study’s test aircraft—in the

early 1970’s. The first study compared the non-dimensional aerodynamic parameters

obtained from various analytical measurements, wind-tunnel testing, and flight-test

4

data.7 That study found large differences between the wind-tunnel and flight-test

values. The second study extracted non-dimensional aerodynamic parameters from

flight data using a maximum-likelihood minimum-variance technique.8 Results from

the study showed that the values of the parameters were affected by the data and math

model used during the extraction process. The third study compared the longitudinal

derivatives estimated from several different parameter identification methods.9 The

study concluded that the type of identification method only weakly affected the

estimated short-period dynamics, and that results for the aerodynamic coefficients

were dependant on the type of math model and the type of test data. The study further

showed that the Navion has stick-fixed longitudinal stability for angles of attack up to

and through stall at 25% MAC, and that power had a small destabilizing effect.

2.2 STATE-SPACE MODELING

State-space modeling is a mathematical characterization of the [coupled] aircraft

dynamics in terms of ordinary linear differential equations with constant coefficients.

The coefficients of the equations are the force and moment (stability and control)

derivatives of the “linearized” equations of motion, as commonly derived using

small-perturbation assumptions. For example, pitching moment is expressed in terms

of the perturbation variables using a Taylor series expansion similar to the form of

Equation 1. The remaining linear velocities and angular rates are similarly expressed.

⋅⋅⋅+∆∂

∂+⋅⋅⋅+∆

∂

∂+∆

∂

∂+∆

∂

∂= p

pMw

wMv

vMu

uMrqpwvuwvuM rea ),,,,,,,,,,,( δδδDDD (1)

5

For the problem of parameter identification of a linear system, the process is assumed

to be governed by the linear matrix differential equations (state and observation

equations) presented in Equation 2 and Equation 3. Using this system structure, the

problem then becomes one of estimating the parameter values of the coefficient

matrix F (the stability derivatives) and the coefficient matrix G (the control

derivatives) that describe the aircraft’s aerodynamic response to changes in the state

variables, x, and the control variables, u. It should be pointed out that violation of the

small-perturbation assumptions negates the validity of the results—this necessarily

includes violation of the assumptions during test maneuvers to collect flight data.

GuFxxM +=� (2)

xHxHy C21 += (3)

2.3 RIGID-BODY DYNAMICS

The dynamic behavior of an aircraft is significantly shaped by its stability and control

characteristics, which in turn have their roots in the aerodynamics of the airframe.10

6-DOF “linearized” state-space models, based only the rigid-body states u, v, w, p, q

and r, are usually sufficient to describe the aerodynamic response of conventional,

fixed-wing aircraft, such as the Navion (N66UT). Limiting the model to (6) degrees

of freedom results in a total of (60) possible derivatives, including (36) stability

derivatives and (24) control derivatives. Many of the derivates, however, are often

negligible, and can be dropped from the model. This includes longitudinal cross-

coupling derivates for “uncoupled” aircraft, as well as the thrust derivatives (when

6

changes in forward velocity are assumed negligible). As such, the dynamics of an

aircraft may be described by simpler longitudinal and lateral/directional subsystems,

typically of the form presented by Equation 4 and Equation 5, respectively.

∆

∆⋅

+++

∆

∆

∆

∆

⋅

+++

−

=

∆

∆

∆

∆

T

e

TwTewe

Te

Te

wqwwwuwu

wu

wu

ZMMZMMZZXX

qwu

uMMZMMZMMuZZ

gXX

qwu

δ

δ

θθ

δδδδ

δδ

δδ

00010000

0

0

0

DDDDD

D

D

D

D

(4)

( )

∆

∆⋅

+

∆

∆

∆

∆

⋅

−−

=

∆

∆

∆

∆

r

a

ra

ra

r

rpv

rpv

rpv

NNLLY

rpv

NNNLLL

gYuYY

rpv

δ

δ

φ

θ

φ

δδ

δδ

δ

00

0

001000

cos 00

D

D

D

D

(5)

2.4 IDENTIFICATION

System identification is a procedure for accurately characterizing the dynamic

response of a complete aircraft, subsystem, or individual component from measured

data.11 Parameter identification, the determination of stability and control derivatives,

from flight data is a special application of system identification. In simple terms,

identification is a process by which computer model outputs are matched with time

histories (in this case flight data). Other aerospace identification applications include:

� Simulation model development;

� Handling qualities analysis and evaluations; and,

� Control system design and optimization.

7

Analytical approaches to identification may be broadly categorized as either time-

domain or frequency-domain methods. Frequency-domain identification methods,

such as the one used for this study, are based on the spectral analysis of the input and

output time histories using FFT techniques. Unlike time-domain methods, frequency-

domain identification does not require high fidelity analytical modeling with its

inherent assumptions. That is to say, frequency responses (see Equation 6), complex-

valued describing functions that relate the Fourier Transforms of an output parameter

to an input parameter (non-parametric modeling), can be analyzed without the need to

first establish a parametric model structure (e.g., state-space or transfer function

structure). It should be noted that frequency responses are linear approximations of

the response (the actual aircraft/system dynamics may be nonlinear).

)()(

)(fGfG

fHxx

xy= (6)

Frequency-domain methods have a distinct advantage over time-domain approaches

in that calculation of the frequency response is unbiased for uncorrelated effects such

as process and measurement noise. Moreover, unlike time-domain techniques,

frequency-domain techniques are relatively insensitive to input shaping, especially at

higher frequencies.12 It should be pointed out, however, that frequency-domain

techniques, as well as time-domain approaches, can be significantly altered by the

data collection instrumentation/system. In particular, time lags and phase shifts in the

data adversely affect analysis by frequency-domain techniques.

8

2.5 DATA PRESENTATION

Frequency responses are presented in the form of a Bode diagram. A Bode diagram is

comprised of two corresponding plots, the gain plot (input-to-output versus log-

magnitude) and the phase plot (linear-phase versus log-frequency). A Bode diagram

completely describes the response of a system as a linear approximation. An example

Bode diagram is shown in Figure 2-1. (Refer to Figure A-1 for the naming convention

used for the plots.)

In addition to Bode diagrams, Coherence plots (the graphical presentation of the

coherence function) are typically presented, as shown in Figure 2-1. The coherence

function (see Equation 7) provides an indication of the “quality” of the first harmonic

as a linear model of the particular input-output dynamics.13 The coherence may be

physically interpreted as the fraction of the output power that is linearly related to the

input power with the linear effects of the correlated input.14 Thus, a perfectly linear

relationship between two parameters would have coherence equal to unity. In general,

the coherence is less than unity due to:

� Input and output noise in the system (i.e., low signal-to-noise ratio);

� Secondary/off-axis inputs which excite the system; and,

� System nonlinearities (i.e., remnants not accounted for by the first harmonic

describing function).

yyxx

xy

GG

G

⋅

=

2

2γ (7)

9

Figu

re 2

-1. B

ode

and

Coh

eren

ce P

lots

for

Aile

ron

Inpu

t (D

AIL

) to

Rol

l Rat

e (P

), L

ater

al A

ccel

. (A

Y),

and

Rol

l Ang

le

(PH

I).

Unr

elia

ble

Dat

a

10

3. NAVION (N66UT) RESEARCH AIRCRAFT

3.1 AIRCRAFT DESCRIPTION

The research aircraft is a modified Navion (Registration Number N66UT; Serial

Number NAV-42013) manufactured by Ryan Aeronautical Company of San Diego,

California. It is a low-wing, four-place airplane, powered by a single air-cooled

engine with tricycle landing gear. Its fuselage is an all-metal one-piece semi-

monocoque structure; the empennage is a cantilever monoplane type with a

conventional horizontal stabilizer and vertical fin arrangement. A picture and a three-

view drawing of the Navion are shown in Figure 3-1and Figure 3-2, respectively.

Aircraft specifications are presented in Table 3-1. A detailed description of the

Navion may be found in Reference [15].

3.1.1 Aircraft Modifications

The Navion (N66UT) has several modifications that make it uniquely different than a

production Navion. The most notable modifications to the Navion (N66UT) are

flight-test instrumentation (refer to Section 4.2.2) and an analog, variable-stability

(variable-response) computer and fly-by-wire (FBW) control system The FWB

system, which is commanded from the pilot’s station, is comprised of an assortment

of sensors, hydraulically actuated control surfaces (commanded by electrical signals),

cockpit controls and an analog flight control computer. The computer is housed in the

cabin in place of the N66UT’s two rear seats.

11

Figure 3-1. Photo of Ryan Navion (N66UT).

Figure 3-2. Three-View Drawing of a Navion.

12

Table 3-1. Navion Specifications.

AIRCRAFT: Length........................................................................... 27 ft, 6 in WING: Span (b) ........................................................................ 33 ft, 5 in Area (S) ........................................................................ 184 ft2 Sweep, leading edge..................................................... 2.996 deg Aspect Ratio (AR)........................................................ 6.04 Taper Ratio (λ) ............................................................. 0.54 Mean aerodynamic chord (MAC) ................................ 5.7 ft Dihedral........................................................................ 7.5 deg AIRFOIL: Tip ................................................................................ NACA 6410 Root .............................................................................. NACA 4415 Wing loading ................................................................ 14.6 lb/ft2 HORIZONTAL TAIL: Area .............................................................................. 43 ft2 Aspect ratio .................................................................. 4.0 Taper ratio .................................................................... 0.67 Airfoil ........................................................................... NACA 0012 VERTICAL TAIL: Area above horizontal stabilizer................................... 15.5 ft2 Rudder area .................................................................. 8.33 ft2

13

Other notable changes to the Navion (N66UT) include:

� The engine is replaced with the heavier and more powerful Teledyne-

Continental IO-520B engine. This engine is rated at 285 take-off horsepower

(2700 RPM at sea level).

� The standard two-bladed propeller is replaced with a McCauley three-bladed

constant speed propeller.

� The main landing gear are replaced with stronger struts designed for the

Camair twin (a Navion conversion with nearly 40% greater GW). As a result

of this modification, the landing gear are no longer retractable.

� Hinging and actuation modifications are installed that allow the flaps to be

deflected up, as well as down, within a ±30 deg range.

� The mechanical system linking the pilot’s and co-pilot’s flight yoke and

rudder pedals is removed in lieu of the FBW flight controls installed at the

pilot’s station.

� The mechanical flight control coupling system, which applies coordinated

aileron movement with pedal input, is removed.

14

3.2 CONTROL SYSTEM

3.2.1 Cockpit Controls

The conventional flight control system is commanded from the copilot’s station, and

is fully reversible with the FBW system disengaged. The primary cockpit controls

consist of pedals governing rudder action, a control wheel/yoke governing aileron and

elevator action and an engine throttle. As stated previously, the standard rudder-to-

aileron mechanical coupling system, which applies coordinating aileron movement

with pedal input, is removed from the Navion (N66UT).

3.2.2 Moment Controls

Piloted control of the aircraft’s pitching, rolling, and yawing are through conventional

elevator, aileron, and rudder control surfaces, respectively. Additional control of the

aircraft’s pitching is through trim tabs, installed in the trailing edge of each elevator.

The control surfaces are connected to their respective cockpit controls by various

combinations of rods/linkages, bell cranks, cables and pulleys.

3.2.3 Normal Force Control

Control of the aircraft’s normal acceleration is exercised through set of in-board flaps.

As stated previously, the installed flaps have hinging and actuation modifications that

allow the flaps to be deflected over a ±30 deg range. Actuation is hydraulic, with a

maximum available surface rate of 110 deg/sec.

15

4. FLIGHT TEST AND DATA AQUISITION

4.1 FLIGHT TEST

A test flight was conducted on October 9, 1998 to collect short-term frequency

response data of the Navion (N66UT), unaugmented (i.e., with the FBW system

disengaged). The essential test elements were on-axis pilot-generated-frequency-

sweep inputs for each control axis (directional, lateral, longitudinal, and thrust). In

total, data was collected for 12 frequency sweeps; three on-axis frequency sweeps

were conducted for each control axis. The flight crew consisted of two crewmembers,

an experimental test pilot and a flight test engineer. The test pilot flew all

experimental maneuvers. The flight test engineer manually activated the data

recording process, monitored the data on strip charts and provided timing cues to the

test pilot. The duration of the test flight was 1.5 hours.

4.1.1 Flight Test Conditions

The test flight was conducted in calm air under daylight visual meteorological

conditions (VMC) at the Tullahoma Municipal Airport, Tullahoma, Tennessee, on

October 9, 1998. The test was conducted within the existing envelope of the aircraft

with the canopy closed, flaps at 0°, the landing gear extended, the FBW system

disengaged and with the heater and carburetor heat off. The trim configuration was

steady-state horizontal flight at 90 KCAS at about 5000 ft Hd. The average gross

weight and C.G. for the data collection flight were 3,242 pounds and 100.3 inches aft

16

of datum (24.4% M.A.C.), respectively (refer to Figure A-2 for weight and balance

calculations).

4.1.2 Flight Test Technique

The flight test technique to acquire data consisted of on-axis pilot-generated-

frequency-sweep inputs (see Figure 4-1) for each control axis (directional, lateral,

longitudinal, and thrust); three on-axis frequency sweeps were conducted for each

control axis (see Table 4-1). Each sweep was performed by making control

displacements to either side of trim, at frequencies from 0.1 Hz to approximately 2.0

Hz. The control displacements were pseudo-sinusoidal in shape, and were large

enough to affect a noticeable aircraft response, but not so large as to generate large

airspeed changes (+/- 10 knots), translations or changes in attitude.

Figure 4-1. Example Frequency Sweep Input (Aileron Control Deflection).

17

Table 4-1. Test Matrix.

CONTROL INPUT

CONTROL AXIS

NO. OF EVENTS

AIRSPEED[KTS]

ALT. [FT]

Roll 3

Pitch 3

Yaw 3

Piloted Frequency

Sweep

Throttle 3

90 5000

Roll 3

Pitch 3

Yaw 3 Doublet

Throttle 3

90 5000

The sweeps were begun with 3 seconds at trim, followed by two low frequency

cycles, with periods of approximately 20 seconds (lower end of the desired

identification frequency range). The sweep frequency was then increased gradually to

the highest frequency that the pilot could generate (approximately 2 Hz), and then

returned to the trim condition for 3 seconds (trim data preceding and following the

sweep are required for CIFER® to concatenate time histories). Each sweep was

approximately 80 – 100 seconds in duration. Uniformity in the magnitude of control

input throughout the frequencies was of no particular concern, however, care was

taken to make sure that off-axis control movement (needed to maintain attitude, etc.)

were not “in sync” with the on-axis control since correlated movements cannot be

removed (by data conditioning) to create MISO frequency responses.

18

Control doublets, at various frequencies and amplitudes, for each control axis were

performed in addition to the frequency sweeps. The data from the doublets are needed

for time-domain verification of the identification results.

4.1.3 Special Flight Test Precautions

There is a high potential for structural damage during ‘bandwidth’ testing, not

excluding damage severe enough to led to the total loss of the aircraft. More

specifically, there is the possibility of accidentally exciting a vibratory mode at or

near the natural frequency of a structural member or other aircraft component,

particularly if inputs are made with higher magnitudes or frequencies than required.

Because the test aircraft was not instrumented for continuous measurement of real

time structural loads, precautions were taken to start frequency sweeps with small

magnitude inputs at low frequency, increase the magnitude of inputs through the

middle frequency range and decrease the magnitude of the inputs at high frequency.

Additionally, the aircraft’s input and response parameters were monitored to observe

any abnormalities/inconstancies in the data that would suggest a hazardous flight

situation.

4.2 DATA AQUISITION

4.2.1 Cockpit Instrumentation

Air data (airspeed and altitude) and engine parameters (manifold pressure and RPM),

as well as OAT, were recorded by hand on flight data cards referencing the aircrafts

cockpit instrumentation.

19

4.2.2 Flight Test Instrumentation

Potentiometers, gyroscopes, and linear accelerometers were used in conjunction with

a data acquisition system (refer to Section 4.2) to record the input and output

parameters presented in Table 4-2. Five (5) potentiometers were used to measure

surface positions and body attitudes; the potentiometers were located at their

respective control surfaces. Three (3) gyroscopes were used to measure body rates;

the gyros were located in the rear of the cabin “near” the C.G.. Three (3)

accelerometers were used to measure body accelerations. The accelerometers were

mounted together 2.3 inches forward of the aircraft’s longitudinal C.G. or 98 inches

aft of datum (see Figure A-2). The placement of the accelerometer package relative to

the aircraft’s vertical C.G. was “backed out” during the model identification process

using basic kinematic relationships. The package was estimated to be about 2.4 inches

(0.1977 ft) below the vertical C.G.

4.2.3 Data Acquisition System

A personal computer-based data acquisition system was used for the test flights. The

system was comprised of a 12-bit, multi-plexed analog-to-digital (A/D) converter (a

IoTech DaqBook® 120 shown in Figure A-3) and a 120 Mhz Pentium® notebook

computer. Using IoTech’s DaqView® software, the computer served three functions:

(1) to control the DaqBook; (2) to provide real-time storage of all the measured input

and output parameters (see Table 3-1) for post acquisition retrieval; and (3) to display

real-time strip charts of each input channel.

20

Table 4-2. Measured Input and Output Parameters.

INPUT PARAMETERS OUTPUT PARAMETERS

� Aircraft control deflections -Aileron deflection (δa) -Elevator deflection (δe) -Rudder deflection (δr) -Thrust (δT)

� Body attitudes -Roll attitude (φ) -Pitch attitude (θ) -Yaw attitude (ψ), not instrumented

� Body angular rates -Roll rate (p) -Pitch rate (q) -Yaw rate (r)

� Linear accelerations -Longitudinal acceleration (ax) -Lateral acceleration (ay) -Normal acceleration (az)

* Reconstructed from body angular rates

using basic kinematic relationships

� Speed data * -Longitudinal speed component (u) -Lateral speed component (v) -Normal speed component (w)

21

The system was setup to measure voltage differential across 16 single-ended, bipolar

input channels (see Table A-1). Total data transfer rates for the test were on the order

of 800 Kbytes/sec at a sampling rate of 100 Hz for all channels—channel sequencing

was at 10 millisecond intervals.

A sufficient number of anti-aliasing filters (one for each channel) were not available;

thus, filters were not used so as not to induce phase shifts in the data of specific

channels, or otherwise condition specific data in anyway. It should be noted,

however, that “pre-filtering” of the data is typically done to remove high frequency

“noise”, such as structural and engine vibrations. High frequency content, outside the

range of interest, was digitally filtered out during data conditioning (refer to Section

5.1).

22

5. PARAMETER IDENTIFICATION AND MODELING

Parameter identification and model structure determination, including all necessary

data conditioning and analysis, for this study were accomplished using CIFER® 3.0

(Comprehensive Identification from FrEquency Responses), a system identification

and verification software facility developed by Mark Tischler and Mavis Cauffman of

the U.S. Army’s Aeroflightdynamics Directorate (AFDD) and Mavis Cauffman of

Sterling Software, Inc, respectively. The identification process followed for this study

is presented in Figure 5-1. A detailed description of the identification methodology,

as well as the specific capabilities of CIFER®, may be found in Reference [14].

5.1 DATA CONDITIONING AND COMPATABILITY

The starting point for frequency-domain identification is data conditioning. Unlike

time-domain approaches, frequency-domain analysis requires that the time-based data

be converted to frequency-based data used to create frequency responses of each

input/output pair. The extraction of frequency data from the parameter time histories

is achieved with CIFER® using the Chirp-Z or “Zoom” FFT. It should be noted that

the data is also typically pre-processed for dropouts using a “smoother” (e.g., Kalman

filtering) to avoid computational problems. However, the capability to “smooth” the

data was not available for this study—no computational problems were encountered

during the subsequent analysis. Also, the time history data for each parameter were

first converted to consistent engineering units and digitally filtered at

23

Figure 5-1. Frequency-Domain Identification/Modeling Methodology.

24

5 Hz to remove high frequency data (noise). The time history sets were additionally

conditioned by digitally resampling the data at 50 Hz—this was done to reduce the

overall computational burden.

5.2 SPECTRAL ANALSIS

The foundation of frequency-domain parameter identification is the creation of

frequency responses using the responses for each input/output [parameter] pair. Three

steps are performed to create the conditioned frequency-responses used for model

structure determination, and ultimately the estimation of the model coefficients (in

this case the stability and control derivatives). The first step is to generate single-

input/single-output (SISO) frequency-responses for each input/output pair for all of

time history records for each “sweep” axis. Of the three (3) sweep data sets collected

for each axis, the “best” two (2) records were selected based on a qualitative

assessment of the spectral functions for select frequency response pairs. The two (2)

time history records were then concatenated to form a single record. SISO frequency

responses were then created for the concatenated records. Also included in the

analysis are frequency responses for pseudo measurements of udot, vdot, and wdot

(used for model identification). These pseudo states are reconstructed from the

[measured] inputs/outputs using the basic kinematic relationships presented in

Equation 8. Next, “poor” quality frequency-responses, as well as responses that

showed no correlation, are removed based on analysis of the spectral functions

(coherence, cross-spectrum, input-autospectrum, and output-autospectrum) for each

frequency-response pair. Finally, the remaining frequency responses are regenerated

25

for window lengths. For this study, window lengths of 10, 20 and 30 seconds were

used.

qUpVaw

gpWrUavgrVqWau

oozcg

ooycg

ooxcg

+−=

++−=

−+−=

D

D

D

φ

θ

(8)

The second step is to condition the frequency responses by removing the effect of

secondary/off-axis control inputs to form a set of multi-input/single-output (MISO)

responses The MISO frequency responses are created by effectively “filtering” out

secondary/off-axis inputs (e.g., aileron input to maintain wings level during the

elevator sweep) using a multivariable spectral analysis technique (a unique capability

of CIFER®). The resulting MISO frequency responses are equivalent to SISO

frequency responses that would have been obtained with only a “single” input.

The third step is to combine the MISO results of [the same] data calculated from the

various window sizes. For this study, the MISO frequency responses for window

lengths of 10, 20, and 30 seconds were combined to create a set of [optimal]

composite frequency responses. Combining windows is a unique capability of

CIFER®; it is achieved by combining the auto- and cross-spectral density estimates

using a non-linear, least-squares optimization method. The windows can be thought

of as sub records of the FFT for the total time history record. The term refers to a

technique called “windowing” or “tapering” which reduces the erroneous frequency

content that results from computation of a FFT. Selection of a window length

26

necessarily limits the frequency content of a window (i.e., a sub-record), in that the

lowest frequency possible is that frequency with a corresponding period the size of

the window. Thus, use of a single window is a compromise between having low

frequency content and having a low random error (i.e., erroneous frequency content).

These steps are repeated until composite responses are created for each input/output

pair. An example CIFER® output of the spectral analysis and frequency response

conditioning steps described above is presented in Figure A-4.

5.3 MODEL SETUP

The 6-DOF model structure is assumed to be governed by Equation 2 and Equation 3,

where:

[ ]TTreau δδδδ=

[ ]Trqpwvux θφ=

[ ]Tzyx aaarqpwvuy =

For this study, however, analysis of the frequency response pairs for the thrust

parameters lead to the removal of all of the thrust derivatives from the model

structure due to poor coherence throughout the entire frequency range of interest.

Furthermore, analysis of the frequency-response pairs for the longitudinal cross-

coupling parameters showed that there was either no cross-correlation between

parameters or a negligible response. Therefore, the cross-coupling derivatives listed

in Table 5-1 were “dropped” from the model. Dropping these parameters resulted in

27

Table 5-1. Parameters Removed During Initial Model Setup.

uvwpqr

Mδr, MrMδa, Mp

δe δa δr

Xδr, Xr

Zδr, Zr

Xδa, Xp

Zδa, Zp

Nδe, Nq

Xδe, Xq

Yδe,Yq

Lδe, Lq

two (2) 3-DOF subsystem models, similar to Equation 4 and Equation 5 for the

longitudinal and lateral/directional dynamics, respectively.

In addition to stipulating the initial model structure, it is necessary to specify the

frequency ranges of “good” data for each of the input/output frequency-response pairs

that correspond to the model parameters—this is a critical step since the inclusion of

“bad” data can adversely affect the determination of the coefficient values. The low

and upper frequency range is selected based on an overall range of “good” coherence

(greater than 0.6)—each frequency point is weighted during the optimization process

based on the coherence value at that point. The frequency ranges selected for this

study are presented in Table 5-2.

Another necessary step is the estimation of the initial model parameters. Like any

technique that uses an optimization routine, estimates for the initial parameter vales

should be reasonably close to the expected final values to ensure that the converged

solution isn’t for local minima that do not have any physical correlation to stability

28

Table 5-2. Frequency Ranges for Selected Frequency-Response Pairs.

ω min ω max ω min ω max ω min ω max ω min ω max

u 0.3 4

v 0.4 17 0.2 20

w 0.2 10

p 0.2 20 0.2 15

q 0.3 20

r 0.2 8 0.9 20

ax

ay 0.2 15 0.2 15

az 0.2 7

φ 0.2 20 0.2 7

θ 0.25 15

δe δa δr δT

and control derivatives.

For this study, wind tunnel results from Reference [16] were used to “initialize” both

the longitudinal and lateral/directional subsystem models (see Table A-2 and Table

A-3 for initial values). Initial values for the model could have also been obtained in

part or in whole from three possible sources:

1) Modal characteristics from direct inversion of transfer-function model fits of

frequency-response data (see Figure A-5 for an example);

2) Equation-error identification results; and,

3) A priori values based on simulation models or other data sources.7

29

5.4 MODEL STRUCTURE DETERMINATION

The purpose of model structure determination is to identify and remove parameters

that either do not contribute or only marginally contribute to the fidelity of the model.

Using CIFER®, the state-space models of some specified general structure and of

high-order are simultaneously fitted for all the frequency responses. The process is

iterative; it involves minimizing the weighted square-error between the measured

frequency responses (see Figure A-4) and the model’s frequency responses. The

parameter values are determined with CIFER® using a nonlinear pattern search

(Secant) algorithm that matches MISO frequency-responses. The final model

structure is determined by eliminating parameters one-by-one and reconverging the

solution until the error function (i.e., the “cost”) for the model is reduced to an

acceptable level or begins to increase. Parameters that the model is insensitive to or

that are highly correlated to other parameters are identified by their large individual

“cost.” Parameters that the model is insensitive to are removed first (one-by-one), and

the model is reconverged until the solution has insensitivities less than 10%. Then,

parameters that are highly correlated to another parameters or multiple parameters, as

measured by their Cramer-Rao bounds, are eliminated from the model one-by-one,

and the model is reconverged. The judgment as to which parameter to eliminate is

based on confidence ellipsoids and/or physical principals. A minimally parameterized

model is achieved once all the parameters in the converged solution have Cramer-Rao

bounds less than 20%.

30

The converged longitudinal and lateral/directional model parameters, and associated

Cramer Rao bounds and cost functions, are presented in Table A-2 and Table A-3,

respectively. Bode diagrams and coherence plots for the resulting frequency

responses for the longitudinal and lateral/directional models are presented in Figure

A-6 and Figure A-7, respectively.

5.5 TIME DOMAIN VERIFICATION

The final step in the identification process (see Figure 5-1) is model verification.

Ultimately, this step demonstrates the predictive capability of a model by showing the

aircraft’s dynamic behavior, in both magnitude and phase. Verification is

accomplished with CIFER® by driving the identified state-space model(s) with

dissimilar flight (i.e., data not used during the identification process). For this study

control doublets collected during the flight test are used. The unknown [time-domain]

state equation biases and reference (or zero) shifts are determined by performing a

non-iterative minimization of the weighted least square error between the model and

vehicle responses.

The time history plots for the longitudinal and lateral/directional models are presented

in Section 6.2 and Section 6.3 , respectively.

31

6. RESULTS

6.1 GENERAL

3-DOF models for the decoupled longitudinal and lateral/directional, rigid-body

dynamics of the Navion (N66UT) were extracted from flight data for the cruise flight

condition (V = 90 KCAS). The models were derived using the frequency-domain

identification methodology described in Section 1. The resulting model structure (two

3-DOF subsystem models instead of a 6-DOF) was arrived at “a priori” based on

analysis of the MISO frequency responses of the [measured] parameters. The analysis

showed that there was either no cross-correlation between parameters or a negligible

response for the longitudinal cross-coupling parameters. Similarly, analysis of the

thrust parameters lead to the removal of all thrust derivatives due to poor coherence

throughout the entire frequency range of interest. In total, (17) “conditioned”

frequency response pairs were used to determine the model structure. Bode diagrams

of these frequency response pairs, along with a few selected pairs with “poor”

coherence throughout the frequency range, are presented in Figure A-8, Figure A-9

and Figure A-10. The model parameters, the extracted longitudinal and

lateral/directional derivatives, are presented along with the identified models in the

following sections.

6.2 LONGITUDINAL DYNAMICS

The minimally parameterized model identified for the decoupled longitudinal

dynamics is presented in Equation 9. The validity of the model is demonstrated by

32

time history plots of the model versus flight data for an elevator control doublet, as

shown in Figure 6-1. The time histories show that that the model closely predicts, in

both magnitude and phase, the aircraft’s on- and off-axis response for dissimilar flight

data.

∆

∆⋅

−+

∆

∆

∆

∆

⋅

−−

−

−−

=

∆

∆

∆

∆

T

e

qwu

qwu

δ

δ

θθ 00064.120000

01000114.205581.02054.00152066.351.10

17.32004242.009148.0

�

�

�

�

(9)

The derivatives extracted from the model are presented in Table 6-1, along with

estimated values (based on non-dimensional coefficients) obtained from the wind

tunnel test results presented in Reference [16] for the landing gear up. The identified

values do not compare well with the wind tunnel results. In particular, the Zu, Mu and

Mw differ substantially in magnitude and/or sign. These derivatives, along with Zδe

(which was dropped during model convergence due its large insensitivity), are

considered to be questionable since there was insufficient low frequency content in

the flight data. Analysis of the corresponding frequency-responses shows that the

coherence is poor at frequencies between 0.2 rad/sec and 0.8 rad/sec, indicating that

the aircraft’s dynamics were not sufficiently excited (and possible signal-to-noise

issues for the measurements for ax and az). Also apparent is the fact that the identified

values are for the landing gear [fixed] down, not up as in the wind tunnel case.

Additional flight data, with adequate low frequency content, is necessary to better

determine these values.

33

Flight Data

Identified Model

Figure 6-1. Time Histories of the Identified Longitudinal Model for an Elevator Doublet.

34

Table 6-1. Identified Longitudinal Dimensional Derivatives.

Name UnitsXu 1/sec -0.092 -0.045

Xw 1/sec 0.0424 0.0361

Zu 1/sec 10.51 -0.37

Zw 1/sec -3.066 -2.024

Zq ft/sec 51.01 --

Mu+MwdotZu 1/(sec ⋅ ft) 0.2054 0.0018

Mw+MwdotZw 1/(sec ⋅ ft) -0.056 -0.039

Mq+Mwdotu o 1/sec -2.114 -2.861

Xδe ft/(sec2) 0 † --

Zδe ft/(sec2) 0 †† -28.17

Mδe+MwdotZδe 1/sec2 -12.67 -11.19

* Body-axis coordinate system† Not included in model structure bases on spectral analysis

†† Eliminated during model structure determination

Derivative * Results from Reference 16

Long

itudi

nal D

eriv

ativ

es

Results from present study

Note: Descriptions of the longitudinal dimensional derivatives can be found in Table A-4.

35

For comparison, the eigenvalues of the stability matrix for both the identified model

and the wind tunnel results are presented in Table 6-2 and Table 6-3, respectively.

The corresponding Root Locus diagrams are presented in Figure 6-2 and Figure 6-3.

The phugoid eigenvalues do not agree well with the estimated phugoid mode.

Specifically, the phugoid damping ratio is too large—a short period estimation of the

damping ratio from flight data suggest that the damping should be between 0.2 and

0.3. The speed derivatives Zu and Mu are directly related to the phugoid motion; not

surprisingly, the identified values for these derivatives differ greatly from the

estimated values.

Table 6-2. Longitudinal Dynamic Stability Modes for the Identified Longitudinal Model.

Eigenvalue Damping Ratio (ζζζζ) Frequency (ωωωω) Mode -0.188+0.237i 0.620 0.303 Phugoid -0.188-0.237i 0.620 0.303 Phugoid -2.45+3.04i .627 3.90 Short period -2.45-3.04i .627 3.90 Short period

Table 6-3. Longitudinal Dynamic Stability Modes for the Comparison Model Based On the Wind Tunnel Results.

Eigenvalue Damping Ratio (ζζζζ) Frequency (ωωωω) Mode -0.242E-01+0.221i 0.108 0.223 Phugoid -0.242E-01-0.221i 0.108 0.223 Phugoid

-2.48+2.40i 0.718 3.455 Short period -2.48-2.40i 0.718 3.455 Short period

36

-20 -15 -10 -5 0-15

-10

-5

0

5

10

15

17.5 15

0.42

0.91

0.68

0.14

12.5

0.68

10 2.57.5

0.8

5

0.975

0.280.56

0.14

0.8

0.28

0.975

0.420.56

0.91

Real Axis

Imaginary Axis

Figure 6-2. Root Locus Diagram of the Identified Longitudinal Model.

-20 -15 -10 -5 0-15

-10

-5

0

5

10

15

17.5 15

0.42

0.91

0.68

0.14

12.5

0.68

10 2.57.5

0.8

5

0.975

0.280.56

0.14

0.8

0.28

0.975

0.420.56

0.91

Real Axis

Imaginary Axis

Figure 6-3. Root Locus Diagram of the Comparison Model Based on the Wind

Tunnel Results.

37

6.3 LATERAL/DIRECTIONAL DYNAMICS

The minimally parameterized model identified for the lateral/directional dynamics is

presented in Equation 10. The validity of the model is demonstrated by time history

plots of the model versus flight data for an aileron control doublet and rudder control

doublet, as shown in Figure 6-4 and Figure 6-5, respectively. The time histories show

that the model closely predicts, in both magnitude and phase, the aircraft’s on- and

off-axis response for dissimilar flight data. The model is valid over a frequency range

of 0.2 to 12 rad/sec.

∆

∆⋅

−+

∆

∆

∆

∆

⋅

−−

−−

−−

=

∆

∆

∆

∆

r

a

rpv

rpv

δ

δ

φφ 00288.4486.0

380.174.15469.30

00100953.0548.0030.00854.1816.5059.017.326.1520093.0

D

D

D

D

(10)

The lateral/directional derivatives extracted from the model are presented in Table

6-4, along with estimated values from the wind tunnel results presented in Reference

[16]. The derivatives compare well; Nδa and Yδr are the only two notable exceptions.

The positive value for the identified Nδa suggests that the aircraft exhibits a direct

couple, or “proverse” yaw, which is contrary to the “adverse” yaw indicated by the

wind tunnel results. In-flight observation of the aircraft’s response confirmed that the

Navion (N66UT) has a “proverse” yaw response. Furthermore, the magnitude of both

Nδa and Yδr may be explained by the fact that the sweep input is dynamic and of small

amplitude, as opposed to a static, large amplitude input for the wind tunnel case.

38

Flight Data

Identified Model

Figure 6-4. Time Histories of the Identified Lateral/Directional Model for an

Aileron Doublet.

39

Figure 6-5. Time Histories of the Identified Lateral/Directional Model for a

Rudder Doublet.

40

Table 6-4. Identified Lateral/Directional Dimensional Derivatives.

Name Units

Yv 1/sec -0.093 -0.254

Yp ft/sec 0 †† --

Yr ft/sec 1.617 --

Lv 1/(sec ⋅ ft) -0.059 -0.091Lp 1/sec -5.816 -8.402

Lr 1/sec 1.854 2.193

Nv 1/(sec ⋅ ft) 0.0299 0.025

Np 1/sec -0.548 -0.35Nr 1/sec -0.953 -0.761

Yδa ft/(sec2) 0 † --

Yδr ft/(sec2) 3.469 12.45

Lδa 1/sec2 15.74 28.98

Lδr 1/sec2 1.38 2.548

Nδa 1/sec2 0.4862 -0.222

Nδr 1/sec2 -4.288 -4.597

* Body-axis coordinate system† Not included in model structure bases on spectral analysis

†† Eliminated during model structure determination

Late

ral/D

irect

iona

l Der

ivat

ives

Derivative * Results from Reference 16

Results from present study

Note: Descriptions of the lateral/direction dimensional derivatives can be found in Table A-5.

41

For comparison, the eigenvalues of the stability matrix for both the identified model

and the wind tunnel results are presented in Table 6-5 and Table 6-6, respectively.

The corresponding Root Locus diagrams are presented in Figure 6-6 and Figure 6-7.

The natural frequency and damping ratio of the Dutch roll mode, and time delay for

the spiral mode, for the identified model are essentially identical to wind tunnel

results. The time delay for the roll mode, however, is about 45% less for the identified

model. This is likely due to the larger vertical fin on the test aircraft.

Table 6-5. Lateral/Directional Dynamic Stability Modes for the Identified Model.

Eigenvalue Damping Ratio (ζζζζ) Frequency (ωωωω) Mode -0.0111 N/A N/A Spiral

-0.551-2.17i 0.246 2.24 Dutch roll -0.551+2.17i 0.246 2.24 Dutch roll

-8.30 N/A N/A Roll

Table 6-6. Lateral/Directional Dynamic Stability Modes for the Comparison Model Based on the Wind Tunnel Results.

Eigenvalue Damping Ratio (ζζζζ) Frequency (ωωωω) Mode -0.00101 N/A N/A Spiral

-0.562-2.35i 0.232 2.42 Dutch roll -0.562+2.35i 0.232 2.42 Dutch roll

-5.74 N/A N/A Roll

42

-20 -15 -10 -5 0-15

-10

-5

0

5

10

15

17.5 15

0.42

0.91

0.68

0.14

12.5

0.68

10 2.57.5

0.8

5

0.975

0.280.56

0.14

0.8

0.28

0.975

0.420.56

0.91

Real Axis

Imaginary Axis

Figure 6-6. Root Locus Diagram for the Identified Lateral/Directional Model.

-20 -15 -10 -5 0-15

-10

-5

0

5

10

15

17.5 15

0.42

0.91

0.68

0.14

12.5

0.68

10 2.57.5

0.8

5

0.975

0.280.56

0.14

0.8

0.28

0.975

0.420.56

0.91

Real Axis

Imaginary Axis

Figure 6-7. Root Locus Diagram for the Comparison Model Based on the Wind

Tunnel Results.

43

7. CONCLUDING REMARKS

The identified longitudinal and lateral/directional models closely predict the Navion's

(N66UT) on- and off-axis responses. The extracted lateral/directional derivatives

closely match previous wind tunnel results. The longitudinal derivatives, however,

differ, in some cases significantly, from the previous values. Differences for the

longitudinal derivatives are attributed to insufficient excitation of key parameters

throughout the frequency range of interest, and to differences in the test conditions

(most notably that N66UT has its gear fixed down). Further investigation, including

additional data collection, is required to better identify these derivatives. Such future

efforts should be conducted using a Navion with retractable gear.

44

8. RECOMMENDATIONS

The identification analysis should be re-accomplished for the longitudinal dynamic

model to better characterize Zu, Zδe, Mu and Mw. The additional flight data needed for

this effort should be collected using a Navion with retractable gear so that a more

direct comparison with previous results can be made. The location of the Navion’s

(N66UT) C.G. (vertical, lateral, and longitudinal) should also be determined prior to

collecting any data so that the accelerometers can be [properly] placed at the C.G.—

this will avoid having to “back” out the location, and prevent any unnecessary data

collection errors from being introduced into the analysis.

45

LIST OF REFERENCES

46

WORKS CITED

1) Kimberlin, Ralph, “Parameter Identification Techniques”, Class Notes from a course taught at the University of Tennessee Space Institute, No date.

2) Smetana, F. O., Summey, D. C., and Johnson, W. D., “Flight Testing Techniques for the Evaluation of Light Aircraft Stability Derivatives”, NASA CR2106, May 1972.

3) Eykhoff, Pieter, System Identification, Parameter and State Estimation, John Wiley & Sons, London, 1974.

4) Tischler, Mark B., “Identification and Verification of Frequency-Domain Models for XV-15 Tilt-Rotor Aircraft Dynamics in Cruising Flight”, AIAA Journal of Guidance, Control, and Dynamics, July 1986.

5) Tischler, Mark B., “Frequency-Response Method for Rotorcraft System Identification: Flight Applications to BO-105 Coupled Rotor/Fuselage Dynamics”, Journal of the American Helicopter Society, Vol 37, No 3, pp.. 3-17, July 1992.

6) Tischler, Mark B., “Time and Frequency-Domain Identification and Verification of BO 105 Dynamic Models”, 15th European Rotorcraft Forum, Amsterdam, September 1989.

7) Seckel, E., and Morris, J. J. “The Stability Derivatives of the Navion Aircraft Estimated by Various Methods and Derived From Flight Test Data,” Report No. FAA-RD-71-6, Princeton University, January 1971.

8) Seckel, E. and Morris, J. J., “Full Scale Wind Tunnel Tests of a North American Navion Airframe with Positive and Negative Propeller Thrust and Up and Down Flap Deflection,” Princeton University AMS Report 922, July 1970.

9) R. C. Windgrove, “Estimation of Longitudinal Aircraft Characteristics”, 9th Annual Symposium of the Society of Flight Test Engineers, Arlington, Texas, October 4-6, 1978.

10) Ham, Johnnie A., Tischler, Mark B., “Flight Testing and Frequency Domain Analysis for Rotorcraft Handling Qualities Characteristics”, AHS Specialists’ Meeting on Piloting Vertical Flight Aircraft, 20-23 Jan. 1993, San Francisco, California.

11) Tischler, Mark B., “Identification and Verification of Frequency-Domain Models for XV-15 Tilt-Rotor Aircraft Dynamics in Cruising Flight,” AIAA Journal of Guidance, Control, and Dynamics, July 1986.

WORKS CITED (CONT’D)

47

12) ANON. Technical Memorandum, NASA TM 89422, USAAVSCOM TM 87-A-1, “Demonstration of Frequency-Sweep Testing Technique Using a Bell 214-ST Helicopter”, April 1987.

13) Tischler, Mark B., “Flight Test Manual, Rotorcraft Frequency Domain Flight Testing,” U.S. Army Aviation Technical Test Center, AQTD Project No. 93-14, September 1995.

14) ANON., NASA Conference Publication, NASA CP 10149, USAATCOM TR-94-A-017, "Comprehensive Identification from Frequency Responses", Class Notes from a short course held at NASA Ames Research Center, Moffett Field, California, September 19-23, 1994.

15) ANON., “Navion Service Manual”, First Edition, revised January 1951.

16) Teper, Gary L., “Aircraft Stability and Control Data,” NASA CR-96008, April 1969.

48

WORKS REFERENCED

ANON., Advisory Report, AGARD-AR-280, “Rotorcraft System Identification,” North Atlantic Treaty Organization, September 1991.

ANON., Flight Test Manual, USNTPS-FTM-No. 107, “Rotary Wing Stability and Control,” Patuxent River, Maryland: U.S. Naval Test Pilot School, Naval Air Warfare Center, No Date.

ANON., Military Handbook, MIL-HDBK-1797, “Flying Qualities of Piloted Aircraft”, December 1997.

ANON., Technical Memorandum, NASA TM 110369, USAATCOM TR 95-A-007, “System Identification Methods for Aircraft Flight Control Development and Validation”, October 1995.

ANON., Technical Note, NASA TN D-5857, “Full-Scale Wind-Tunnel Investigation of the Static Longitudinal and Lateral Characteristics of a Light Single-Engine Low-Wing Airplane”, June 1970.ANON., Military Specification, MIL-F-8785C, “Flying Qualities of Piloted Airplanes”, November 1980.

ANON., “The Variable-Response Research Aircraft”, Princeton University, no date.

Basile, P.S., Gripper, S.F., and Kline, G.F., “An Experimental Investigation of the Lateral Dynamic Stability of a Navion Airplane”, Princeton University, June 1970.

Bolding, Randy, “Handling Qualities Evaluation of a Variable Stability Navion Airplane (N66UT) Using Frequency-Domain Test Techniques,” M.S. Thesis, The University of Tennessee Space Institute, December 1998.

Buxo, Arturo Manual, “Performance of the Variable-Stability Navion with Flap Deflection”, MS thesis, The University of Tennessee Space Institute, May 1995.

Catterall, R. Charles, and Lewis, William D., “Determination of Navion Stability and Control Derivatives Using Frequency-Domain Techniques”, American Institute of Aeronautics and Astronautics Conference Publication, AIAA-99-4036, August, 1999.

Cook, M.V., Flight Dynamics Principles, New York, NY: John Wiley & Sons, Inc., 1997.

Fletcher, Jay W., “Obtaining Consistent Models of Helicopter Flight-Data Measurement Errors Using Kinematic-Compatibility and State-Reconstruction

WORKS REFERENCED (CONT’D)

49

Methods”, 46th Annual Forum of the American Helicopter Society, Washington, D.C., May 1990.

Hoh, Roger H., Mitchell, D. G., Ashkenas, I. L., Klein, R. H., Heffley, R. K., and Hodgkinson, J., “Proposed MIL Standard and Handbook-Flying Qualities of Air Vehicles”. AFWAL-TR-82-3081, vol. 2, 1982.

Iliff, K. W., Klein, V., Maine, R. E., and Murray, J., Parameter Estimation Analysis for Aircraft, AIAA Professional Study Series, Williamsburg, VA, 1986.

Kimberlin, Ralph D., “Stability and Control Flight Testing Lecture Notes”, The University of Tennessee Space Institute, No Date.

Lewis, William D., "An Aeroelastic Model Structure Investigation for a Manned Real-Time Rotorcraft Simulation," American Helicopter Society 48th Annual Forum, St. Louis, MO, May 1993.

Nelson, Robert C., Flight Stability and Automatic Control, Second Edition, New York, NY: McGraw-Hill, 1998.

Pierson, Brett M., “Flight Test Safety: Lessons Learned from AN S-3B Flight Test Mishap During a Frequency Domain Test”, M.S. Thesis, The University of Tennessee Space Institute, August 1998.

Tischler, Mark B., “Frequency-Domain Identification of XV-15 Tilt-Rotor Aircraft Dynamics in Hovering Flight”, Paper presented at the AIAA/AHS/IES/SETP/DGLR 2nd Flight Testing Conference, Las Vegas, Nevada, November 1983.

Tischler, Mark B., “Frequency-Response Method for Rotorcraft System Identification: Flight Applications to BO-105 Coupled Rotor/Fuselage Dynamics”, Journal of the American Helicopter Society, Vol 37, No 3, pgs 3-17, July 1992.

Tischler, Mark B., “Flying Quality Analysis and Flight Evaluation of a Highly Augmented Combat Rotorcraft”, Journal of Guidance, Control, and Dynamics, Vol 14, No. 5, pg 954-964, Sep-Oct 1991.

Tischler, Mark B., “Time and Frequency-Domain Identification and Verification of BO 105 Dynamic Models”, 15th European Rotorcraft Forum, Amsterdam, September 1989.

50

APPENDIX

51

Frequency response data within CIFER® is identified according to the following naming convention:

CASENAME_PGM_WINDS_CTRL_OUTC The CASENAME is a user defined 8-character alpha-numeric case name. PGM is a 3-character abbreviation of the CIFER® sub-program that generated the frequency responses. The applicable abbreviations are as follows:

FRESPID spectral analysis and frequency-response identification program

MISOSA multi-input/single-output [frequency-response] conditioning program

COMPOSITE multi-window averaging [frequency-response] program DERIVID generalized stability derivative identification from

frequency responses program VERIFY state-space model verification program

WINDS denotes which user defined windows, labeled A, B, C, D and/or E comprise the frequency response. For this study, the window lengths are defined as follows: A = 30 sec, B = 20 sec, and C = 10 sec; windows D and E were not defined. For example, a frequency response comprised of windows A and C would be indicated as A0C00. CTRL and OUTC are user defined 4-character alpha-numeric input/control parameters names; the names for each parameter are identified in the Nomenclature section of report. For example, the composite (COM) frequency response for three different window sizes (A, B, and C) of the aileron input (DAIL) to roll rate (P), based on the time history data for the aileron axis sweep (CCLATSWP), would labeled as follows:

CCLATSWP_COM_ABC00_DAIL_P In addition to the naming convention described above, “Ph: 360” may appear on plots next to a frequency response name; this indicates that a phase shift was required for data analysis. Similarly, “Pw: -1” may also appear on plots; this indicates the power of a Laplacian integration (recall, s is the complex frequency in the Laplace domain) multiplied into the transfer function.

Figure A-1. CIFER® Naming Convention for Plots.

52

Navion (N66UT) Weight and Balance Flight Test: Open-Loop Frequency Response Crew: William Lewis, pilot, and Charles Catterall, test engineer Date: 9 Oct 98

TAKEOFF CONDITIONS

Description Weight (lbs) Arm (in) Moment (in-lbs)

Basic Aircraft 2546 100.79 256,605.69 Pilot/Copilot* 430 96.0 41,280

Laptop Computer 10 96.0 960 DaqBook Shelf 20 116.0 2,320

Fuel (39.5 gallons) 237 103.0 24,411 Total 3242 lbs 325,577 in-lbs

CG = Moment/weight = 100.39 in CG Location 100.39 in. – 83.66 in = 16.73 in/68.35 in x 100 = 24.5% M.A.C

LANDING CONDITIONS

Description Weight (lbs) Arm (in) Moment (in-lbs) Takeoff Conditions 3242 100.4 325,576.69 Fuel Burned (21.7

gallons) (130.2) 103.0 (13,410.6)

Total 3112 lbs 312,166 in-lbs

CG = Moment/weight = 100.32 in CG Location 100.32 in. – 83.66 in = 16.66 in/68.35 in x 100 = 24.4% M.A.C

Figure A-2. Weight and Balance Sheet for Frequency Response Flight Test.

53

Figure A-3. Photo of DaqBook Installation in N66UT’s Cabin.

54

Table A-1. Input and Output/Response Parameters

INPUT AND OUTPUT/RESPONSE PARAMETERS

CHANNEL

Control Positions (measured by potentiometers*)

Pilot’s aileron wheel position (δA) 00 Pilot’s elevator wheel position (δE) 01 Pilot’s pedal position (δP) 02 Pilot’s throttle position (δT) 03 Angular Body Rates (measured by gyroscopes)

Roll rate (p) 04 Pitch rate (q) 05 Yaw rate (r) 06 Linear Body Accelerations (measured by linear accelerometers)

Longitudinal acceleration (ax) 07 Lateral acceleration (ay) 08 Normal acceleration (az) 09 Control Surface Positions (measured by potentiometers)

Aileron position (δa) 10 Elevator position (δe) 11 Rudder position (δr) 12 Throttle position (δt) 13 Attitudes (measured by potentiometers)

Roll (φ) 14 Pitch (θ) 15 * Except for the throttle, which was measured as a linear displacement from the FBW system.

55

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le A

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eriv

ativ

e V

alue

s and

Ass

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ted

Cos

ts fo

r th

e M

inim

ally

Par

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ongi

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nal M

odel

. ** F-Matrix **

INITIAL SOLUTION

UBEST CR Percent % Insens UBEST CR Percent % Insens

XU -0.04500 0.000 0.000 -0.09148 113.1 11.84

XW 0.03600 0.000 0.000 0.04242 61.33 6.474

XQ 0.000 + ------ ------ 0.000 + ------ ------

G 0.000 + ------ ------ 0.000 + ------ ------

ZU -0.3690 0.000 0.000 10.51 10.33 0.3663

ZW -2.020 0.000 0.000 -3.066 6.945 0.3287

ZQ 0.000 + ------ ------ 0.000 + ------ ------

MU 1.900E-03 0.000 0.000 0.2054 9.056 0.3328

MW -0.03960 0.000 0.000 -0.05581 7.482 0.3169

MQ -2.948 0.000 0.000 -2.114 14.68 2.061

KIN 1.000 ! ------ ------ 1.000 ! ------ ------

ZCG 0.000 + ------ ------ 0.000 + ------ ------

** G-Matrix **

INITIAL SOLUTION

UBEST CR Percent % Insens UBEST CR Percent % Insens

XDE 0.000 + ------ ------ 0.000 + ------ ------

ZDE -28.17 0.000 0.000 0.000 + ------ ------

MDE -11.04 0.000 0.000 -12.67 6.131 0.9801

DELE 0.000 ------ ------ 0.000 ------ ------

** Transfer Function Costs **

INITIAL SOLUTION

U /DELE 77.653 156.311

W /DELE 2848.265 160.715

Q /DELE 125.819 17.227

AZ /DELE 861.892 58.540

Average 978.407 98.198

109

Tab

le A

-3. D

eriv

ativ

e V

alue

s and

Ass

ocia

ted

Cos

ts fo

r th

e M

inim

ally

Par

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ater

al/D

irec

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l Mod

el.

** M-Matrix **

INITIAL SOLUTION

UBEST CR Percent % Insens UBEST CR Percent % Insens

Mzcg -1.000 * ------ ------ 0.1977 * ------ ------

** F-Matrix **

INITIAL SOLUTION

UBEST CR Percent % Insens UBEST CR Percent % Insens

YV -0.2540 0.000 0.000 -0.09305 3.698 1.304

YP 0.000 0.000 0.000 0.000 + ------ ------

YR 0.000 0.000 0.000 1.617 14.25 5.714

G 0.000 + ------ ------ 0.000 + ------ ------

LV -0.09100 0.000 0.000 -0.05932 7.249 1.869

LP -8.402 0.000 0.000 -5.816 6.830 0.9673

LR 2.193 0.000 0.000 1.854 11.69 4.765

NV 0.02500 0.000 0.000 0.02991 2.667 0.8274

NP -0.3498 0.000 0.000 -0.5475 8.680 1.562

NR -0.7605 0.000 0.000 -0.9526 5.703 2.031

KIN 0.000 + ------ ------ 0.000 + ------ ------

ZCG 1.000 0.000 0.000 -0.1977 10.16 4.550

** G-Matrix **

INITIAL SOLUTION

UBEST CR Percent % Insens UBEST CR Percent % Insens

YDA 0.000 + ------ ------ 0.000 + ------ ------

YDR 0.07080 0.000 0.000 3.469 9.036 4.029

LDA 28.98 0.000 0.000 15.74 6.402 0.8759

LDR 2.548 0.000 0.000 1.380 13.66 6.306

NDA -0.2218 0.000 0.000 0.4862 23.84 4.143

NDR -4.597 0.000 0.000 -4.288 2.824 0.9534

DAIL 0.000 0.000 0.000 0.000 ------ ------

DRUD 0.000 0.000 0.000 0.000 ------ ------

110

Tabl

e A

-3. C

ontin

ued.

** Transfer Function Costs **

INITIAL SOLUTION

V /DAIL 441.385 22.034

P /DAIL 148.912 4.103

R /DAIL 232.521 22.563

AY /DAIL 4521.113 19.533

V /DRUD 67.466 25.191

P /DRUD 166.893 16.522

R /DRUD 62.248 14.734

AY /DRUD 2258.725 10.944

Average 987.408 16.953

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129

Table A-4. Relevant Longitudinal Dimensional Derivatives.

Stability Derivative Description Units

Yβ Dimensional variation of Y force with sideslip angle ft/sec2

Yp Dimensional variation of Y force with roll rate ft/sec

Yr Dimensional variation of Y force with yaw rate ft/sec

Yδa Dimensional variation of Y force with aileron angle ft/sec2

Yδr Dimensional variation of Y force with rudder angle ft/sec2

Lβ Dimensional variation of rolling moment with sideslip angle 1/sec2

Lp Dimensional variation of rolling moment with roll rate 1/sec

Lr Dimensional variation of rolling moment with yaw rate 1/sec

Lδa Dimensional variation of rolling moment with aileron angle 1/sec2

Lδr Dimensional variation of rolling moment rudder angle 1/sec2

Nβ Dimensional variation of yawing moment with sideslip angle 1/sec2

Np Dimensional variation of yawing moment with roll rate 1/sec

Nr Dimensional variation of yawing moment with yaw rate 1/sec

Nδa Dimensional variation of yawing moment with aileron angle 1/sec2

Nδr Dimensional variation of yawing moment with rudder angle 1/sec2

130

Table A-5. Relevant Lateral/Directional Dimensional Derivatives.

Stability Derivative Description Units

Xu Dimensional variation of X force with speed along the X-axis 1/sec

Xw Dimensional variation of X force with speed along the Z-axis 1/sec

Zu Dimensional variation of Z force with speed along the X-axis 1/sec

Zw Dimensional variation of Z force with speed along the Z-axis 1/sec

Zq Dimensional variation of Z force with pitch rate ft/sec

Zα Dimensional variation of Z force with angle of attack ft/sec2

Zwdot Dimensional variation of Z force with rate change of pitch rate (i.e., vertical acceleration) 1

Zαdot Dimensional variation of Z force with rate change of angle of attack ft/sec

Zδe Dimensional variation of Z force with elevator angle ft/sec2

Mu Dimensional variation of pitching moment with speed along the X-axis 1/ft-sec

Mw Dimensional variation of pitching moment with speed along the Z-axis 1/ft-sec

Mq Dimensional variation of pitching moment with pitch rate 1/sec

Mα Dimensional variation of pitching moment with angle of attack 1/sec2

Mwdot Dimensional variation of pitching moment with rate change of pitch rate (i.e., vertical acceleration) 1/ft

Mαdot Dimensional variation of pitching moment with rate change of angle of attack 1/sec

Mδe Dimensional variation of pitching moment with elevator angle 1/sec2

131

VITA

Robert Charles Catterall was born in Rockford, Illinois, on May 23, 1974. In 1986 he

moved to Mt. Juliet, Tennessee, where he later graduated from Mt. Juliet Christian

Academy in 1992. Thereafter, he attended Tennessee Technological University, in

Cookeville, Tennessee, where he received a Bachelor of Science in Mechanical

Engineering with a minor in Mathematics in 1997. From there, he attended the

University of Tennessee Space Institute (UTSI), in Tullahoma, Tennessee, to pursue a

Master of Science degree, with a major in Aviation Systems.

Robert is currently a Sr. Engineer with the Enabling Technologies group at Delta Air

Lines, in Atlanta, Georgia.