Automatic tracking control of aircraft

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April 2, 1947

Professor Joseph S. Newell Secretary of the Faculty

Massachusetts Institute of Technology Cambridge, Massachusetts

Dear Professor Newell:

In accordance with the regulations of the faculty, I hereby submit a thesis entitled "Automatic Tracking Control of Aircraft" in partial fulfillment of the requirements for the degree of Doctor of Science in Aeronautical Engineering.

Sincerely,

Signature Redacted

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BIOGR

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James B. Rea was born in Honolulu, Hawaii, on April 2, 1916. He attended primary schools in Honolulu, graduating from Punahou

School in 1933. He then studied Sugar Technology at the University of Hawaii until 1934, when he transferred to the University of Cali-fornia at Berkeley to study Music and Mathematics.

In 1938 he transferred to the Massachusetts Institute of Technology to study Aeronautical Engineering. _{In 1939 he was appointed} Research Assistant in Aeronautical Engineering. In 1940 he was awarded

the Sloan Automotive Engineering Fellowship, and was elected to Tau Beta Pi. _{As a member of the Honors Group in Aeronautical Engineering} he received the degrees of S.B. and S.M. simultaneously in the spring of 1941.

He was then hired by Pan American Airways as an Engineering-pilot, later becoming Flight Captain. He flew scheduled flights over the Atlantic, the Pacific, and South American routes.

In the fall of 1943 he left Pan American Airways to work as a Research Test-Pilot and Design Spedialist for Consolidated Vultee

Aircraft Corporation.

In the fall of 1945, he was appointed Consolidated Vultee Fellow, and was given a leave of absence to return to the Massachusetts Institute of Technology as a candidate for the degree of Doctor of Science in Aeronautical Engineering. In the summer of 1946 he was appointed Research Associate in Aeronautical Engineering.

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ABSTRACT

The primary object of the work described in this thesis was to develop equations suitable for predicting the performance of a system for automatic tracking control of aircraft, and to use these equations to determine the requirements and analyze the performance of a system for automatic tracking control of the A-26 Airplane. The secondary object was to develop equations suitable for predicting the steady-state response of aircraft to forced sinusoidal motion of controls, and to use these equations as a basis for indicating flight-test methods for determining aircraft dynamic stability coefficients.

Equations suitable for predicting the performance of a system for automatic tracking control of aircraft were developed, and applied to a system for automatic tracking control of the A-26 Airplane. These equations were analyzed by the M.I.T. Rockefeller Differential Analyzer to determine the servo stability coefficients required for practical solution time of the overall system.

The basic problem given to the Differential Analyzer was as

follows: Two airplanes were assumed to be flying level, each trimmed at 300 mph TAS, one behind the other. The rear (tracking) airplane was assumed to be an A-26 Airplane, flying at

10,000

ft. density altitude, while the forward (target) airplane was assumed to be separated from

the tracking airplane by an initial range of 1000 yards. In addition, the target airplane was assumed to be 150 ft. above and 150 ft. to the right of the tracking airplane, so that the initial line of sight from the tracking airplane would have an elevation component of +50 mils as well as a deflection component of +50 mils. It was then assumed that the automatic control system was suddenly turned on, thus applying a step-function input to the radar system.

Analysis of the longitudinal loop of the overall system indicated

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the following:

Solution time to establish pure pursuit course ^' l second.

Solution time to establish aerodynamic lead pursuit course 5 2 seconds.

A stability number.6 was found to be best.

The effect on solution time of neglecting certain of the aircraft stability coefficients was also found. For instance the effect of neglecting zu was negligible, whereas the effect of neglecting m was

not.

Equations suitable for predicting the steady-state response of aircraft to forced sinusoidal motion of controls, were developed. These equations were used to compute the steady-state response of the A-26 Airplane to forced sinusoidal motion of its controls. The steady-state computed response agreed very well with steady-state flight-test response for the cases investigated. The response equa-tions were also used to indicate flight-test methods for determining aircraft stability coefficients. The damping coefficient and the spring constant for the short-period component of the transient lon-gitudinal response of the A-26 Airplane were computed theoretically, and determined from flight-test data by the circle-diagram method. The computed and flight-test values agreed very well.

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The author wishes to acknowledge his gratitude and appre-ciation to the persons who rendered assistance in the development of this thesis.

To Professor C. S. Draper and Professor R. C. Seamans, Jr., for their inspiration and help during the entire project. To

Pro-fessor 0. C. Koppen for his helpful suggestions, especially with regard to problems in aircraft stability and control.

To Consolidated Vultee Aircraft Corporation for assistance throughout the project.

To the members of the Instrumentation Laboratory at the Massachusetts Institute of Technology for their cooperation, and especially to Miss A. Boissevain and Mrs. J. Fisher for help with mathematical analyses, and to Miss S. Knight and Miss G. Sloan

for help with computations. To the Ralph C. Coxhead Corporation, and especially to Mrs. A. Sakakeeny, for typing the manuscript. To Mr. Earle Payne and staff for reproducing the manuscript, and especially to Mr. R. Hofmann for preparing the figures. To my wife for proof-reading and compiling the manuscript.

To the members of the Flight Research Department of the Cornell Research Laboratories at Buffalo, and especially to Mr. I.M. Ross, Mr. W. F. Milliken, Jr., and Mr. E. V. Laitone for their help-ful suggestions regarding the practical application of the theory, and for their cooperation in supplying the author with flight data.

To the members of the Differential Analyzer staff at the Massachusetts Institute of Technology, and especially to Mr. B. Svihel and Mr. John Lof, for solving the automatic tracking control network

equations.

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OBJECT

. . . .

INTRODUCTION

. . . .

PART ONE, OVERALL TRACKING CONTROL SYSTEM . . . .

SECTION I

SECTION II

SECTION III

SECTION IV

SECTION V SECTION VI

- Requirements of a System for Automatic Tracking Control of Aircraft . . . .

- Equations of Motion for the Primary Components of a System for Automatic Tracking Control of Aircraft . . . .

- Stability Coefficients for a System for Automatic Tracking Control of Aircraft

- Network Equations for a System for Automatic Tracking Control of Aircraft. Application to A-26 Airplane . . . .

- Solution Time of a System for Automatic Tracking Control of the A-26 Airplane, using Differential Analyzer to Solve

Network Equations . . . . . . . . . . . . .

- Conclusions.... ...

PART TUO(, AIRCRAFT . . . . . . . . .. . #. .. .. . .. .

SECTION I

SECTION II

SECTION III

- Steady-State Response of Aircraft to

Forced Sinusoidal Motion of Controls.

Application to A-26 Airplane . . . . . .

- Fixed-Control Transient Response of Aircraft to Step-Functions. Application

to A-26 Airplane . . . . . . . . . .

- Transient Response of Aircraft to Lon-gitudinal Step-Function when Aircraft is equipped with Automatic Pilot which uses Displacement, Integral and

Deriva-tive Pitch-Control. Application to

A-26 Airplane, with Ideal Displacement Pitch Control in Automatic Pilot . . .

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27

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Page No.

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Page No. SECTION IV

SECTION V

- Methods for Determining Dynamic Stability Coefficients for Aircraft from Steady-State Flight-Test Response of Aircraft to Forced'Sinusoidal Motion of

Application to A-26 Airplane .

- Conclusions . . . . . . . . .

Controls.

51 56

BIBLIOGRAPHY . . . . . . . . . . . . . .

NOTATION AND CONVENTIONS FOR AUTOMATIC

TRACKING CONTROL OF AIRCRAFT . . . . . . . . . . .

BASIC CONSTANTS FOR AN AUTOMATIC TRACKING

CONTROL SYSTEM FOR THE A-26 AIRPLANE . . . . . . . . .

APPENDICES . . . (Each Appendix has its own page numbering system)

59

62

100 Viii

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APPENDIX A Der Pri mat

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A

P P E N D I C E S

Page No.

I

ivation of Equations of Motion for the mary Components of a System for

Auto-ic Tracking Control of Aircraft . . . .

APPENDIX B

Derivation of Formulae for Stability Coefficients for a System for Automatic

Tracking Control of Aircraft . . . .

APPENDIX C

Development of Network Equations for an Automatic Tracking Control System for

Aircraft. . . . . . . . . . . . . . . .

APPENDIX D

Determination of Stability Coefficients for A System for Automatic Tracking Control of

the A-26 Airplane. . . . . . . . . . . . . . . . . .

APPENDIX E

Equations of Motion for the Primary

Compo-nents of a System for Automatic Tracking

Control of the A-26 Airplane., . . . . . . . . . . .

APPENDIX F

Network Equations for an Automatic Tracking

Control System for the A-26 Airplane, Based upon a System Characteristic Time Equal to

the Initial Time of Flight of the Airplane... APPENDIX G

Determination of Solution Time of an

Auto-matic Tracking Control System for the A-26 Airplane, Using Differential Analyzer to

Solve Network Equations . . . . . .

Ix A-1 B-1 C- 1 D-1 E- 1

F-

1 G-1

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LIST OF APPENDICES (Continued)

Page No. APPENDIX H

Derivation of Equations for Steady-State Response of Aircraft to Forced Sinusoidal

Motion of Control Surfaces . . . . . . . . H-1 APPENDIX I

Derivation of Equations for Fixed-Control Transient Response of Aircraft to

Step-Functions . . . . . . . . . . . I-1 APPENDIX J

Derivation of Equations for Steady-state Response of Aircraft to Forced Sinusoidal Variation in Output 'Signal from Oscillator which Drives Elevator Servo, with Ideal Displacement, Integral and Derivative Pitch

Control in Automatic Pilot . . . . . . ... J-1 APPENDIX K

Derivation of Equations for Transient Response of Aircraft to Longitudinal Step-Function, with Ideal, or Actual, Displacement, Integral

and Derivative Pitch Control in Automatic Pilot . . . . K-i APPENDIX L

Analyses for Steady-state Response of A-26 Airplane to Forced Sinusoidal Motion of Control

Surfaces. . . . . . . . . . . . . . . . .

.L-1

APPENDIX M

Analyses for Fixed Control Transient Response

of A-26 Airplane to Step Functions . . . . . . M-1 APPENDIX N

Analysis for Steady-state Response of A-26 Airplane to Forced Sinusoidal Variation in Output Signal from Oscillator which Drives Elevator Servo, with Ideal Displacement Pitch

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LIST OF APPENDICES (Continued)

Page No. APPENDIX 0

Analysis for Transient Response of A-26 Airplane to Longitudinal Step Function, with Ideal

Dis-placement Pitch Control in Automatic Pilot . . . . 0-1 APPENDIX P

Methods for Determining Dynamic Stability Coef-ficients for Aircraft from Steady-State Flight-test Response of Aircraft to Forced Sinusoidal

Motion of Control Surfaces . . . . P-1 APPENDIX Q

Derivation of Equations for Steady-State Response of Aircraft to Forced Sinusoidal Motion of Throttle

and Elevator . . . . . . . . . . . . . . . . . Q-1

APPENDIX R

Analysis for Steady-State Response of A-26 Airplane

to Forced Sinusoidal Motion of Throttle and Elevator . R1 APPENDIX S

Methods for Determining Longitudinal Dynamic Sta-bility Coefficients for Aircraft from Steady-State Flight-Test Response of Aircraft to Forced Sinusoidal

Motion of Throttle and Elevator . . . . . . . . S-1 APPENDIX T

Determination of Damping Coefficient and Spring Constant for Short Period Component of Longitudinal Motion of A-26 Airplane, from Flight-Test Data, using Circle Diagram Method. Comparison of Com-puted Response with Flight-Test Response, for A-26

Airplane . . . . . . . . . . . . . . . T-1

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LIST

OF

FIGURES

FIGURES IN TEXT:

FIGURE NO. TITLE PAGE NO.

I Mechanical Arrangement of Components in a System for Automatic Tracking Control

of Aircraft . . . . . . . . . . . . .. . 8

II Antenna Assembly of G.E. AN/APG-3(XA-1)

Radar Unit . . . . . . . . . . . . .

.10

III TCS-1 Computer . . ... ... . . . . ... 12 IV Minneapolis-Honneywell C-lA Servo Unit and

Balance Potentiometer . . . . . . . . .. . 14 V Douglas A-26 Airplane, 3/4 Front View . . . . . . 16 VI Douglas A-26 Airplane, Front View . . . . . . .. 17 VII Functional Diagram of System for Automatic

Tracking Control of Aircraft (Tight Loop

System) . . .. . * . .. . . -.. & . - - 19 VIII Block Diagram of System for Automatic

Tracking Control of Aircraft . . . . 21 IX Angle and Angular Velocity Notation and

Conventions for Aircraft . . . . 65 X Linear Velocity Notation and Conventions

for Airctaft. . . . . .

....66

XI Moment and Force Notation and Conventions

for Aircraft , . . . . . . . . . . . . . 67

XII Schematic Sketches of Elevator-Servo and Rudder-Servo Systems . . . ...... 68

XIII Schematic Sketch of Aileron-Servo System . .... 69 XIV Tracking Notation and Conventions for

Automatic Tracking Control of Aircraft . . . . . 70 XV Tracking Notations and Conventions for

Aircraft (For Elevation and Deflection

Tracking with Ideal Radar) . . . . . .. . 72

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F

Differential Analyzer Circuit for Complete Longitudinal Network

Equations . . . . . . . .

Differential Analyzer Circuit for Simplified Longitudinal Network

Equations . . . . . . . . .

Time History of Elevation Component of Ideal Radar Angle, for Computed Response of A-26 Airplane to Step-Function Imposed on Automatic Tracking-Control System, Effect of Neglecting Certain Stability Coefficients, Analysis by Differential Analyzer . . . .

Time History of Elevation Component of Tracking Error Angle, for Computed

Response of A-26 Airplane to Step-Function Imposed on Automatic Tracking Control System, Computer Caged until fe"= +5 Miliradians, Analysis by Differential Analyzer . . . . . . . . . . .

Time History of Prediction Angle of

Elevation Computer, for Computed Response of A-26 Airplane to Step-Function imposed

on Automatic Tracking Control System,

Computer Caged until fe = 5 Miliradians, Analysis by Differential Analyzer . . . . .

G-17

. . . G-18

. . . G-19

. . G-20

G-21 Differential Analyzer Circuit for Simplified

Lateral Network Equations . . . . . . . . . . . G-22

Time History of Deflection Component of Ideal Radar Angle, for Computed Response of A-26 Airplane to Step-Function Imposed

on Automatic Tracking-Control System,

Analysis by Differential Analyzer . . . . G-23 Effect of Change in Elevator Frequency on

the Phase Angle Between Change in Longitu-dinal Velocity per Airplane Trim Speed and Change in Elevator Angle, Computed for

A-26 Airplane . . . . . . . . a. . . .. .. . .L-25

IGURES IN APPENDICES:

FIGURE NO. TITLE PAGE NO.

G-1-a

G-

1-b

G-2

G-3

G-4

G-5

G-6

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FIGURE NO.

L-2

L-3

L-4

L-5 L-6 L-7

L-8

L-9 xlv TITLE

Effect of Change in Elevator Frequency on the Phase Angle Between Change in Normal Velocity per Airplane Trim Speed and Change in Elevator Angle,

Computed for A-26 Airplane . . . .

Effect of Change in Elevator Frequency on the Phase Angle Between Change in Pitch Angle and Change in Elevator Angle, Computed for A-26 Airplane . . . . . . .

Effect of Change in Elevator Frequency on Amplitude Response Ratio for Change in Longitudinal Velocity per Airplane

Trim Speed, Computed for A-26 Airplane . . .

Effect of Change in Elevator Frequency on Amplitude Response Ratio for Change in Normal Velocity per Airplane Trim Speed, Computed for A-26 Airplane Effect of Change in Elevator Frequency on Amplitude Response Ratio for Change

in Pitch Angle, Computed for A-26 Airplane Effect of Change in Elevator Frequency on Ratio of Amplitude of Change in Pitch Angle to Amplitude of Change in Normal Velocity per Airplane Trim Speed, computed

for A-26 Airplane . . . . . . . .

Time History of Change in Elevator Angle, Change in Pitch Angle, Change in Normal Velocity per Trim Speed, and Change in Longitudinal Velocity per Trim Speed, per Amplitude of Change in Elevator Angle, for Computed Response of A-26 Airplane to Forced Sinusoidal Motion of Elevator, Ele-vator Frequency = .1 Cycle per Second . . .

Time History of Change in Elevator Angle, Change in Pitch Angle, Change in Normal Velocity per Trim Speed, and Change in Longitudinal Velocity per Trim Speed, per Amplitude of Change in Elevator Angle, for Computed Response of A-26 Airplane to Forced Sinusoidal Motion of Elevator, Ele-vator Frequency = .7 Cycle per Second . . .

PAGE NO. L-26 L-27 L-28 L-29 L-30 L-31 L-32 L-33

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FIGURE NO. L- 10 L- 11

L- 12

L- 13

L- 14

L-15

Time History of Change in Elevator Angle, Change in Pitch Angle, Change

in Normal Velocity per Trim Speed, and Change in Longitudinal Velocity per

Trim Speed, per Amplitude of Change in Elevator Angle, for Computed Response

of A-26 Airplane to Forced Sinusoidal Motion of Elevator, Elevator Frequency

= 1.3 Cycles per Second . .. .. .... Effect of Change in Rudder Frequency on

the Phase Angle Between Change in Angle of Bank and Change in Rudder Angle, the Phase Angle Between Change in Aerodynamic Yaw and Change in Rudder Angle, and the Phase Angle Between Change in Geometric Yaw and Change Rudder Angle, Computed for A-26 Airplane, with Fixed Ailerons

Effect of Change in Rudder Frequency on

Amplitude Response Ratios Computed for

A-26 Airplane, with Fixed Ailerons . . . Effect of Change in Rudder Frequency on Ratio of Amplitude Response Ratio for Angle of Aerodynamic Yaw to Amplitude Response Ratio for Angle of Geometric Yaw, Computed for A-26 Airplane, with Fixed Ailerons . . . . . . .

Time History of Change in Rudder Angle, Change in Angle of Bank, Change in Angle of Geometric Yaw, and Change in Angle of Aerodynamic Yaw, per Amplitude of Change in Rudder Angle, for Computed Response of A-26 Airplane to Forced Sinusoidal Motion of Rudder, with Fixed Ailerons, Rudder Frequency,= .1 Cycle per Second. Time History of Change in Rudder Angle, Change in Angle of Bank, Change in Angle of Geometric Yaw, and Change in Angle of Aerodynamic Yaw, per Amplitude of Change in Rudder Angle, for Computed Response of A-26 Airplane to Forced Sinusoidal Motion of Rudder, with Fixed Ailerons, Rudder Frequency = .7 Cycle per Second . . . . .

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PAGE NO. TITLE

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L-16 _{Time History of Change in Rudder Angle,} Change in Angle of Bank, Change in Angle of Geometric Yaw, and Change in Angle of Aerodynamic Yaw, per Amplitude of Change in Rudder Angle, for Computed Response of A-26 Airplane to Forced Sinusoidal Motion of Rudder, with Fixed Ailerons,

Rudder Frequency

=

1.3 Cycles per Second. . . L-40 L-17 _{Effect of Change in Aileron Frequency on}

the Phase Angle between Change in Aero-dynamic Yaw and Change in Aileron Angle, The Phase Angle between Change in Geo-metric Yaw and Change in Aileron Angle,

and the Phase Angle between Change in Angle of Bank and Change in Aileron Angle, Computed for A-26 Airplane, with

Fixed Rudder . . . L-41

L-18 _{Effect of Change in Aileron Frequency on}

Amplitude Response Ratios, Computed for

A-26 Airplane, with Fixed Rudder . . . . L-42 L-19 _{Effect of Change in Aileron Frequency on}

Ratio of Amplitude Response Ratio for Angle of Aerodynamic Yaw to Amplitude Response Ratio for Angle of Bank, Computed

for A-26 Airplane, with Fixed Rudder ... L-43 L-20 _{Time History of Change in Aileron Angle,}

Change in Angle of Aerodynamic Yaw, Change in Angle of Geometric Yaw, and Change in Angle of Bank, per Amplitude of Change in Aileron Angle, for Computed Response of

A-26 Airplane to Forced Sinusoidal Motion

of Ailerons, with Fixed Rudder, Aileron

Frequency = .1 Cycle per Second ..0. . ..L-44.

L-21 _{Time History of Change in Aileron Angle,} Change in Angle of Aerodynamic Yaw, Change in Angle of Geometric Yaw, and Change in Angle of Bank, per Amplitude of Change in Aileron Angle, for Computed Response of A-26 Airplane to Forced Sinusoidal Motion of Ailerons, with Fixed Rudder, Aileron

Frequency.= .7 Cycle per Second . . . . . L-45

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L-22 Time History of Change in Aileron Angle, Change in Angle of Aerodynamic Yaw, Change in Angle of Geometric Yaw, and Change in Angle of Bank, per Amplitude of Change in Aileron Angle, for Computed Response of A-26 Airplane to Forced Sinusoidal Motion of Ailerons, with Fixed Rudder, Aileron

Frequency = 1.3 Cycles per Second . . . . L-46 L-23 Effect of Change in Aileron-Rudder Frequency

on the Phase Angle Between Change in Aileron Angle and Change in Rudder Angle, the Phase Angle between Change in Angle of Bank and Chan'ge in Rudder Angle, and the Phase Angle between Change in Geometric Yaw and Change in Rudder Angle, Computed for A-26 Airplane,

Adjusted for Zero Aerodynamic Yaw . . . . L-47 L-24 Effect of Change in Aileron-Rudder Frequency

on Amplitude Response Ratios, Computed for

A-26 Airplane, Adjusted for Zero Aerodynamic

Yaw . . . . . . . . . . . . .. . . . . . L-48

L-25 Effect of Change in Aileron-Rudder Frequency on Ratio of Amplitude Response Ratio for Angle of Bank to Amplitude Response Ratio for Angle of Geometric Yaw, Computed for A-26 Airplane, Adjusted for Zero Aerodynamic Yaw-. . . . . . . . . . . .

...L-49

L-26 Time History of Change in Rudder Angle, Change in Aileron Angle, Change in Bank Angle, and Change in Angle of Geometric Yaw, per Ampli-tude of Change in Rudder Angle, for Computed Response of A-26 Airplane to Forced Sinusoidal Motion of Ailerons and Rudder, Adjusted for Zero Aerodynamic Yaw, Frequency of Ailerons

and Rudder

=

.1 Cycle per Second . . . . L-50 L-27 Time History of Change in Rudder Angle, Change

in Aileron Angle, Change in Bank Angle, and Change in Angle of Geometric Yaw, per Ampli-tude of Change in Rudder Angle, for Computed Response of A-26 Airplane to Forced Sinusoidal Motion of Ailerons and Rudder, Adjusted for Zero Aerodynamic Yaw, Frequency of Ailerons and

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FIGURE NO. L- 28 L- 29 L-30 L-31 L-32 L-33 PAGE NO. Time History of Change in Rudder Angle,

Change in Aileron Angle, Change in Bank Angle, and Change in Angle of Geometric Yaw, per Amplitude of Change in Rudder Angle, for Computed Response of A-26 Airplane to Forced Sinusoidal Motion of Ailerons and Rudder, Adjusted

for Zero Aerodynamic Yaw, Aileron-Rudder

Frequency = 1.3 Cycles per Second . . . . . L-52 Effect of Change in Aileron-Rudder

Fre-quency on the Phase Angle between Change in Aileron Angle and Change in Rudder Angle, the Phase Angle between Change in Angle of Bank and Change in Rudder Angle, and the Phase Angle between Change in Lateral

Velocity per Airplane Trim speed and Change in Rudder Angle, Computed for A-26 Airplane,

Adjusted for Zero Geometric Yaw . . . . . . . . L-53 Effect of Change in Aileron-Rudder

Fre-quency on Amplitude Response Ratios, Computed for A-26 Airplane, Adjusted for Zero

Geo-metric Yaw _{. . . .} L-54 Effect of Change in Aileron-Rudder Frequency

on Ratio of Amplitude Response Ratio for Change in Lateral Velocity per Airplane Trim Speed to Amplitude Response Ratio for Change in Angle of Bank, Computed for A-26 Airplane,

Adjusted for Zero Geometric Yaw . . . . L-55 Time History of Change in Rudder Angle, Change

in Aileron Angle, Change in Bank Angle, and Change in Angle of Aerodynamic Yaw, per Am-plitude of Change in Rudder Angle, for Computed Response of A-26 Airplane to Forced Sinusoidal Motion of Ailerons and Rudder, Adjusted for Zero Geometric Yaw, Aileron-Rudder Frequency

.1 Cycle per Second . . . . . ... L-56 Time History of Change in Rudder Angle, Change

in Aileron Angle, Change in Bank Angle, and Change in Angle of Aerodynamic Yaw, per Amplitude of Change in Rudder Angle, for Computed Response of

A-26 Airplane to Forced Sinusoidal Motion of

Ailerons and Rudder, Adjusted for Zero Geometric Yaw, Aileron-Rudder Frequency = .7 Cycle per

Second . . . . . L-57

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FIGURE NO.

L-34

Time History of Change in Rudder Angle Change in Aileron Angle, Change in Angle of Bank, and Change in Angle of Aerodynam#iYav-;

yer

Amplitude of Change in Rudder Angle, for Computed Response of A-26 Airplane to Forced Sinusoidal Motion of Ailerons and Rudder, Adjusted for Zero Geometric Yaw,

Aileron-Rudder Frequency

=

1.3 Cycles per Second . . . . L-58 Effect of Change in Aileron-Rudder Frequency

on the Phase Angle between Change in Aero-dynamic Yaw and Change in Rudder Angle, the Phase Angle between Change in Geometric Yaw

and Change in Rudder Angle, and the Phase Angle between Change in Aileron Angle and Change in Rudder Angle, Computed for A-26

Airplane, Adjusted for Zero Angle of Bank . . . L-59 Effect of Change in Aileron-Rudder Frequency

on Amplitude Response Ratios, Computed for A-26 Airplane, Adjusted for Zero Angle of

Bank . . . . . . . . .. *. .. .. .. .. L-60 Effect of Change in Aileron-Rudder Frequency

on Ratio of Amplitude Response Ratio for

Anigle of,- Aerodynamic Yaw to Amplitude Res-ponse Ratio for Angle of Geometric Yaw,

Computed for A-26 Airplane, Adjusted for Zero

Angle of Bank . . . . . . .. .0. .. . . . . . . L-61

Time History of Change in Aileron Angle, Change in Angle of Aerodynamic Yaw, Change in Angle of Geometric Yaw, and Change in Rudder Angle, per Amplitude of Change in

Rudder Angle, for Computed Response of A-26 Airplane to Forced Sinusoidal Motion of Ailerons and Rudder, Adjusted for Zero Angle of Bank, Aileron-Rudder Frequency = .1 Cycle per Second . . . . . . . . .. . . . .

....L-62

Time History of Change in Aileron Angle, Change in Angle of Aerodynamic Yaw, Change in Angle of Geometric Yaw, and Change in Rudder Angle, per Amplitude of Change in Rudder Angle, for Computed Response of A-26 Airplane to Forced Sinusoidal Motion of Ailerons and Rudder, Adjusted for Zero Angle of Bank, Aileron-Rudder Frequency = .7 Cycle per Second . . . . . ...

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L-36

L-37

L-38

L-39

TITLE

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FIGURE NO.

L-40

M-1

M-2

M-3

M-4

M-5

M-6

.M-7

M-8

PAGE NO.

L-64

Short-Period Component of Change in

Longitudinal Velocity per Trim Speed, for Computed Transient Response of A-26

Air-plane to Longitudinal Step-Function . . . . . . M-11 Long-Period Component of Change in

Longitudinal Velocity per Trim Speed, for Computed Transient Response of A-26

Air-plane to Longitudinal Step-Function . . . . M-12 Short-Period Component of Change in

Normal Velocity per Trim Speed, for Com-puted Transient Response of A-26 Airplane

to Longitudinal Step Function . . . . . . . . . M-13 Long-Period Component of Change in Normal

Velocity per Trim Speed, for Computed Transient Response of A-26 Airplane to Longitudinal Step-Function . . . . . . . .

Short-Period Component of Change in Pitch Angle, for Computed Transient Response of A-26 Airplane to Longitudinal Step-Function Long-Period Component of Change in Pitch Angle for Computed Transient Response of A-26 Airplane to Longitudinal Step-Function

M- 14

M-15

M- 16

Time History of Subsidence -Components of Angle of Bank, Angle of Geometric Yaw, and Angle of Aerodynamic Yaw, for Computed Tran-sient Response of A-26 Airplane to Lateral

Step-Function . . . . M-17

Time History of Spiral Stability Components of Angle of Bank, Angle of Geometric Yaw, and Angle of Aerodynamic Yaw, for Computed Transient Response of A-26 Airplane to Lateral

Step-Function . . . . . . .. . . .. .. .. . M-18

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TITLE

Time History of Change in Aileron Angle, Change in Angle of Aerodynamic Yaw, Change in Angle of Geometric Yaw, and Change in Rudder Angle, for Computed Response of A-26 Airplane to Forced

Sinusoidal Motion of Ailerons and Rudder, Adjusted for Zero Angle of Bank, Aileron-Rudder Frequency

=

1.3 Cycles per Second . . .

(22)

M-9 Time History of Dutch Roll Components of Angle of Bank, Angle of Geometric Yaw, and Angle of Aerodynamic Yaw, for Computed Transient Response of A-26 Airplane to

Lateral Step-Function . . . . . . . . M-19 M-1O Time History of Total Response of Angle of

Bank, Angle of Geometric Yaw, and Angle of Aerodynamic Yaw, for Computed Transient Response of A-26 Airplane to Lateral Step-Function . . . . . . . .

....M-20

N-1 Effect of Ideal Displacement Pitch Control on Lead Angle between Change in Longitudinal Velocity per Airplane Trim Speed and Change in Elevator Angle, at Various Values of Ele-vator Frequency, for Computed Response of A-26 Airplane to Forced Sinusoidal Motion of

Elevator, for ko in Automatic Pilot

=

0 and

.5, Elevator Motion for ko = 0 Basis for

Comparison . . . . . . . . . . . . . . . . . . . N-6

N-2 Effect of Ideal Displacement Pitch Control on Lead Angle between Change in Angle of Attack and Change in Elevator Angle, at

Various Values of Elevator Frequency, for

Computed Response of A-26 Airplane to Forced Sinusoidal Motion of Elevator, for k6 in

Automatic Pilot Equal 0 and .5, Elevator

Motion for ko = 0 Basis for Comparison . . . ..N-7 N-3 Effect of Ideal Displacement Pitch Control

on Lead Angle between Change in Pitch Angle and Change in Elevator Angle, at Various Values of Elevator Frequency, for Computed Response of A-26 Airplane to Forced Sinu-soidal Motion of Elevator, for ko in Automatic Pilot = 0 and .5, Elevator Motion for ko = 0

Basis for Comparison . . . . . . . . .. . N8

N-4 Effect of Ideal Displacement Pitch Control on Amplitude Response Ratio for Change in Longitudinal Velocity per Trim Speed, at Various Values of Elevator Frequency, for Computed Response of A-26 Airplane to Forced Sinusoidal Motion of Elevator, for ko in

Automatic Pilot

=

0 and .5, Elevator Motion

(23)

FIGURE NO.

N-5

xxii

Effect of Ideal Displacement Pitch Control on Amplitude Response Ratio for Change in Angle of Attack, at Various Values of Elevator Frequency, for Com-puted Response of A-26 Airplane to Forced Sinusoidal Motion of Elevator, for ke in Automatic Pilot Equal 0 and

.5, Elevator Motion for k6 0 Basis for

Comparison . . . . ... .. .. .. .. . N-10

Effect of Ideal Displacement Pitch Control on Amplitude Response Ratio for Change in Pitch Angle, at Various Values of Elevator Frequency, for Computed Response of A-26 Airplane to Forced Sinusoidal Motion of Elevator, for k9 in Automatic Pilot Equal

o

and .5, Elevator Motion for k. 0 Basis

for Comparison . . . . . . . . . . . . . . . . . N-li

Effect of Elevator Frequency on Ratio of Amplitude Response Ratio for Change in Pitch Angle to Amplitude Response Ratio for Change in Normal Velocity per Airplane Trim Speed, for Computed Response of A-26 Airplane to Forced Sinusoidal Motion of

Elevator . . . . . . . . . . . . N-12 Time History of Change in Longitudinal

Vel-ocity per Trim Speed, for Computed Transient Response of A-26 Airplane to Longitudinal

Step-Function, with Ideal Displacement Pitch Control in Automatic Pilot, k= .5 eompared

to ko = 0. ... . ... 0-6

Time -History of Change in Angle of Attack, for Computed Transient Response of A-26 Airplane to Longitudinal Step-Function, with Ideal Displacement Pitch Control in Automatic

Pilot, ko = .5 Compared to k, = 0 .. . . . . . 0-7 Time History of Change in Angle of Pitch, for

Computed Transient Response of A-26 Airplane to Longitudinal Step-Function, with Ideal Displacement Pitch Control in Automatic

Pilot, k₆ ='.5 Compared to ko

=

0 . . . . . . 0-8 N-6

N-7

0-1

0-2

0-3

(24)

TITLE PAGE NO.

R-1

xxiii

:1

FIGURE NO.

Effect of Change in Elevator-Throttle Frequency on the Phase Angle Between Change in Normal Velocity per Trim Speed and Change in Elevator Angle, the Phase Angle between Change in Pitch Angle and Change in Elevator Angle,and the Phase Angle between Change in Longitudinal Thrust

and Change in Elevator Angle, Computed for A-26 Airplane, Adjusted for Zero Change

in Longitudinal Velocity _{. . . R-11} Effect of Change in Elevator-Throttle

Frequency on Amplitude Response Ratios, Computed for A-26 Airplane, Adjusted for

Zero Change in Longitudinal Velocity . . . . . . . R-12 Effect of Change in Elevator-Throttle

Fre-quency on Ratio of Amplitude Response Ratio for Angle of Pitch to Amplitude Response Ratio for Normal Velocity per Trim Speed, Computed for A-26 Airplane, Adjusted for

Zero Change in Longitudinal Velocity . . . . . R-13 Time History of Change in Elevator Angle,

Change in Pitch Angle, Change in Normal Velocity per Trim Speed, and Change in Nondimensional Longitudinal Thrust, per Amplitude of Change in Elevator Angle, for Computed Response of A-26 Airplane to Forced Sinusoidal Motion of Elevator and Throttle, Adjusted for Zero Change in Longitudinal Velocity, Elevator-Throttle

Frequency = .1 Cycle per Second . . . . . . . R-14 Time History of Change in Elevator Angle,

Change in Pitch Angle, Change in Normal Velocity per Trim Speed, and Change in Nondimensional Longitudinal Thrust, per Amplitude of Change in Elevator Angle,

for Computed Response of A-26 Airplane to Forced Sinusoidal Motion of Elevator and Throttle, Adjusted for Zero Change in Longitudinal Velocity, Elevator-Throttle

Frequency = .7 Cycle per Second . . . . . . . . . R-15

R-2

R-3

R-4

(25)

FIGURE NO.

R-6

xxiv

Time History of Change in Elevator Angle, Change in Pitch Angle, Change in Normal Velocity per Trim Speed, and Change in Nondimensional Longitudinal Thrust, per Amplitude of Change in Elevator Angle, for Computed Response of A-26 Airplane to Forced Sinusoidal Motion of Elevator and Throttle, Adjusted for Zero Change in Longitudinal Velocity, Elevator-Throttle

Frequency = 1.3 Cycles per Second . . . . . . . . R-16 Effect of Change in Elevator-Throttle

Fre-quency on the Phase Angle Between Change in Longitudinal Velocity per Trim Speed and Change in Elevator Angle, the Phase Angle Between Change in Pitch Angle and Change in

Elevator Angle, and the Phase Angle between Change in Longitudinal Thrust and Change in Elevator Angle, Computed for A-26 Airplane,

Adjusted for Zero Change in Angle of Attack . . . R-17 Effect of Change in Elevator-Throttle

Fre-quency on Amplitude Response Ratios, Com-puted for A-26 Airplane, Adjusted for Zero

Change in Angle of Attack . . . . . . . . R-18 Time History of Change in Elevator Angle,

Change in Pitch Angle, Change in Longitu-dinal Velocity per Trim Speed, and Change in Nondimensional Longitudinal Thrust, per Amplitude of Change in Elevator Angle, for Computed Response of A-26 Airplane to Forced Sinusoidal Motion of Elevator and Throttle, Adjusted for Zero Change in Angle of Attack, Elevator-Throttle Frequency.= .1 Cycle per

Second . . . . . . . . . . . . . . . R-19 Time History of Change in Elevator Angle,

Change in Pitch Angle, Change in Longitudi-nal Velocity per Trim Speed, and Change in Nondimensional Longitudinal Thrust, per Amplitude of Change in Elevator Angle, for Computed Response of A-26 Airplane to Forced Sinusoidal Motion of Elevator and Throttle, Adjusted for Zero Change in Angle of Attack, Elevator-Throttle Frequency

= .7 Cycle per Second . ..o *. . .. ... .. . . R-20

R-7

R-8

R-9

R- 10

(26)

FIGURE NO.

R-11

R- 12

R- 13

R- 14

Time History of Change in Elevator Angle, Change in Pitch Angle, Change in Longitudinal Velocity per Trim Speed, and Change in Nondimensional Longitudinal Thrust, per Amplitude of Change in Ele-vator Angle, for Computed Response of A-26 Airplane to Forced Sinusoidal Motion of Elevator and Throttle, Adjusted for Zero Change in Angle of Attack,

Elevator-Throttle Frequency = 1.3 Cycles per Second . . . . R-21 Effect of Change in Elevator-Throttle

Fre-quency on The Phase Angle between Change in Longitudinal Velocity per Trim Speed and Change in Elevation Angle, The Phase Angle between Change in Angle of Attack and Change in Elevator Angle, and the Phase Angle between Change in Longitudinal Thrust and Change in Elevator Angle, Computed for A-26 Airplane, Adjusted for Zero Change

in Pitch Angle ... . . . . . . . . . . . . . . . . R-22

Effect of Change in Elevator-Throttle Fre-quency on Amplitude Response Ratios, Computed for A-26 Airplane, Adjusted for Zero Change

in Pitch Angle . . . . . . . . . . . . R-23 Effect of Change in Elevator-Throttle

Fre-quency on Ratio of Amplitude Response Ratio for Change in Longitudinal Velocity per Trim Speed to Amplitude Response Ratio for Change in Angle of Attack, Computed for A-26 Airplane, Adjusted for Zero Change in Pitch

Angle . . . . . .. . . . . . . .R-24 Time History of Change in Elevator Angle,

Change in Angle of Attack, Change in Longi-tudinal Velocity per Trim Speed, and Change in Nondimensional Longitudinal Thrust, per

Amplitude of Change in Elevator Angle, for Computed Response of A-26 Airplane to Forced Sinusoidal Motion of Elevator and Throttle, Adjusted for Zero Change in Pitch Angle,

Ele-vator-Throttle Frequency = .1 Cycle per Second . . R-25 PAGE NO.

R- 15

(27)

FIGURE NO.

R-16

R- 17

T-1

T-2

T-3

XIV1

Time History of Change in Elevator

Angle, Change in Angle of Attack, Change in Longitudinal Velocity Per Trim Speed, and Change in Nondimensional Longitudinal Thrust, per Amplitude of Change in Elevator Angle, for Computed Response of A-26 Airplane to Forced Sinusoidal Motion of Elevator and Throttle, Adjusted for Zero Change in Pitch Angle, Elevator-Throttle Frequency = .-L. Cycle

per Second . . . . R-26 Time History of Change in Elevator Angle,

Change in Angle of Attack, Change in Longi-tudinal Velocity per Trim Speed, and Change in Nondimensional Longitudinal Thrust, per Amplitude of Change in Elevator Angle, for Computed Response of A-26 Airplane to Forced Sinusoidal Motion of Elevator and Throttle, Adjusted for Zero Change in Pitch Angle, Elevator-Throttle Frequency 1.3 Cycles per

Second. . . . . . . . . . . . . . . . . . R-27 Effect of Elevator Frequency on Phase and

Amplitude of Change in Reading of Normal Accelerometer, per Amplitude of Change in Elevator Angle, Flight-Test Data for A-26

Airplane . . . . .... .. .. T-9 Effect of Elevator Frequency on Phase and

Amplitude of Pitching Velocity, per Ampli-tude of Change in Elevator Angle, Flight-Test Data Compared with Computed Data, for

A-26 Airplane . . . . . . . . . . . . . . . . . T-lO

Circle Diagram for Determining Damping Coefficient and Spring Constant for Short Period Component of Longitudinal Motion of A-26 Airplane, Based on

Flight-Test Data . . . . . . . . . . . . . . . . . . .. T-11

(28)

TABLE NO. G-1 . G-2 . . . .

L

I

ST

0 F

T ABL ES

G-3 . . . .

G-4

. .

G-5 . . . . G-6 . . . . G-7 . . G-8 . . . .

G-9-a, G-9-

b TITLE

Differential Analyzer Data, Longitudinal, for Establishing a Pure Pursuit Course,

Effect of neglecting various Stability Coeffi-cients: .Run 19 . . . . . . . . . .Run 20 . . . . . . . . . . . . .Run 21 . . . . . . . . . . . . .Run 22 . . . . . .Run 23 . . . . . . . .Run 24 . . . . . . . . .Run 25 . . . . . . . . . .Run 26 . . . . . . . . .Run 29 .. .. . . ... .. PAGE NO. . . . G-24 G-25 G-26 G-27 G-28 G-29 G-30 G-31 G-32, G-33

Differential Analyzer Data, Longitudinal,

for Establishing an Aerodynamric Lead

Pursuit Course:

... Run 38, SNe = 0 . . . . ... Run 39, SNe = .2 . . . .

. . . . .. . . Run 40, SNe = .4 . . . . * . . .. . .. . . Run 41, SNe = .6 . . .

Differential Analyzer Data, Lateral,

for Establishing Fire-Control Corrections: . .. . .. .. Run 28 . . . . . . G-34 . . . G-35 . . . . G-36 . . . . G-37 G-38 xxvii G-10 . . .

G-11

. G-12 .

G-13

. G-14 . . .

.

(29)

LIST OF TABLES (Continued)

TABLE PAGE

NO. TITLE NO.

L-1 _{Analysis for Steady-State Response of A-26} Airplane to Forced Sinusoidal Motion of Elevator, when the Effect of Change in Longitudinal Velocity is included in all Three Equations of Motion

Computations for LAUSe, LAWse, and LAO₈e . L-65

L-2 Analysis for 'Steady-State Response of A-26 Airplane to Forced Sinusoidal Motion of Elevator, when the Effect of Change in Lon-gitudinal Velocity is Included in All Three Equations of Motion

Computations for p Je e , 68e and piq j .. L-66

L-3 Analysis for Steady-State Response of A-26 Airplane to Forced Sinusoidal Motion of Elevator

CASE I: When the Effect of Change in Longi-tudinal Velocity is Included in All Three Equations of Motion, CASE II: When the Effect of Change in Longitudinal Velocity is

Neglec-ted in the Equations of Motion for Normal

Force and Pitching Moment

_e u w

Time Histories of ea , 8 ea, e and

6

, for DE = .1 cycle/second ... L-67

ea

L-4 Analysis for Steady-State Response of A-26 Airplane to Forced Sinusoidal Motion of

Elevator

CASE I: When the Effect of Change in Longi-tudinal Velocity is Included in All Three Equations of Motion, CASE II: When the Effect of Change in Longitudinal Velocity is Neglec-ted in the Equations of Motion for Normal Force and Pitching Moment

Be u w

Time Histories of , i-, i-, and

ea I e. e.

6

0-, for E = .7 cycle/second ... .. L-68

xxviii

(30)

LIST OF TABLES (Continued)

TABLE NO.

L-5

L-6

L-7

Analysis for Steady-State Response of A-26 Airplane to Forced Sinusoidal Motion of Rudder, with Fixed Ailerons,

Computations for uv ra r, and ap '

Analysis for Steady-State Response of A-26 Airplane to Forced Sinusoidal Motion of Rudder, with Fixed Ailerons

Computations for LAVSr, LApsr, and LA . .

PAGE NO. TITLE

Analysis for Steady-State Response of A-26 Airplane to Forced Sinusoidal Motion of Elevator

CASE I: When the Effect of Change in Longi-tudinal Velocity is Included in All Three Equations of Motion, CASE II: When the Effect of Change in Longitudinal Velocity is Neglec-ted in the Equations of Motion for Normal Force and Pitching Moment

Se u w

Time Histories of , 8 e and

6

0e, for E = 1.3 cycles/second. . . .

Analysis for Steady-State Response of A-26 Airplane to Forced Sinusoidal Motion of Elevator, when the Effect of Change in Longitudinal Velocity is Neglected in the

Equations of Motion for Normal Force and Pitching Moment

Computations for LAus e, LAw e, and LAse. Analysis for Steady-State Response of A-26 Airplane to Forced Sinusoidal Motion of

Elevator, when the Effect of Change in Longi-tudinal Velocity is Neglected in the Equations of Motion for Normal Force and Pitching Moment

Computations for pus e., 'We, "G e , and p-q

'

. L-72 . . L-73 . . . L-69 . . L-70 . . L-71 L-8

L-9

(31)

LIST OF TABLES (Continued)

TABLE PAGE

NO. TITLE NO.

L-10 _{Analysis for Steady-State Response of A-26} Airplane to Forced Sinusoidal Motion of Rudder, with Fixed Ailerons

Time Histories of -y- , -- , -- , and

ra _r. _r

8 _{for 'R} _{= .1}_{cycle/second '} _'

. . . . . .- -. L-74 L-11 _{Analysis for Steady-State Response of A-26}

Airplane to Forced Sinusoidal Motion of Rudder, with Fixed Ailerons

Sr _v ₀

Time Histories of -, r , r , and

r a r, r

, for OR = .7 and 1.3 cycles/second ... '.L-75'

L-12 _{Analysis for Steady-State Response of A-26}

Airplane to Forced Sinusoidal Motion of Ailerons, with Fixed Rudder

Computations for LAvs _, LAOSa:; _LA, _{a Ya}

11,8a' Yt a, and 'V4. ... .. ... ... .. L-76

L-13 Analysis for Steady-State Response of A-26 Airplane to Forced Sinusoidal Motion of Ailerons, with Fixed Rudder

Sa v

Time Histories of s , , , and a,

a a

a

a a, a,

for 0A = .1 and .7 cyclesecond _{. . .} L-77 ..- 14 Analysis for Steady-State Response of A-26

Airplane to Forced Sinusoidal Motion of Ailerons, with Fixed Rudder

Sa v 'p ,

Time Histories of

a

, '

a

a and g-- for

a= a1.a c ya,

f2A = 1.3 cycles/second o - * ' ' ' ' ' ' ' ' ' ' ' L-78

(32)

TABLE NO. L-15

L-16

L- 17

PAGE NO. TITLE

Analysis for Steady-State Response of A-26 Airplane to Forced Sinusoidal Motion of Rudder and Ailerons, Adjusted for Zero

Aerodynamic Yaw

Computations for LASa8 r, LA , LA ,

S a Sr' , 8r' I 3P8r, and 4. .. ... . . . . . .. L-79 Analysis for Steady-State Response of A-26

Airplane to Forced Sinusoidal Motion of Rudder and Ailerons, Adjusted for Zero Aerodynamic Yaw

Time Histories of r -- -'ra' 8

r,' Sr.

_dSr.'

for QAR -1 cycle/second . . . . . . . . L-80 Analysis for Steady-State Response of A-26

Airplane to Forced Sinusoidal Motion of Rudder and Ailerons, Adjusted for Zero Aerodynamic Yaw

8

_r

8

_a

_9$

₄

Time Histories of Sr ,a s a and ,

r r, ra r.

for QAR = .7 cycle/second' ... L-81

Analysis for Steady-State Response of A-26 Airplane to Forced Sinusoidal Motion of Rudder and Ailerons, Adjusted for Zero Aerodynamic Yaw

Sr Sa 0

Time Histories of r ' f '

r r, ands ,

for DAR = 1.3 cycle/second _{. .}* * . . . . . . . . L-82

Analysis for Steady-State Response of A-26 Airplane to Forced Sinusoidal Motion of Rudder and Ailerons, Adjusted for Zero

Geometric Yaw

Computations for LASa₈ _, _LA _, _LAgSr

A8ar' $Sr. YVSr, and ... ... L-83

LIST OF TABLES (Continued)

L- 18

(33)

LIST OF TABLES (Continued)

TABLE PAGE

NO. TITLE NO.

L-20 Analysis for Steady-State Response of A-26 Airplane to Forced Sinusoidal Motion of Rudder and Ailerons, Adjusted for Zero Geometric Yaw

Sr Sa v

Time Histories of

y,

, S and

S,

r,

r

for DAR= .1 cycle/second . . . . . . L-84

L-21 Analysis for Steady-State Response of A-26 Airplane to Forced Sinusoidal Motion of Rudder and Ailerons, Adjusted for Zero Geometric Yaw

Sr Sa v

Time Histories of j-, , s , and₈ ,

r, ra

ra

for )AR = .7 cycle/second *. .. . . . .. .. . L-85 L-22 Analysis for Steady-State Response of A-26

Airplane to Forced Sinusoidal Motion of Rudder and Ailerons, Adjusted for Zero Geometric Yaw

8

_r

_Sa

₄

_V

Time Histories of r,

S a

, and v,

ra

r

a

r,

for DAR = 1.3 cycles/second . . . . L-86

L-23 Analysis for Steady-State Response of A-26 Airplane to Forced Sinusoidal Motion of Rudder and Ailerons, Adjusted for Zero

Angle of Bank

Computations for LA_{8 8}_a_{r, LAq8}_r _{LAvs r'}

p98a8r

, 40y8r) rVS , and p...

. .. .. ..

L-87

L-24 Analysis for Steady-State Response of A-26 Airplane to Forced Sinusoidal Motion of Rudder and Ailerons, adjusted for Zero Angle of Bank 8_r _Sa

'

v

Time Histories of 8, , --

,

and ,

for 'AR,= .1 cycle/second ' s ' * '_{... ..}* @ # L-88 xxxII

(34)

LIST OF TABLES (Continued)

PAGE

TITLE NO.

Analysis for Steady-State Response of A-26 Airplane to Forced Sinusoidal Motion of Rudder and Ailerons, Adjusted for Zero Angle of Bank

8

_r

8

_a

_'P

_V

Time Histories of ,; , 8 , and ,

ra ra ra

for nAR = .7 cycles/second # e ,

.

*.

..

L-89

Analysis for Steady-State Response of A-26 Airplane to Forced Sinusoidal Motion of Rudder and Ailerons, Adjusted for Zero Angle of Bank

8

_r

_Sa

_'P

V Time Histories of , , Ei,

and

,

ra r. ra ra

for DAR = 1.3 cycles/second . .. .. .. .. . L-90

Analysis for Fixed-Control Transient Response of A-26 Airplane to Longitudinal Step-Function

Determinants for Au, Bu' Cu' Du, Aw, Bw, Cw, Dw, Ag, Be, Co, Do .@. . . .

.# #.. .. . .. ... M-21

Determinants for Au' Bu' Cu, Du, Aw, Bw, Cw' Dw, A9, Bo, Ce, Do . .. . @. . .a. .. .. . .. M-22

Calculations for Short-Period Transient Response for Change in Longitudinal

Velocity . . . . . . . . . . . . . .

.M-23

Calculations for Long-Period Transient Response for Change in Longitudinal

(35)

LIST OF TABLES (Continued)

TITLE TABLE NO.

M-5

Xxxiv

Calculations for Long-Period Transient Response for Change in Longitudinal

Velocity . . . . . .

Calculations for Short-Period Transient Response for Change in Normal Velocity . . .

Calculations for Long-Period Transient Response for Change in Normal Velocity

Calculations for Short-Period Transient Response for Change in Pitch Angle... Analysis for Fixed-Control Transient Response of A-26 Airplane to Longitudinal Step-Function

Calculations for Long-Period Transient Response for Change in Pitch Angle . . . . . Analysis for Fixed-Control Transient Response of A-26 Airplane to Lateral Step-Function

Determinants for v., v2, v3, Av, Bv, 4i, 02. 4p3, AO, Bo,

'02

$ 9!3, AO, B, . . . .

Analysis for Fixed-Control Transient Response of A-26 Airplane to Lateral Step-Function

Computations for Time History of v . . . . .

Analysis for Fixed-Control Transient Response of A-26 Airplane to Lateral Step-Function

Computations for Time History of 4.... . . Analysis for Fixed-Control Transient Response of A-26 Airplane to Lateral Step-Function

Computations for Time History of 4 . . . . .

PAGE NO.

M-6

M-7

M-8

M-9

M- 10

. . .M-25 . . .M-26 M-27 M-28 M-29 M-30 . . .M-31 M-32 M-33

M- 11

M-

12

M- 13 wood

(36)

LIST OF TABLES (Continued)

PAGE NO. TITLE

Analysis for Steady-State Response of A-26 Airplane to Forced Sinusoidal Variation in Output Signal from Oscillator which Drives Elevator Servo, with Ideal Displacement Pitch Control in Automatic Pilot, ko = .5

Computations for LAu e, LAws , LAOe

e

p"use , S , I /ls, and 4, based upon

e

q

8_e_{for ko .= 0 . . . .} _{. .} _{. . . .} _N-13

Analysis for Transient Response of A-26 Airplane to Longitudinal Step-Function, with

Ideal Displacement Pitch Control in Auto-matic Pilot, ko = .5

Determinants for Au, Aw, A9, Bu, Bw'

Bo, u2

,

w2; 02, u3, w3, and 03

...

0-9

Analysis for Transient Response of A-26 Airplane to Longitudinal Step-Function, with Ideal Displacement Pitch Control in

Automatic Pilot, k

=

.5

Computations for Time History of u . . . . . . . 0-10 Analysis for Transient Response of A-26

Airplane to Longitudinal Step-Function, with Ideal Displacement Pitch Control in Automatic Pilot ko = .5

Computations for Time History of w . . . . . . . 0-11 Analysis for Transient Response of A-26

Airplane to Longitudinal Step-Function,

with Ideal Displacement Pitch Control in

Automatic Pilot, ko = .5

Computations for Time History of 9 . . . . . . . 0-12

Analysis for Steady-State Response of A-26 Airplane to Forced Sinusoidal Motion of Throttle and Elevator, Adjusted for Zero Change in Longitudinal Velocity

Computations for LAs e, LASe , and LAt . . R-28 TABLE

(37)

TABLE NO.

R-2

R-3

PAGE NO. TITLE

Analysis for Steady-State Response of A-26 Airplane to Forced Sinusoidal Motion of Throttle and Elevator Adjusted for Zero Change in Longitudinal Velocity

w 8 tx Se

Time Histories of 8_ea'a , Se 8_ea

_j,

8 ,a W

_n

_e

_a, for nE-TH .1 cycle/second * - -...

w 6 x Se

Time Histories of -, --

,

-,

and ,

ea ea ea ea

for DE-TH _-.7 cycle/second _{. . . .}

w

e

_X Se

Time Histories of S , s , ,

and

8 ,

ef ea e a e a

for DE-TH

=

1.3 cycles/second -

_{. . .}

-

_{. . .}

R-29

R-30

. . . . R-31

Analysis for Steady-State Response of A-26 Airplane to Forced Sinusoidal Motion of Throttle and Elevator, Adjusted for Zero Change in Normal Velocity (for Small Oscilla-tions, Same as Zero Change in Angle of Attack)

Computations for LAuse, LAOeSe LAt xse, PqSe,

Pse' /Itx 8e' 4qu, and /lose ... . .. .. ... . .R-32

Xxxvi

LIST OF TABLES (Continued)

R-4

R-5

(38)

LIST OF TABLES (Continued)

PAGE

TITLE NO.

Analysis for Steady-State Response of A-26 Airplane to Forced Sinusoidal Motion of

Throttle and Elevator, Adjusted for Zero Change in Normal Velocity (for 'Small Oscilla-tions, Same as Zero Change in Angle of Attack)

u 6 x S

e

Time Histories of ' , ' , w

,and ,

ea

e,

e

for 0E-TH = .1 cycle/second . . . . . . . .. . . . R-33

Analysis for Steady-State Response of A-26 Airplane to Forced Sinusoidal Motion of Throttle and Elevator, Adjusted for Zero Change in Normal Velocity (for Small Oscilla-tions, Same as Zero Change in Angle of Attack)

u 6 tx Be

Time Histories of , 8 , , t --- ,

e

ea

e

a

for fE-TH = .7 cycle/second

...

'..

.

....

R-34 Analysis for Steady-State Response of A-26

Airplane to Forced Sinusoidal Motion of Throttle and Elevator, Adjusted for Zero Change in Normal Velocity (for Small Oscilla-tions, Same as Zero Change in Angle of Attack)

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