Aerodynamic Modeling of an Unmanned Aerial Vehicle Using a Computational Fluid Dynamics Prediction Code

AERODYNAMIC MODELING OF AN UNMANNED AERIAL VEHICLE USING A

COMPUTATIONAL FLUID DYNAMICS PREDICTION CODE

A thesis presented to

the faculty of

the Russ College of Engineering and Technology of Ohio University

In partial fulfillments

of the requirements for the degree

Master of Science

Isaac D. Rose

March 2009

AERODYNAMIC MODELING OF AN UNMANNED AERIAL VEHICLE USING A

COMPUTATIONAL FLUID DYNAMICS PREDICTION CODE

ISAAC D. ROSE

has been approved for

the School of Electrical Engineering and Computer Science

and the Russ College of Engineering and Technology by

______

Douglas A. Lawrence

Professor of Electrical Engineering and Computer Science

______

Dennis Irwin

Dean, Russ College of Engineering and Technology Abstract

ROSE, ISAAC D., M.S., March 2009, Electrical Engineering

AERODYNAMIC MODELING OF AN UNMANNED AERIAL VEHICLE USING A

COMPUTATIONAL FLUID DYNAMICS PREDICTION CODE (192 pp.)

Director of Thesis: Douglas A. Lawrence

The process of creating a six degree-of-freedom model for an aerospace

vehicle requires detailed knowledge of the aerodynamic characteristics. This thesis

presents an implementation of a Computational Fluid Dynamics (CFD) prediction

computer code to generate aerodynamic coefficients for the Brumby Mk. I Unmanned

Aerial Vehicle (UAV). The aerodynamic coefficients include both the force and

moment coefficients. These values are verified by creating a Matlab/Simulink six

degree-of-freedom model.

Approved: ______

Douglas A. Lawrence

Professor of Electrical Engineering and Computer Science Acknowledgments

I would like to thank GOD through whom all things are possible. I would like to thank my wife and son for their understanding and encouragement. The hours spent working on this thesis were hours spent away from them. I would also like to thank Dr.

Lawrence for the guidance and direction that he has given me over the years. The research presented in this thesis is a tribute to his resolve in the autonomous control of the

Brumby Unmanned Arial Vehicle. Finally, I would like to thank the faculty of the department of Electrical Engineering and Computer Science for their help throughout the years. Table of Contents

Abstract...... 3

Acknowledgments...... 4

Glossary of Variables...... 14

Chapter 1: Introduction...... 19

1.1 Overview...... 19

1.2 Motivation ...... 20

1.3 Modeling Aerodynamic Forces and Moments...... 21

1.4 Objectives...... 22

1.5 Thesis Organization...... 22

1.5.1 Missile DATCOM Input Parameters...... 22

1.5.2 Missile DATCOM Model of the Brumby UAV...... 23

1.5.3 Equations of Motion...... 23

1.5.4 Brumby UAV Model Simulation...... 23

Chapter 2: Missile DATCOM Modeling Parameters Overview...... 24

2.1 Flight Conditions...... 26

2.2 Fuselage...... 28

2.3 Primary Lifting Surface ...... 29

2.4 Horizontal Stabilizer...... 33

2.5 Vertical Stabilizer...... 35

2.6 Control Surfaces...... 36 2.7 Generating Additional Data...... 38

2.8 File Format and Content...... 39

2.9 Missile DATCOM Example...... 41

Chapter 3: Missile DATCOM Model of the Brumby Unmanned Aerial Vehicle ...... 72

3.1 Flight Conditions...... 72

3.2 Fuselage...... 75

3.3 Wing Planform...... 76

3.4 Vertical Stabilizer...... 77

3.5 Control Surfaces ...... 80

Chapter 4: Equations of Motion and Rigid Body Modeling ...... 86

4.1 Equations of Motion for A Rigid Body ...... 86

4.2 Aerodynamic Coefficients...... 91

4.3 Six Degree-of-Freedom Aircraft Model...... 101

Chapter 5: Simulation...... 103

5.1 Simulink Nonlinear Aircraft Model...... 103

5.2 Trimmed Aircraft Flight...... 107

5.3 Linearized Aircraft Model...... 108

5.4 Nonlinear Simulation Results...... 112

5.4 Control Surface Doublet Simulation Results...... 123

Chapter 6: Conclusions and Future Work...... 138

References...... 140

Appendix A.1: for005.dat File...... 141 Appendix A.2 : Truncated for006.dat File...... 145

Appendix A.3 : for003.dat File...... 154

Appendix A.4 : for021.dat File...... 155

Appendix B.1: Equations of Motion s-function...... 184

Appendix B.2: forces_moments.m...... 189

Appendix B.3: datcomderive.m...... 192 List of Tables

Table 2.1: Missile DATCOM Control Cards...... 26

Table 2.2: Missile DATCOM Namelist FLTCON ...... 27

Table 2.3: Missile DATCOM Namelist REFQ...... 28

Table 2.4: Missile DATCOM Namelist AXIBOD...... 29

Table 2.5: Missile DATCOM Namelist FINSET...... 30

Table 2.6: Missile DATCOM Namelist DEFLCT...... 38

Table 2.7: Missile DATCOM File Definitions...... 41

Table 3.1: Brumby UAV Flight Conditions (FLTCON) ...... 73

Table 3.2: Brumby UAV Reference Values (REFQ)...... 75

Table 3.3: Brumby UAV Body Definition (ASYM)...... 76

Table 3.4: Brumby UAV Twin Vertical Tail Planform Definition (FINSET2)...... 79

Table 4.5: Brumby UAV Wing Planform Definition (FINSET1)...... 88

Table 5.1: Brumby UAV Mass Properties...... 104

Table 5.2: S-function Functionality...... 104

Table 5.3: DATCOMTableMex.dll Functionality...... 106

Table 5.4: Brumby UAV Control Input Trimmed Values (Case 1)...... 107

Table 5.5: Brumby UAV State Variables Initial Condition Values (Case 1)...... 108

Table 5.6: Brumby UAV Trimmed Aerodynamic Values (Case 1)...... 108

Table 5.7: Brumby UAV Control Input Trimmed Values (Case 2)...... 112

Table 5.8: Brumby UAV Control Effector Doublet Values...... 124 List of Figures

Figure 2.1: Missile DATCOM Axes Definition ...... 25

Figure 2.2: Missile DATCOM Body Variables...... 30

Figure 2.2: Missile DATCOM Finset...... 31

Figure 2.3: NACA Airfoil Number Decomposition...... 31

Figure 2.4: Missile DATCOM Fin Panel Location Definition...... 33

Figure 2.5:Twin-Horizontal Stabilzer Fin Panel Location Definition...... 34

Figure 2.6:V-Tail Stabilzer Fin Panel Location Definition...... 35

Figure 2.7:Twin-Vertical Stabilzer Fin Panel Location Definition...... 36

Figure 2.8: Missile DATCOM Control Surface Definition...... 38

Figure 2.9: for005.dat File Example...... 43

Figure 2.10: for006.dat File Example (Data Input)...... 44

Figure 2.11: for006.dat File Example (Error Checking)...... 45

Figure 2.12: for006.dat File Example (Case 1 Output, Page 2)...... 46

Figure 2.13: for006.dat File Example (Case 1 Output, Page 3)...... 47

Figure 2.14: for006.dat File Example (Case 1 Output, Page 4)...... 48

Figure 2.15: for006.dat File Example (Case 1 Output, Page 5)...... 49

Figure 2.16: for006.dat File Example (Case 1 Output, Page 6)...... 50

Figure 2.17: for006.dat File Example (Case 1 Output, Page 7)...... 51

Figure 2.18: for006.dat File Example (Case 1 Output, Page 8)...... 52

Figure 2.19: for006.dat File Example (Case 1 Output, Page 9)...... 53

Figure 2.20: for006.dat File Example (Case 1 Output, Page 10)...... 54 Figure 2.21: for006.dat File Example (Case 1 Output, Page 11)...... 55

Figure 2.22: for006.dat File Example (Case 1 Output, Page 12)...... 56

Figure 2.23: for006.dat File Example (Case 1 Output, Page 13)...... 57

Figure 2.24: for006.dat File Example (Case 2 Output, Page 1 and 2)...... 58

Figure 2.25: for006.dat File Example (Case 2 Output, Page 3)...... 59

Figure 2.26: for006.dat File Example (Case 2 Output, Page 4)...... 60

Figure 2.27: for006.dat File Example (Case 2 Output, Page 5)...... 61

Figure 2.28: for006.dat File Example (Case 2 Output, Page 6)...... 62

Figure 2.29: for006.dat File Example (Case 2 Output, Page 7)...... 63

Figure 2.30: for006.dat File Example (Case 2 Output, Page 8)...... 64

Figure 2.31: for006.dat File Example (Case 2 Output, Page 9)...... 65

Figure 2.32: for006.dat File Example (Case 2 Output, Page 10)...... 66

Figure 2.33: for006.dat File Example (Case 2 Output, Page 11)...... 67

Figure 2.34: for006.dat File Example (Case 2 Output, Page 12)...... 68

Figure 2.35: for006.dat File Example (Case 2 Output, Page 13)...... 69

Figure 2.36: for003.dat File Example...... 70

Figure 2.37: for021.dat File Example...... 71

Figure 3.1: Brumby UAV Fuselage...... 75

Figure 3.2: Brumby UAV Wing Planform...... 77

Figure 3.3: Brumby UAV Vertical Planform...... 79

Figure 3.4: Brumby UAV Vehicle Description Case for005.dat File...... 82

Figure 3.5: Brumby UAV Wing Control Deflection Cases for005.dat File...... 83 Figure 3.6: Brumby UAV Twin Vertical Tail Control Deflection Cases for005.dat File. 84

Figure 3.7: Brumby UAV Side-Slip Angle and Altitude Cases for005.dat File...... 85

Figure 4.1: Aerodynamic Angles...... 88

Figure 4.2: Lift Coefficient (a) and Drag Coefficient (b)...... 94

Figure 4.3: Force(a, b, c) and Moment Coefficients(d, e, f)...... 95

Figure 4.4: Brumby UAV Moment Definition...... 98

Figure 4.5: Axial and Normal Forces...... 100

Figure 4.6: Lift and Drag Forces...... 100

Figure 5.1: Simulink Nonlinear Aircraft Model...... 103

Figure 5.2: Navigation Position Output (SLF)...... 113

Figure 5.3: Euler Angles Output (SLF)...... 113

Figure 5.4: Translational Velocities Output (SLF)...... 114

Figure 5.5: Angular Velocities Output (SLF)...... 114

Figure 5.6: Velocity Magnitude Output (SLF)...... 115

Figure 5.7: Aerodynamic Angles Output (SLF)...... 115

Figure 5.8: Flight-Path Angle Output (SLF)...... 116

Figure 5.9: Rate-of-Climb Output (SLF)...... 116

Figure 5.10: Navigation Position Ground Track Output (SLF)...... 117

Figure 5.11: Navigation Position 3-Dimensional Output (SLF)...... 117

Figure 5.12: Navigation Position Output (CTROC)...... 118

Figure 5.13: Euler Angles Output (CTROC)...... 118

Figure 5.14: Translational Velocities Output (CTROC)...... 119 Figure 5.15: Angular Velocities Output (CTROC)...... 119

Figure 5.16: Velocity Magnitude Output (CTROC)...... 120

Figure 5.17: Aerodynamic Angles Output (CTROC)...... 120

Figure 5.18: Flight-Path Angle Output (CTROC)...... 121

Figure 5.19: Rate-of-Climb Output (CTROC)...... 121

Figure 5.20: Navigation Position Ground Track Output (CTROC)...... 122

Figure 5.21: Navigation Position 3-Dimensional Output (CTROC)...... 122

Figure 5.22: Doublet Response Navigation Position Output (SLF)...... 125

Figure 5.23: Doublet Response Euler Angles Output(SLF)...... 125

Figure 5.24: Doublet Response Translational Velocities Output(SLF)...... 126

Figure 5.25: Doublet Response Angular Velocities Output (SLF)...... 126

Figure 5.26: Doublet Response Velocity Magnitude Output (SLF)...... 127

Figure 5.27: Doublet Response Aerodynamic Angles Output (SLF)...... 127

Figure 5.28: Flight-Path Angle Output (SLF)...... 128

Figure 5.29: Rate-of-Climb Output (SLF)...... 128

Figure 5.30: Doublet Response Navigation Ground Track Output (SLF)...... 129

Figure 5.31: Doublet Response Navigation 3-Dimensional Output (SLF)...... 129

Figure 5.32: Control Surface Deflection Input Angles (SLF)...... 130

Figure 5.33: Aerodynamic Control Surface Deflection Input Angles (SLF)...... 130

Figure 5.34: Doublet Response Navigation Position Output (CTROC)...... 131

Figure 5.35: Doublet Response Euler Angles Output (CTROC)...... 131

Figure 5.36: Doublet Response Translational Velocities Output (CTROC)...... 132 Figure 5.37: Doublet Response Angular Velocities Output (CTROC)...... 132

Figure 5.38: Doublet Response Velocity Magnitude Output (CTROC)...... 133

Figure 5.39: Doublet Response Aerodynamic Angles Output (CTROC)...... 133

Figure 5.40: Flight-Path Angle Output (CTROC)...... 134

Figure 5.41: Rate-of-Climb Output (CTROC)...... 134

Figure 5.42: Doublet Response Navigation Ground Track Output (CTROC)...... 135

Figure 5.43: Doublet Response Navigation 3-Dimensional Output (CTROC)...... 135

Figure 5.44: Control Surface Deflection Input Angles (CTROC)...... 136

Figure 5.45: Aerodynamic Control Surface Deflection Input Angles (CTROC)...... 136 Glossary of Variables

D Drag Force

Y Side Force

L Lift Force

A Axial Force

N Normal Force

C A Axial Force Coefficient

C Y Side Force Coefficient

C N Normal Force Coefficient

C L Lift Force Coefficient

C D Drag Force Coefficient

C l Rolling Moment Coefficient (Body axis)

C m Pitching Moment Coefficient(Body axis)

C n Yawing Moment Coefficient (Body axis)

C n Normal Force Coefficient derivative with respect to angle-of-attack

C m Pitching Moment Coefficient derivative with respect to angle-of-attack

C y  Side Force Coefficient derivative with respect to side-slip angle

C n  Yawing Moment Coefficient derivative with respect to side-slip angle (Body

axis)

C l  Rolling Moment Coefficient derivative with respect to side-slip angle (Body

axis) X cp Center of Pressure in calibers from the moment reference center l Rolling Moment m Pitching Moment n Yawing Moment l Rolling Moment m Pitching Moment n Yawing Moment l Rolling Moment m Pitching Moment n Yawing Moment

C l p Rolling Moment derivative with respect to Roll Rate

C mq Pitching Moment derivative with respect to Pitch Rate

C nr Yawing Moment derivative with respect to Yaw Rate

C l r Rolling Moment derivative with respect to Yaw Rate

C n p Yawing Moment derivative with respect to Roll Rate

C Lr Lift Force derivative with respect to Pitch Rate

C Y P Side Force derivative with respect to Roll Rate

C Y r Side Force derivative with respect to Yaw Rate

C q ele Pitching Moment derivative with respect to Elevator Deflection Angle

C L ele Lift Force derivative with respect to Elevator Deflection Angle

C l  ail Rolling Moment derivative with respect to Aileron Deflection Angle C n ail Yawing Moment derivative with respect to Aileron Deflection Angle

C l  Rolling Moment derivative with respect to Rudder Deflection Angle rud

C n rud Yawing Moment derivative with respect to Rudder Deflection Angle

∇ C  Dynamic Derivative q Dynamic Pressure b Wing Span c Mean Aerodynamic Chord

S Wing Span

 Mass Density

V T Free Stream Velocity k Dimensionless Rate Scale Factor rate Angular Rotation Rate Corresponding to Aerodynamic Force

Coefficient Derivative

 Angle-of-Attack

 Side-Slip Angle

 Flight-Path Angle

U Body-Frame Translational Velocity X-Axis Component

V Body-Frame Translational Velocity Y-Axis Component

W Body-Frame Translational Velocity Z-Axis Component p Body-Frame Rotational Velocity X-Axis Component q Body-Frame Rotational Velocity Y-Axis Component r Body-Frame Rotational Velocity Z-Axis Component  Roll Attitude Euler Angle

 Pitch Attitude Euler Angle

 Yaw Attitude Euler Angle

N Inertial Navigation Position X-Axis Component

E Inertial Navigation Position Y-Axis Component

D Inertial Navigation Position Z-Axis Component

C b/n Direction Cosine Matrix of the Body Frame with respect to the Navigation

Frame

p˙ ne Position Vector of Navigation Frame Derivative taken with respect to the Earth

Fixed Frame

b vCM / e Velocity Vector in the body frame of the Center-of-Mass with respect to the

Fixed Earth

 Rotational Rate Vector ˙  Rotational Rate Derivative Vector

b b /e Rotational Rate Vector expressed in the Body Frame of the Body with respect

to the Fixed Earth

˙ bb vCM / e Velocity Vector Body Derivatives in the Body Frame of the Center-of-Mass

with respect to the Fixed Earth m Mass of Vehicle

b F A,T Aerodynamic and Thrust Force Vector expressed in the Body Frame gn Gravity Vector in the Navigation Frame

b b/ e Cross Product Matrix of Rotational Rates in the Body Frame of the Body with

respect to the Fixed Earth

˙bb b/ e Rotational Rate Vector Body Derivative in the Body Frame of the Body with

respect to the Fixed Earth

b M A,T Aerodynamic and Thrust Moment Vector expressed in the Body Frame

J b Mass Moment of Inertia Tensor in the Body Frame

pne Navigation Position Vector expressed in the Earth Fixed Frame 19

Chapter 1: Introduction

The first step in developing a compensation scheme for a dynamic system is to create a mathematical model of the dynamic system or plant. For an aerospace vehicle, developing a mathematical model requires knowledge of the physical characteristics such as weight, mass properties and aerodynamic parameters. Until the 1970's most aerodynamic coefficients were obtained from wind tunnel data or through system identification techniques.[1] Today's computers make it possible to use Computational

Fluid Dynamics (CFD) modeling to generate the aerodynamic coefficients of an aerospace vehicle.

One method to compute the aerodynamic coefficients of the aircraft is to use the

United States Air Force Data Compendium (DATCOM)[2]. DATCOM was introduced in the 1970's as a handbook containing tabular aerodynamic coefficients for different vehicle geometries. The user builds the aerodynamic model from the characteristics of its components, such as the Aspect Ratio of the wing planform, geometry and location of the stabilizing and control surfaces, as well as the shape of the fuselage. The DATCOM handbook was implemented as a computer code, entitled Digital DATCOM, written in the Fortran language.[3]

Digital DATCOM is a set of computer codes that creates a composite aerodynamic model based on the user input geometry. The latest version of the United

States Air Force Data Compendium, Missile DATCOM, allows the user to model more abstract vehicle geometries, as well as expanding the environmental envelope.[4]

1.1 Overview 20

The Avionics Research Center at Ohio University purchased an unmanned aerial vehicle for control and navigation research after the recent expansion in the use of unmanned aerial vehicles for data collection and deployment in hazardous environments.

The unmanned aerial vehicle purchased from the University of Sydney during the 1990's is a Brumby MK I.[1] The Brumby Unmanned Ariel Vehicle (UAV) provides an ideal platform for aircraft control and navigation research. The Brumby UAV has a delta wing planform with twin vertical stabilizers, the contour of the fuselage is that of a cylinder with a blunted ogive nose. This makes creating a Missile DATCOM model a relatively straight forward task. The delta wing also contains the Ailevons (aileron and elevator control on one control surface). The change in deflection angles create changes in the lift, drag, roll, and pitch coefficients of the main lifting planform. This causes the angle-of- attack and side-slip angle to be coupled with the deflection angles of the ailevons. This creates a nonlinear aircraft model, that is an ideal system for a non-linear control research platform.

1.2 Motivation

There are many methods available to obtain an aerodynamic model of an aerospace vehicle. Methods such as system identification require data to be taken while the vehicle is operating over a predefined envelope. This method requires that a physical model be constructed and operated in the environment for which the aerodynamic model is desired.

This can be costly and very time consuming. It may not be possible to fly the model over all the desired flight envelopes. Other options such as wind tunnel data also require that a model be built and tested. Traditionally, full sized aircraft must be scaled down to meet 21 the size constraints of the wind tunnel. Some unmanned UAV's are small enough to fit inside the wind tunnel at full scale. Since the model is full sized and full functioning, forces and moments as well as derivatives for control surface deflection angles may be measured. All of these methods require that a new model be constructed, and retested for changes in vehicle geometry.

By using a computational fluid dynamics prediction code it is possible to obtain aerodynamic coefficients for various vehicle geometries over a wide range of environmental conditions without the cost or inconvenience associated with wind tunnel testing. CFD prediction codes can generate aerodynamic coefficients in a shorter time period and at a lower monetary cost. While computational fluid dynamic prediction codes may not capture all the nonlinearities of the aerodynamics, the model is still valid and useful.

1.3 Modeling Aerodynamic Forces and Moments

The processing power of todays computers make it possible to model the aerodynamic forces and moments using computational fluid dynamics prediction codes.

These codes allow the user to create software models of the aircraft and generate the forces and moments using only a computer. These mathematical models can then be used to analyze the dynamic behavior of the aircraft. These models allow the control system engineer to create compensation schemes that will cause the aircraft to have more desirable dynamics. For example the aircraft may not respond to inputs fast enough, there may be an undesirable steady-state error to a control input, or the systems response to disturbance inputs may need to be analyzed, e.g. wind gusts. 22

1.4 Objectives

The reader of this thesis will be able to generate aerodynamic force and moment coefficient data using USAF Missile DATCOM. The reader will be exposed to the basic definitions and terminology of USAF Missile DATCOM. This data will then be integrated into a six degree-of-freedom Simulink simulation where the model will be analyzed for static as well as dynamic stability. The reader should have an understanding of the basic concepts required for modeling and simulation of an aircraft using computational fluid dynamic modeling.

1.5 Thesis Organization

The control system engineer must create an accurate model of a mechanical system before a control system can be designed. Modeling the aerodynamic behavior of an aircraft typically requires a scale model of the aircraft be built and placed in a wind tunnel where forces are measured. It may be difficult for researchers in aircraft control system design to gain access to a wind tunnel or be able to fund the building of a scale model. Computational fluid dynamics allows the researcher the ability to model the aircraft without the trouble or expense of creating scale models or obtaining testing time in a wind tunnel. This thesis will cover the topic of creating an aerodynamic model using a computational fluid dynamics prediction code.

1.5.1 Missile DATCOM Input Parameters

In order to use the CFD prediction code the user must understand the dimensions and variables that are needed to create the model using USAF Missile DATCOM. The physical dimensions of the aircraft are required, such as the lengths of the planforms, the 23 dimensions of the control surfaces, the location of the center-of-mass to name a few. This will be illustrated through use of an example in Chapter 2.

1.5.2 Missile DATCOM Model of the Brumby UAV

The Ohio University Avionics Center conducts research using a Brumby UAV aircraft. This aircraft model is used to perform guidance and navigation research. The

Brumby UAV will be modeled using Missile DATCOM in Chapter 3.

1.5.3 Equations of Motion

The equations of motion for a moving body in 3-dimensions will be presented in

Chapter 4. These equations describe the effect of the forces and moments on the aircraft.

The equations of motion will be used to create a six degree-of-freedom simulation.

1.5.4 Brumby UAV Model Simulation

The Missile DATCOM model of the Brumby UAV will be simulated, analyzed, and subjected to perturbations from equilibrium in Chapter 5. The model will first be trimmed for straight wings level flight. Wings level flight is typical of an aircraft that is traversing between way points. The eigenvalues of the straight wings level flight trim condition will be evaluated as well as an explanation of the dynamics of the aircraft. The model will then be trimmed for a coordinated turn with a constant rate of climb. Finally, the Brumby

UAV model will be subjected to input perturbations and the aircraft dynamics will be observed. 24

Chapter 2: Missile DATCOM Modeling Parameters

Overview

In this chapter a description of Missile DATCOM terminology and variables will be presented. Due to the large number of possible geometric configurations, only terminology and variables needed to create a model of the Brumby Unmanned Aerial

Vehicle (UAV) in Chapter 3 will be discussed. The reader is directed to Reference [4] for more information on other geometric possibilities or for an expanded list of options for the discussed variables.

User vehicle geometric configuration and flight condition specifications are input to Missile DATCOM using a text file. Missile DATCOM parses the input file looking for predefined “Namelists” that it associates to internal variables. Missile DATCOM requires only a minimal number of Namelists be used to define the vehicle geometry. Over- specification of the geometry can generate numerical instability of some calculations in

Missile DATCOM.[4] Missile DATCOM allows the user to set the units that will be used for the calculations, as well as managing additional output data that can be calculated through the use of control cards. Control cards are valid only in the case in which they appear unless the user saves the current case using the SAVE control card.

This allows the user to use different control cards for different cases. A list of control cards used when creating the model in Chapter 3 is given in Table 2.1. Missile DATCOM will generate output data based on the commands in the input file that is used. The output file will contain aerodynamic coefficients, and may also contain dynamic damping derivative coefficients if the DAMP control card was used. 25

It is important to understand the coordinate system that will be used in describing the geometry of the vehicle in question. Let the center of gravity lie inside the vehicle and let it be at the intersection of the longitudinal plane of symmetry and the lateral plane of symmetry if it exists. Then Missile DATCOM designates the positive x-axis as being positive increasing aft from the tip of the nose, the positive y-axis as increasing along the starboard wing, and the positive z-axis as increasing in a manner that it obeys the right hand rule. This coordinate system is shown in Figure 2.1. Missile DATCOM allows the user to place the origin of the coordinate system a specified distance from the tip of the nose along the x-axis by assigning X0 a non-zero value. If no value is assigned to X0 then Missile DATCOM will use the default value of 0.0 units of distance.

Figure 2.1: Missile DATCOM Axes Definition 26

Table 2.1: Missile DATCOM Control Cards

Control Cards Description Values DIM Sets the system of length dimension Units (L) M,CM,FT,IN DERIV Sets the output derivative Units DEG,RAD INCRMT Calculates correction factors for coefficients on the N/A first run, based on experimental data given in EXPR. NOGO Allows program to cycle through input cases without N/A computing configuration Aerodynamics NO LAT Inhibits computation of lateral-directional derivatives, N/A if DAMP is selected PLOT Creates data file for003.dat, containing aerodynamic N/A data for plotting. BUILD Prints results of a configuration build-up N/A CASEID User supplied title output for that case Brumby Flaps DAMP Computes dynamic damping derivatives. N/A DELETE name Ignore namelist saved from previous case Namelist value NAMELIST Prints all Namelist data N/A NEXT CASE Indicates termination of the case input. N/A PART Prints partial aerodynamic output. N/A PRINT AERO name Prints the incremental aerodynamics for name. BODY,FIN1,etc. For more options see reference Page 23. PRINT GEOM name Prints the geometric characteristics of component BODY,FIN1,etc. name. For more options see reference Page 23. SAVE Saves namelist values from previous case. TRIM Calculates fin deflection angles for longitudinal trim N/A condition. NACA Allows use of predefined NACA airfoil types to be 2412 used as airfoil geometries

2.1 Flight Conditions

Missile DATCOM allows the user to specify the flight conditions in namelist

FLTCON, for which the aerodynamic data will be calculated. The user places the angles- of-attack values in the ALPHA array, and the Mach values in the MACH array. The size 27 of the MACH array is stored in NMACH and the size of the ALPHA array is stored in

NALHPA. For each vehicle scenario that Missile DATCOM executes, aerodynamic coefficients will be computed for all combinations of defined Mach and angle-of-attack.

A matrix will be created for each aerodynamic coefficient with NMACH columns and

NALPHA rows. Only one side-slip angle can be run for each case and is stored in the

BETA variable. In order to simplify data input, only a core set of flight condition data needs to be input by the user. For the model that will be generated in Chapter 3, only values for Mach and altitude are required. From these values Missile DATCOM will calculate the internal variable values needed to perform the aerodynamic calculations. A list of variables from namelist FLTCON that are used are given in Table 2.2.

Table 2.2: Missile DATCOM Namelist FLTCON

Variable Name Array Size Description Units Default Value NALPHA - Number of angles of attack - - ALPHA 20 angle-of-attack Deg - BETA - side-slip angle Deg 0.0 PHI - Aerodynamic roll angle Deg 0.0 NMACH - Number of Mach values - - MACH 20 Mach Values - - ALT 20 Altitude values L 0.0

Missile DATCOM also requires that parameter values be specified by the user for referencing and scaling purposes. Missile DATCOM will generate aerodynamic coefficients that have been non-dimensionalized with respect to the reference values. The reference variables used for the model in Chapter 3 are listed in Table 2.3. Typically, the 28 values used for SREF, LREF, and LATREF on a traditional aircraft are wing planform area, mean wing chord length, and wing span length respectively.

Table 2.3: Missile DATCOM Namelist REFQ

Variable Name Description Units Default Value SREF Reference Area L*L Maximum body cross- sectional area LREF Longitudinal Reference Length L Maximum body diameter LATREF Lateral reference length L LREF XCG Longitudinal position of Center of Gravity (+aft) L 0.0 ZCG Vertical Position of Center of Gravity (+up) L 0.0

2.2 Fuselage

Missile DATCOM allows for axially symmetric or elliptical body shapes. These body shapes can either be input using body diameter and length if the body has a continuous radius along the body, and trailing nozzle sections, or the body geometry can be input at different longitudinal stations. These options provide a great deal of flexibility. The variables used in creating the axial body model in Chapter 3 are listed in

Table 2.4. 29

Table 2.4: Missile DATCOM Namelist AXIBOD

Variable Name Description Units Default Value XO Longitudinal coordinate of nose tip. L 0.0 TNOSE Type of nose shape. - OGIVE LNOSE Nose length L - DNOSE Nose diameter at base L 1.0 BNOSE Bluntness radius L 0.0 LCENTR Center body length L 0.0 DCENTR Center body diameter at base L DNOSE

2.3 Primary Lifting Surface

Traditional aircraft typically have a geometry that consists of: a body, a wing, vertical and horizontal stabilizing and control surfaces. Missile DATCOM describes each planform surface as a finset that is located at a defined position on the body. Missile

DATCOM allows for four finsets, each finset can contain a total of eight panels. Using this method, the fin geometry must only be defined once. Then the position of each fin on the body must be specified. Missile DATCOM will not check to see if Finset1 is fore or aft of Finset2 when it performs an error analysis. Placing Finset2 fore of Finset1 will cause errors in the interference flow calculations from one fin to the next. Finset1 will be the foremost finset, which on a traditional aircraft without canards will be the wing planform. We will start out by describing the planform geometry and then describe the position of each panel around the body. A basic set of variables are listed in Table 2.5. 30

Table 2.5: Missile DATCOM Namelist FINSET

Variable Name Array Size Description Units Default Value XLE 10 Distance from nose to chord L 0.0 leading edge CHORD 10 Panel chord at each semi-span L - station SSPAN 10 Semi-span locations L - CFOC 10 Flap chord to Fin chord ratio - 1.0 NPANEL 8 Number of panels in fin set (1-8) - 4.0 PHIF 8 Roll angle of fin about body, Deg Even spacing Clockwise is positive angle. around body. GAM 8 Dihedral angle of each fin, Deg 0.0 Positive angle when PHIF is Figure 2.2:increased Missile DATCOM Body Variables SECTYPE - Type of airfoil section. - HEX STA 10 Sweep back angle at each span Deg. 0.0 station. SWEEP 10 Chord station used in - 1.0 measuring sweep: STA=0.0 is leading edge STA=1.0 is trailing edge

It is important to note that when a fin panel PHIF value is greater than 180 degrees, see Figure 2.4, and has a SECTYPE of NACA (National Advisory Committee for Aeronautics), the airfoil of the fin will also be rotated. This rotation will cause a positive angle-of-attack to be seen by both the port and starboard panels. The NACA control card uses the form NACA 1-4-2412, where the first number designates the finset, in this case FINSET1. The second number designates the NACA series of airfoil, for this example this is a NACA 4 series. The last number is the NACA airfoil section designation. For a NACA 4 series the first number is the camber in percent of the chord 31 length, the second is the location of maximum camber aft from the leading edge in tens of percent of the chord length, and the last two digits are the maximum chord thickness locate at the point of maximum camber. Figure 2.4 is an example of an airfoil that has 2% camber, with 12% thickness located at 4% aft of the leading edge.[5]

Figure 2.2: Missile DATCOM Finset

Figure 2.3: NACA Airfoil Number Decomposition 32

If the wing has a continuous sweep along its leading edge it is possible to only define XLE for the root chord of the wing. Missile DATCOM only requires that XLE(1) be defined if the user inputs the sweep back angle for each span station using the SWEEP namelist. Missile DATCOM will determine that the planform has continuous sweep between semi-span stations and will calculate the XLE values from one semi-span station to the next. In order to place the fin panels directly on the body mold line, start the semi- span at 0.0 and allow each additional element in the SSPAN array to be the distance from the body mold line to that semi-span station. By setting the first semi-span location at zero Missile DATCOM will place the panel directly on the body. Care must be taken in defining SSPAN(1) to be a distance other than the body mold line, SSPAN(1) = 0.0. The user must ensure that the panel is attached to the body, otherwise there may be a gap between the body and the root chord of the panel. Missile DATCOM will not check to see if the panel is attached to the body. Missile DATCOM will not allow cracked panels or the airfoil shape to change over the panel. In Section 2.6 the method for placing control surfaces on a planform will be discussed. 33

Figure 2.4: Missile DATCOM Fin Panel Location Definition

2.4 Horizontal Stabilizer

It is possible to define a horizontal stabilizing surface using the same method described in Section 2.3. For this reason the reader is referred to Section 2.3 for details on creating horizontal stabilizing planforms. In this section horizontal stabilizers that do not lie in the same horizontal plane as the wing planform will be defined. It is possible to create a horizontal stabilizer that is positioned on top of a vertical stabilizer. Because error analysis used in Missile DATCOM does not check to see if a finset is actually located on the body, it is possible to create what is known as a T-tail configuration. This is accomplished by using two panels and setting the two PHIF values to 0.0 degrees.

Then set the SSPAN(1) value to be the distance from the center of the body to the root chord of the horizontal stabilizer. This value would be 0.0 in most cases so that the root chord would be located on the x-z plane. However, if the SSPAN(1) value is 0.0 Missile 34

DATCOM will place the root chord on the body. This means that the SSPAN(1) value must be arbitrarily small so that it will reside as near as possible to the x-z plane. The starboard fin will have a GAM value of 90.0 degrees and the port fin will have a GAM value of -90.0 degrees. Figure 2.6 contains an illustration of the variables associated with defining a a twin-vertical stabilizer.

It is also possible to create what is typically known as a V-tail configuration. This can be accomplished in a manner similar to the method discussed in Section 2.3, with the exception that the fin planforms are located symmetrically about the x-z plane, and dihedral angle is zero.

Figure 2.5:Twin-Horizontal Stabilzer Fin Panel Location Definition 35

Figure 2.6:V-Tail Stabilzer Fin Panel Location Definition

2.5 Vertical Stabilizer

It is also possible to create a single vertical stabilizer or twin vertical stabilizers.

In the case of a single vertical stabilizer NPANEL would have a value of 1.0 and both

PHIF and GAM would be 0.0 degrees. This situation would indicate that the fin planform is aligned with the z-axis and that the dihedral angle is zero. These twin vertical stabilizers can also be placed off of the body and onto another panel, using a similar method as described in the previous section. The first element in the SSPAN array is the distance from the centerline of the body to the location of the root chord of the stabilizer.

To place the stabilizers on another planform the PHIF angles must be the same, e.g. the

PHIF starboard wing is equal to the PHIF angle of the starboard stabilizer. This ensures that the root chord is placed on the existing panel. The roll angle PHIF would contain values of 90.0 degrees for the starboard fin and -90.0 degrees for the port fin. The dihedral values GAM would be used to roll the fins into vertical positions. This would be accomplished by setting the starboard GAM value to be -90.0 degrees and the port GAM value to be 90.0 degrees. 36

Figure 2.7:Twin-Vertical Stabilzer Fin Panel Location Definition

2.6 Control Surfaces

It is often useful and at times necessary to know the contribution of the deflection angles of the control surfaces to the aerodynamic coefficients. To determine the size of each control surface on the fin panel, care must be given in defining the fin. First, the semi-span stations should be defined for all control surface demarcation points along the planform. It is necessary to define semi-span, chord, and flap chord to fin chord ratio laterally for each control surface. For a fin planform having a single flap located between the root and tip chord, it is necessary to define four semi-station points, four chord values at those station points, and four flap chord to fin chord ratio values. In this particular example, shown in Figure 2.6, the break between the flap and the chord does not lie on either the root or the tip chord. Because the length of the flap at the root and tip of the 37 wing is zero, both the first and last values of the CFOC array will contain zeros. The entire stabilizer can be made a movable control surface by setting the values of CFOC to

1.0. This indicates to Missile DATCOM that the flap chord is the total length of the fin chord, and therefore the entire panel is movable.

In order to set the control surface deflection, Missile DATCOM uses the DEFLCT namelist that can been seen in Table 2.6. Only the control surfaces that have been defined should have their deflection values set, any control surface not defined by the user will have its respective deflection angle set to zero internally by Missile DATCOM. Missile

DATCOM will perform calculations over all eight panels in each of the four finsets. Any undefined panel is assigned zero length and does not contribute to the aerodynamic coefficients being calculated. Assuming the panel is placed with the root chord located on the body and the fin is perpendicular to the x-axis, then the deflection angles are defined as positive if they induce a negative body axis rolling moment. A negative body axis rolling moment is defined as counterclockwise when viewed along the x-axis looking forward toward the nose. This is valid for all flaps regardless of orientation. The deflection angle for flaps that are not located on the body are defined as if the fins are located axially around the x-axis. 38

Figure 2.8: Missile DATCOM Control Surface Definition

Table 2.6: Missile DATCOM Namelist DEFLCT

Variable Name Array Size Description Units Default Value DELTA1 8 Deflection values for Finset1 Deg. 0.0 DELTA2 8 Deflection values for Finset2 Deg. 0.0 DELTA3 8 Deflection values for Finset3 Deg. 0.0 DELTA4 8 Deflection values for Finset4 Deg. 0.0

2.7 Generating Additional Data

Some of the limitations in Missile DATCOM can be overcome by running the vehicle again in a new case while only changing one value. An example would be to handle more than one side-slip angle. Even though Missile DATCOM will only consider one side-slip value per case, by running multiple cases and only changing the side-slip 39

value in each case Missile DATCOM will generate data for those side-slip angles. This

becomes especially useful when data for different control surface deflection angles are

desired. By saving the previous vehicle geometry using the SAVE control card, then

overwriting the deflection angle, Missile DATCOM will calculate the aerodynamic

coefficients for the new vehicle configuration.

2.8 File Format and Content

Missile DATCOM uses space delimiting as a method for distinguishing namelists

from control cards. Only a control card should be placed in the first character of a column

in the input file. Namelists should allow one space for the first column and should should

then start and end with a dollar sign ($). Variables in a namelist are separated using

commas and a comma must precede the terminating dollar sign of the namelist. A row

can only contain eighty characters including symbols and blank spaces. Values assigned

to variables must always contain a decimal point, for a value of zero the leading zero is

necessary, while a zero after the decimal point is not. In order for the case to be executed

a NEXT CASE control card must be inserted at the end of each case, including the last

case. Table 2.7 gives a brief explanation of the input and output files created and required

for execution by Missile DATCOM.

The for005.dat file is the input file to Missile DATCOM and contains the control

cards as well as the namelists that are used to describe the vehicle. The for005.dat file for

the Brumby MK. I is listed in Appendix A.1.

The for006.dat file contains two copies of the for005.dat file as well as the output for the cases to be executed by Missile DATCOM. The first listing is a copy of the 40 for005.dat file and the second is the for005.dat file containing error checking markups.

The for003.dat file contains the aerodynamic coefficients for the cases executed

by Missile DATCOM. The Columns of the for003.dat file are: Angle-of-Attack

(ALPHA), Normal Force Coefficient (CN), Pitching Moment Coefficient (CM), Axial

Force Coefficient (CA), Side-Force Coefficient (CY), Yawing Moment Coefficient

(CLN), Rolling Moment Coefficient (CLL) , Deflection Angle for zero Pitching Moment

(DELTA), Lift Coefficient (CL), and Drag Force Coefficient (CD). The rows correspond

to the angle-of-attack values ALPHA. Missile DATCOM will generate a matrix of

ALPHA rows and coefficient columns for each MACH value specified, for each case that

is executed.

The for0021.dat file contains all of the necessary aerodynamic coefficients that

would be required to create a nonlinear vehicle simulation using a build up of the

individual components. The for021.dat file contains a row of variables: Mach, altitude,

side-slip angle, the deflection angles for the flaps, the number of rows of data, the total

columns of data, and finally the number of columns of derivatives. The variables are

immediately followed by the angle-of-attack (ALPHA) and the aerodynamic coefficients

which are: normal force coefficient (CN), pitching moment coefficient (CM), axial force

coefficient (CA), side force coefficient (CY), yawing moment coefficient (CLN), rolling

moment coefficient (CLL), normal force due to pitch rate (CNQ), pitching moment due

to pitch rate (CMQ), axial force due to pitch rate (CAQ), side force due to yaw rate

(CYR), yawing moment due to yaw rate (CLNR), rolling moment due to yaw rate

(CLLR), side force due to roll rate (CYP), yawing moment due to roll rate (CLNP), 41 rolling moment due to roll rate (CLLP). Aerodynamic derivatives are only calculated for the base model, where the deflection angles for the effectors are set to zero. The base model is immediately followed by coefficients for each case that is executed by Missile

DATCOM.

Table 2.7: Missile DATCOM File Definitions

Filename Description For005.dat User input file. For006.dat Output file containing results from error checking and calculations. For003.dat Output file generated by PLOT control card, containing calculated aerodynamic coefficients. For021.dat Output file to be used with Air Force program DATCOMTableMEX.dll

2.9 Missile DATCOM Example

In this section an example missile from the Missile DATCOM user manual will be presented.[4] This particular missile is axially body symmetric with four panels equally distributed around the body. The dimensions are in inches (DIM IN). The envelope in consideration is MACH values 0.4, 0.8, 2.0 (MACH= 0.4, 0.8, 2.0) and angles of attack -8.00, -4.00, 0.00, 4.00, 8.00 (ALPHA=-8.00, -4.00, 0.00, 4.00, 8.00) at an altitude of zero meters (ALT=0.0) with a side-slip angle of zero degrees (BETA=0.0).

The center of gravity lies 39.0 inches from the origin which is located at the tip of the nose (XCG=39.0). The body of the missile is 54.0 inches long (LCENTR=54.0) and 12.0 inches in diameter (DCENTR=12.0). The nose of the missile is ogive in shape

(TYPE=OGIVE) and is 12.0 inches long (LNOSE=12.0) and has a base diameter of 12.0 42 inches (DNOSE=12.0). The missile has four fins that are evenly distributed around the body. The Fins have a NACA airfoil shape with a NACA number of 0310

(SECTYP=NACA and NACA-1-4-0310). The leading edge of the fin at the first semi- span locate is 64.0 inches from the nose (XLE=64.0).The semi-span values of the fins are

0.0 at the root and 9.0 inches at the tip (SSPAN=0.0, 9.0,). The chord length at the root is

14.0 inches and 8.0 inches at the tip (CHORD=14.0, 8.0). The sweep angle of the fins are

0.0 degrees and are measured with respect to the segment trailing edge (SWEEP=0.0 and

STA=1.0). There are four fin panels located at 45.0, 135.0, 225.0, 315.0 degrees around the body (NPANEL=4.0, PHIF=45.0, 135.0, 225.0, 315.0, GAM=0.00, 0.00, 0.00, 0.00).

The fins have a control flap with a a constant cord to flap ratio of 0.25 that starts at the second station point and runs to the tip of the chord (CFOC=0.0, 0.25, 0.25, 0.25). Data must also be generated for a condition where the two fins that are facing horizontal have a deflection that would cause the missile to pitch nose up (SAVE, NEXT CASE,

CASEID PANEL DEFLECTION, $DEFLCT DELTA1=5.0, 0., 0., -5.0, $, SAVE, NEXT

CASE). This case is presented in Figure 2.8. The for006.dat file is listed in Figures 2.9 through 2.33. Figures 2.34 and 2.35 show listings for the for003.dat and for021.dat files respectively. 43

CASEID Example DAMP PLOT DIM IN DERIV RAD $FLTCON NMACH=3.0,ALT=0.,NALPHA=5.0, MACH =0.4,0.8,2.0, ALPHA = -8.00,-4.00,0.00,4.00,8.00, BETA=0.,$ $REFQ XCG=39.0,$ $AXIBOD TNOSE=OGIVE,LNOSE=12.0,DNOSE=12.0,LCENTR=54.0,DCENTR=12.0,$ $FINSET1 SECTYP=NACA, SSPAN=0.0,9.0, CHORD=14.0,8.0, XLE=64.0, SWEEP=0.0, STA=1.0, NPANEL=4., PHIF=45.0,135.0,225.0,315.0, GAM=0.00,0.00,0.00,0.00, CFOC=0.0,0.25,0.25,0.25,$ NACA-1-4-0310 SAVE NEXT CASE CASEID PANEL DEFLECTION DAMP $DEFLCT DELTA1=5.0,0.,0.,-5.0,$ SAVE NEXT CASE

Figure 2.9: for005.dat File Example 44

1 ***** THE USAF AUTOMATED MISSILE DATCOM * REV 01/06 ***** AERODYNAMIC METHODS FOR MISSILE CONFIGURATIONS

CONERR - INPUT ERROR CHECKING

ERROR CODES - N* DENOTES THE NUMBER OF OCCURENCES OF EACH ERROR A - UNKNOWN VARIABLE NAME B - MISSING EQUAL SIGN FOLLOWING VARIABLE NAME C - NON-ARRAY VARIABLE HAS AN ARRAY ELEMENT DESIGNATION - (N) D - NON-ARRAY VARIABLE HAS MULTIPLE VALUES ASSIGNED E - ASSIGNED VALUES EXCEED ARRAY DIMENSION F - SYNTAX ERROR

************************* INPUT DATA CARDS *************************

1 CASEID Example 2 DAMP 3 PLOT 4 DIM IN 5 DERIV RAD 6 $FLTCON NMACH=3.0,ALT=12*0.,NALPHA=5.0, 7 MACH =0.4,0.8,2.0, 8 ALPHA = -8.00,-4.00,0.00,4.00,8.00, 9 BETA=0.,$ 10 $REFQ XCG=39.0,$ 11 $AXIBOD TNOSE=OGIVE,LNOSE=12.0,DNOSE=12.0,LCENTR=54.0,DCENTR=12.0,$ ** SUBSTITUTING NUMERIC FOR NAME OGIVE 12 $FINSET1 SECTYP=NACA, ** SUBSTITUTING NUMERIC FOR NAME NACA 13 SSPAN=0.0,9.0, 14 CHORD=14.0,8.0, 15 XLE=64.0, 16 SWEEP=0.0, 17 STA=1.0, 18 NPANEL=4., 19 PHIF=45.0,135.0,225.0,315.0, 20 GAM=0.00,0.00,0.00,0.00, 21 CFOC=0.0,0.25,0.25,0.25,$ 22 NACA-1-4-0310 23 SAVE 24 NEXT CASE 25 CASEID PANEL DEFLECTION 26 DAMP 27 $DEFLCT DELTA1=5.0,0.,0.,-5.0,$ 28 SAVE 29 NEXT CASE

Figure 2.10: for006.dat File Example (Data Input) 45

1 ***** THE USAF AUTOMATED MISSILE DATCOM * REV 01/06 ***** CASE 1 AERODYNAMIC METHODS FOR MISSILE CONFIGURATIONS PAGE 1 CASE INPUTS FOLLOWING ARE THE CARDS INPUT FOR THIS CASE

CASEID Example DAMP PLOT DIM IN DERIV RAD $FLTCON NMACH=3.0,ALT=12*0.,NALPHA=5.0, MACH =0.4,0.8,2.0, ALPHA = -8.00,-4.00,0.00,4.00,8.00, BETA=0.,$ $REFQ XCG=39.0,$ $AXIBOD TNOSE=1.,LNOSE=12.0,DNOSE=12.0,LCENTR=54.0,DCENTR=12.0,$ $FINSET1 SECTYP=1., SSPAN=0.0,9.0, CHORD=14.0,8.0, XLE=64.0, SWEEP=0.0, STA=1.0, NPANEL=4., PHIF=45.0,135.0,225.0,315.0, GAM=0.00,0.00,0.00,0.00, CFOC=0.0,0.25,0.25,0.25,$ NACA-1-4-0310 SAVE NEXT CASE * WARNING * THE REFERENCE AREA IS UNSPECIFIED, DEFAULT VALUE ASSUMED * WARNING * THE REFERENCE LENGTH IS UNSPECIFIED, DEFAULT VALUE ASSUMED THE BOUNDARY LAYER IS ASSUMED TO BE TURBULENT THE INPUT UNITS ARE IN INCHES, THE SCALE FACTOR IS 1.0000

Figure 2.11: for006.dat File Example (Error Checking) 46

1 ***** THE USAF AUTOMATED MISSILE DATCOM * REV 01/06 ***** CASE 1 AERODYNAMIC METHODS FOR MISSILE CONFIGURATIONS PAGE 2 Example STATIC AERODYNAMICS FOR BODY-FIN SET 1

******* FLIGHT CONDITIONS AND REFERENCE QUANTITIES ******* MACH NO = 0.40 REYNOLDS NO = 2.827E+06 /FT ALTITUDE = 0.0 FT DYNAMIC PRESSURE = 237.02 LB/FT**2 SIDESLIP = 0.00 DEG ROLL = 0.00 DEG REF AREA = 113.097 IN**2 MOMENT CENTER = 39.000 IN REF LENGTH = 12.00 IN LAT REF LENGTH = 12.00 IN

----- LONGITUDINAL ------LATERAL DIRECTIONAL -- ALPHA CN CM CA CY CLN CLL

-8.00 -1.200 1.360 0.032 0.000 0.000 0.000 -4.00 -0.585 0.682 0.091 0.000 0.000 0.000 0.00 0.000 0.000 0.113 0.000 0.000 0.000 4.00 0.585 -0.682 0.091 0.000 0.000 0.000 8.00 1.200 -1.360 0.032 0.000 0.000 0.000

ALPHA CL CD CL/CD X-C.P.

-8.00 -1.183 0.199 -5.943 -1.134 -4.00 -0.578 0.131 -4.399 -1.165 0.00 0.000 0.113 0.000 -1.165 4.00 0.578 0.131 4.399 -1.165 8.00 1.183 0.199 5.943 -1.134

X-C.P. MEAS. FROM MOMENT CENTER IN REF. LENGTHS, NEG. AFT OF MOMENT CENTER

Figure 2.12: for006.dat File Example (Case 1 Output, Page 2) 47

1 ***** THE USAF AUTOMATED MISSILE DATCOM * REV 01/06 ***** CASE 1 AERODYNAMIC METHODS FOR MISSILE CONFIGURATIONS PAGE 3 Example STATIC AERODYNAMICS FOR BODY-FIN SET 1

------DERIVATIVES (PER RADIAN) ------ALPHA CNA CMA CYB CLNB CLLB -8.00 9.0007 -9.6911 -9.3921 11.8173 0.2469 -4.00 8.5908 -9.7420 -8.7265 10.6220 -0.0021 0.00 8.3861 -9.7675 -8.1787 9.3495 0.0000 4.00 8.5908 -9.7420 -8.7265 10.6220 0.0021 8.00 9.0007 -9.6911 -9.3921 11.8173 -0.2469

PANEL DEFLECTION ANGLES (DEGREES) SET FIN 1 FIN 2 FIN 3 FIN 4 FIN 5 FIN 6 FIN 7 FIN 8 1 0.00 0.00 0.00 0.00

Figure 2.13: for006.dat File Example (Case 1 Output, Page 3) 48

1 ***** THE USAF AUTOMATED MISSILE DATCOM * REV 01/06 ***** CASE 1 AERODYNAMIC METHODS FOR MISSILE CONFIGURATIONS PAGE 4 Example BODY + 1 FIN SET DYNAMIC DERIVATIVES

------DYNAMIC DERIVATIVES (PER RADIAN) ------ALPHA CNQ CMQ CAQ CNAD CMAD -8.00 42.648 -93.625 3.424 26.852 -13.587 -4.00 40.731 -88.833 0.820 26.852 -13.587 0.00 42.257 -92.660 -2.175 26.852 -13.587 4.00 44.700 -98.776 -4.956 26.852 -13.587 8.00 44.992 -99.495 -7.183 26.852 -13.587

PITCH RATE DERIVATIVES NON-DIMENSIONALIZED BY Q*LREF/2*V

Figure 2.14: for006.dat File Example (Case 1 Output, Page 4) 49

1 ***** THE USAF AUTOMATED MISSILE DATCOM * REV 01/06 ***** CASE 1 AERODYNAMIC METHODS FOR MISSILE CONFIGURATIONS PAGE 5 Example BODY + 1 FIN SET DYNAMIC DERIVATIVES

------DYNAMIC DERIVATIVES (PER RADIAN) ------ALPHA CYR CLNR CLLR CYP CLNP CLLP -8.00 43.645 -96.560 0.057 -0.036 0.090 -13.514 -4.00 42.541 -93.804 0.022 -0.003 0.008 -12.558 0.00 42.082 -92.660 0.000 0.000 0.000 -11.496 4.00 42.541 -93.804 -0.022 0.003 -0.008 -12.558 8.00 43.645 -96.560 -0.057 0.036 -0.090 -13.514

YAW AND ROLL RATE DERIVATIVES NON-DIMENSIONALIZED BY R*LATREF/2*V

Figure 2.15: for006.dat File Example (Case 1 Output, Page 5) 50

1 ***** THE USAF AUTOMATED MISSILE DATCOM * REV 01/06 ***** CASE 1 AERODYNAMIC METHODS FOR MISSILE CONFIGURATIONS PAGE 6 Example STATIC AERODYNAMICS FOR BODY-FIN SET 1

******* FLIGHT CONDITIONS AND REFERENCE QUANTITIES ******* MACH NO = 0.80 REYNOLDS NO = 5.655E+06 /FT ALTITUDE = 0.0 FT DYNAMIC PRESSURE = 948.07 LB/FT**2 SIDESLIP = 0.00 DEG ROLL = 0.00 DEG REF AREA = 113.097 IN**2 MOMENT CENTER = 39.000 IN REF LENGTH = 12.00 IN LAT REF LENGTH = 12.00 IN

----- LONGITUDINAL ------LATERAL DIRECTIONAL -- ALPHA CN CM CA CY CLN CLL

-8.00 -1.233 1.421 0.053 0.000 0.000 0.000 -4.00 -0.604 0.721 0.110 0.000 0.000 0.000 0.00 0.000 0.000 0.132 0.000 0.000 0.000 4.00 0.604 -0.721 0.110 0.000 0.000 0.000 8.00 1.233 -1.421 0.053 0.000 0.000 0.000

ALPHA CL CD CL/CD X-C.P.

-8.00 -1.214 0.224 -5.416 -1.152 -4.00 -0.595 0.152 -3.922 -1.192 0.00 0.000 0.132 0.000 -1.192 4.00 0.595 0.152 3.922 -1.192 8.00 1.214 0.224 5.416 -1.152

X-C.P. MEAS. FROM MOMENT CENTER IN REF. LENGTHS, NEG. AFT OF MOMENT CENTER

Figure 2.16: for006.dat File Example (Case 1 Output, Page 6) 51

1 ***** THE USAF AUTOMATED MISSILE DATCOM * REV 01/06 ***** CASE 1 AERODYNAMIC METHODS FOR MISSILE CONFIGURATIONS PAGE 7 Example STATIC AERODYNAMICS FOR BODY-FIN SET 1

------DERIVATIVES (PER RADIAN) ------ALPHA CNA CMA CYB CLNB CLLB -8.00 9.1791 -9.8874 -9.5764 12.0985 0.2753 -4.00 8.8313 -10.1771 -8.9663 11.0908 -0.0019 0.00 8.6575 -10.3220 -8.4680 9.9592 0.0000 4.00 8.8313 -10.1771 -8.9663 11.0908 0.0019 8.00 9.1791 -9.8874 -9.5764 12.0986 -0.2753

PANEL DEFLECTION ANGLES (DEGREES) SET FIN 1 FIN 2 FIN 3 FIN 4 FIN 5 FIN 6 FIN 7 FIN 8 1 0.00 0.00 0.00 0.00

Figure 2.17: for006.dat File Example (Case 1 Output, Page 7) 52

1 ***** THE USAF AUTOMATED MISSILE DATCOM * REV 01/06 ***** CASE 1 AERODYNAMIC METHODS FOR MISSILE CONFIGURATIONS PAGE 8 Example BODY + 1 FIN SET DYNAMIC DERIVATIVES

------DYNAMIC DERIVATIVES (PER RADIAN) ------ALPHA CNQ CMQ CAQ CNAD CMAD -8.00 43.540 -102.509 3.557 26.878 -12.015 -4.00 41.891 -98.416 0.865 26.878 -12.015 0.00 43.305 -101.938 -2.236 26.878 -12.015 4.00 45.350 -107.020 -5.121 26.878 -12.015 8.00 45.141 -106.492 -7.429 26.878 -12.015

PITCH RATE DERIVATIVES NON-DIMENSIONALIZED BY Q*LREF/2*V

Figure 2.18: for006.dat File Example (Case 1 Output, Page 8) 53

1 ***** THE USAF AUTOMATED MISSILE DATCOM * REV 01/06 ***** CASE 1 AERODYNAMIC METHODS FOR MISSILE CONFIGURATIONS PAGE 9 Example BODY + 1 FIN SET DYNAMIC DERIVATIVES

------DYNAMIC DERIVATIVES (PER RADIAN) ------ALPHA CYR CLNR CLLR CYP CLNP CLLP -8.00 44.153 -104.500 0.007 -0.032 0.078 -13.994 -4.00 43.433 -102.718 0.020 -0.003 0.008 -13.173 0.00 43.117 -101.938 0.000 0.000 0.000 -12.190 4.00 43.433 -102.718 -0.020 0.003 -0.008 -13.173 8.00 44.153 -104.501 -0.007 0.032 -0.078 -13.994

YAW AND ROLL RATE DERIVATIVES NON-DIMENSIONALIZED BY R*LATREF/2*V *** NOSE TIP ANGLE GREATER THAN MACH ANGLE, HYBRID THEORY INVALID SECOND ORDER SHOCK EXPANSION TO BE USED

*** NOSE TIP ANGLE GREATER THAN MACH ANGLE, HYBRID THEORY INVALID SECOND ORDER SHOCK EXPANSION TO BE USED

Figure 2.19: for006.dat File Example (Case 1 Output, Page 9) 54

1 ***** THE USAF AUTOMATED MISSILE DATCOM * REV 01/06 ***** CASE 1 AERODYNAMIC METHODS FOR MISSILE CONFIGURATIONS PAGE 10 Example STATIC AERODYNAMICS FOR BODY-FIN SET 1

******* FLIGHT CONDITIONS AND REFERENCE QUANTITIES ******* MACH NO = 2.00 REYNOLDS NO = 1.414E+07 /FT ALTITUDE = 0.0 FT DYNAMIC PRESSURE = 5925.45 LB/FT**2 SIDESLIP = 0.00 DEG ROLL = 0.00 DEG REF AREA = 113.097 IN**2 MOMENT CENTER = 39.000 IN REF LENGTH = 12.00 IN LAT REF LENGTH = 12.00 IN

----- LONGITUDINAL ------LATERAL DIRECTIONAL -- ALPHA CN CM CA CY CLN CLL

-8.00 -1.108 1.049 0.771 0.000 0.000 0.000 -4.00 -0.533 0.569 0.782 0.000 0.000 0.000 0.00 0.000 0.000 0.786 0.000 0.000 0.000 4.00 0.533 -0.569 0.782 0.000 0.000 0.000 8.00 1.108 -1.049 0.771 0.000 0.000 0.000

ALPHA CL CD CL/CD X-C.P.

-8.00 -0.990 0.918 -1.079 -0.946 -4.00 -0.478 0.817 -0.584 -1.067 0.00 0.000 0.786 0.000 -1.067 4.00 0.478 0.817 0.584 -1.067 8.00 0.990 0.918 1.079 -0.946

X-C.P. MEAS. FROM MOMENT CENTER IN REF. LENGTHS, NEG. AFT OF MOMENT CENTER

Figure 2.20: for006.dat File Example (Case 1 Output, Page 10) 55

1 ***** THE USAF AUTOMATED MISSILE DATCOM * REV 01/06 ***** CASE 1 AERODYNAMIC METHODS FOR MISSILE CONFIGURATIONS PAGE 11 Example STATIC AERODYNAMICS FOR BODY-FIN SET 1

------DERIVATIVES (PER RADIAN) ------ALPHA CNA CMA CYB CLNB CLLB -8.00 8.5313 -6.2311 -8.3955 8.8365 0.2574 -4.00 7.9368 -7.5101 -7.8241 8.6319 -0.0007 0.00 7.6400 -8.1515 -7.4963 8.0310 0.0000 4.00 7.9368 -7.5101 -7.8242 8.6319 0.0007 8.00 8.5313 -6.2311 -8.3955 8.8365 -0.2574

PANEL DEFLECTION ANGLES (DEGREES) SET FIN 1 FIN 2 FIN 3 FIN 4 FIN 5 FIN 6 FIN 7 FIN 8 1 0.00 0.00 0.00 0.00

BODY ALONE LINEAR DATA GENERATED FROM SECOND ORDER SHOCK EXPANSION METHOD

Figure 2.21: for006.dat File Example (Case 1 Output, Page 11) 56

1 ***** THE USAF AUTOMATED MISSILE DATCOM * REV 01/06 ***** CASE 1 AERODYNAMIC METHODS FOR MISSILE CONFIGURATIONS PAGE 12 Example BODY + 1 FIN SET DYNAMIC DERIVATIVES

------DYNAMIC DERIVATIVES (PER RADIAN) ------ALPHA CNQ CMQ CAQ CNAD CMAD -8.00 39.342 -108.127 0.000 28.285 -9.526 -4.00 39.022 -107.260 0.000 28.285 -9.526 0.00 40.288 -110.734 0.000 28.285 -9.526 4.00 40.755 -112.006 0.000 28.285 -9.526 8.00 39.578 -108.773 0.000 28.285 -9.526

PITCH RATE DERIVATIVES NON-DIMENSIONALIZED BY Q*LREF/2*V

Figure 2.22: for006.dat File Example (Case 1 Output, Page 12) 57

1 ***** THE USAF AUTOMATED MISSILE DATCOM * REV 01/06 ***** CASE 1 AERODYNAMIC METHODS FOR MISSILE CONFIGURATIONS PAGE 13 Example BODY + 1 FIN SET DYNAMIC DERIVATIVES

------DYNAMIC DERIVATIVES (PER RADIAN) ------ALPHA CYR CLNR CLLR CYP CLNP CLLP -8.00 39.051 -107.399 0.017 -0.003 0.008 -9.839 -4.00 39.480 -108.582 0.003 -0.002 0.004 -9.678 0.00 39.880 -109.682 0.000 0.000 0.000 -9.191 4.00 39.480 -108.582 -0.003 0.002 -0.004 -9.678 8.00 39.051 -107.399 -0.017 0.003 -0.008 -9.839

YAW AND ROLL RATE DERIVATIVES NON-DIMENSIONALIZED BY R*LATREF/2*V

Figure 2.23: for006.dat File Example (Case 1 Output, Page 13) 58

1 ***** THE USAF AUTOMATED MISSILE DATCOM * REV 01/06 ***** CASE 2 AERODYNAMIC METHODS FOR MISSILE CONFIGURATIONS PAGE 1 CASE INPUTS FOLLOWING ARE THE CARDS INPUT FOR THIS CASE

CASEID PANEL DEFLECTION DAMP $DEFLCT DELTA1=5.0,0.,0.,-5.0,$ SAVE NEXT CASE * WARNING * THE REFERENCE AREA IS UNSPECIFIED, DEFAULT VALUE ASSUMED * WARNING * THE REFERENCE LENGTH IS UNSPECIFIED, DEFAULT VALUE ASSUMED THE BOUNDARY LAYER IS ASSUMED TO BE TURBULENT THE INPUT UNITS ARE IN INCHES, THE SCALE FACTOR IS 1.0000 1 ***** THE USAF AUTOMATED MISSILE DATCOM * REV 01/06 ***** CASE 2 AERODYNAMIC METHODS FOR MISSILE CONFIGURATIONS PAGE 2 PANEL DEFLECTION STATIC AERODYNAMICS FOR BODY-FIN SET 1

----- LONGITUDINAL ------LATERAL DIRECTIONAL -- ALPHA CN CM CA CY CLN CLL

ALPHA CL CD CL/CD X-C.P.

-8.00 -1.183 0.199 -5.943 -1.134 -4.00 -0.578 0.131 -4.399 -1.165 0.00 0.000 0.113 0.000 -1.165 4.00 0.578 0.131 4.399 -1.165 8.00 1.183 0.199 5.943 -1.134

X-C.P. MEAS. FROM MOMENT CENTER IN REF. LENGTHS, NEG. AFT OF MOMENT CENTER

Figure 2.24: for006.dat File Example (Case 2 Output, Page 1 and 2) 59

1 ***** THE USAF AUTOMATED MISSILE DATCOM * REV 01/06 ***** CASE 2 AERODYNAMIC METHODS FOR MISSILE CONFIGURATIONS PAGE 3 PANEL DEFLECTION STATIC AERODYNAMICS FOR BODY-FIN SET 1

FLAP DEFLECTION ANGLES (DEGREES) SET FIN 1 FIN 2 FIN 3 FIN 4 FIN 5 FIN 6 FIN 7 FIN 8 1 5.00 0.00 0.00 -5.00 EQUIVALENT PANEL DEFLECTION ANGLES (DEGREES) SET FIN 1 FIN 2 FIN 3 FIN 4 FIN 5 FIN 6 FIN 7 FIN 8 1 0.00 0.00 0.00 0.00

Figure 2.25: for006.dat File Example (Case 2 Output, Page 3) 60

1 ***** THE USAF AUTOMATED MISSILE DATCOM * REV 01/06 ***** CASE 2 AERODYNAMIC METHODS FOR MISSILE CONFIGURATIONS PAGE 4 PANEL DEFLECTION BODY + 1 FIN SET DYNAMIC DERIVATIVES

PITCH RATE DERIVATIVES NON-DIMENSIONALIZED BY Q*LREF/2*V

Figure 2.26: for006.dat File Example (Case 2 Output, Page 4) 61

1 ***** THE USAF AUTOMATED MISSILE DATCOM * REV 01/06 ***** CASE 2 AERODYNAMIC METHODS FOR MISSILE CONFIGURATIONS PAGE 5 PANEL DEFLECTION BODY + 1 FIN SET DYNAMIC DERIVATIVES

YAW AND ROLL RATE DERIVATIVES NON-DIMENSIONALIZED BY R*LATREF/2*V

Figure 2.27: for006.dat File Example (Case 2 Output, Page 5) 62

1 ***** THE USAF AUTOMATED MISSILE DATCOM * REV 01/06 ***** CASE 2 AERODYNAMIC METHODS FOR MISSILE CONFIGURATIONS PAGE 6 PANEL DEFLECTION STATIC AERODYNAMICS FOR BODY-FIN SET 1

----- LONGITUDINAL ------LATERAL DIRECTIONAL -- ALPHA CN CM CA CY CLN CLL

ALPHA CL CD CL/CD X-C.P.

-8.00 -1.214 0.224 -5.416 -1.152 -4.00 -0.595 0.152 -3.922 -1.192 0.00 0.000 0.132 0.000 -1.192 4.00 0.595 0.152 3.922 -1.192 8.00 1.214 0.224 5.416 -1.152

X-C.P. MEAS. FROM MOMENT CENTER IN REF. LENGTHS, NEG. AFT OF MOMENT CENTER

Figure 2.28: for006.dat File Example (Case 2 Output, Page 6) 63

1 ***** THE USAF AUTOMATED MISSILE DATCOM * REV 01/06 ***** CASE 2 AERODYNAMIC METHODS FOR MISSILE CONFIGURATIONS PAGE 7 PANEL DEFLECTION STATIC AERODYNAMICS FOR BODY-FIN SET 1

Figure 2.29: for006.dat File Example (Case 2 Output, Page 7) 64

1 ***** THE USAF AUTOMATED MISSILE DATCOM * REV 01/06 ***** CASE 2 AERODYNAMIC METHODS FOR MISSILE CONFIGURATIONS PAGE 8 PANEL DEFLECTION BODY + 1 FIN SET DYNAMIC DERIVATIVES

PITCH RATE DERIVATIVES NON-DIMENSIONALIZED BY Q*LREF/2*V

Figure 2.30: for006.dat File Example (Case 2 Output, Page 8) 65

1 ***** THE USAF AUTOMATED MISSILE DATCOM * REV 01/06 ***** CASE 2 AERODYNAMIC METHODS FOR MISSILE CONFIGURATIONS PAGE 9 PANEL DEFLECTION BODY + 1 FIN SET DYNAMIC DERIVATIVES

YAW AND ROLL RATE DERIVATIVES NON-DIMENSIONALIZED BY R*LATREF/2*V *** NOSE TIP ANGLE GREATER THAN MACH ANGLE, HYBRID THEORY INVALID SECOND ORDER SHOCK EXPANSION TO BE USED

*** NOSE TIP ANGLE GREATER THAN MACH ANGLE, HYBRID THEORY INVALID SECOND ORDER SHOCK EXPANSION TO BE USED

Figure 2.31: for006.dat File Example (Case 2 Output, Page 9) 66

1 ***** THE USAF AUTOMATED MISSILE DATCOM * REV 01/06 ***** CASE 2 AERODYNAMIC METHODS FOR MISSILE CONFIGURATIONS PAGE 10 PANEL DEFLECTION STATIC AERODYNAMICS FOR BODY-FIN SET 1

----- LONGITUDINAL ------LATERAL DIRECTIONAL -- ALPHA CN CM CA CY CLN CLL

ALPHA CL CD CL/CD X-C.P.

-8.00 -0.990 0.918 -1.079 -0.946 -4.00 -0.478 0.817 -0.584 -1.067 0.00 0.000 0.786 0.000 -1.067 4.00 0.478 0.817 0.584 -1.067 8.00 0.990 0.918 1.079 -0.946

X-C.P. MEAS. FROM MOMENT CENTER IN REF. LENGTHS, NEG. AFT OF MOMENT CENTER

Figure 2.32: for006.dat File Example (Case 2 Output, Page 10) 67

1 ***** THE USAF AUTOMATED MISSILE DATCOM * REV 01/06 ***** CASE 2 AERODYNAMIC METHODS FOR MISSILE CONFIGURATIONS PAGE 11 PANEL DEFLECTION STATIC AERODYNAMICS FOR BODY-FIN SET 1

BODY ALONE LINEAR DATA GENERATED FROM SECOND ORDER SHOCK EXPANSION METHOD

Figure 2.33: for006.dat File Example (Case 2 Output, Page 11) 68

1 ***** THE USAF AUTOMATED MISSILE DATCOM * REV 01/06 ***** CASE 2 AERODYNAMIC METHODS FOR MISSILE CONFIGURATIONS PAGE 12 PANEL DEFLECTION BODY + 1 FIN SET DYNAMIC DERIVATIVES

PITCH RATE DERIVATIVES NON-DIMENSIONALIZED BY Q*LREF/2*V

Figure 2.34: for006.dat File Example (Case 2 Output, Page 12) 69

1 ***** THE USAF AUTOMATED MISSILE DATCOM * REV 01/06 ***** CASE 2 AERODYNAMIC METHODS FOR MISSILE CONFIGURATIONS PAGE 13 PANEL DEFLECTION BODY + 1 FIN SET DYNAMIC DERIVATIVES

YAW AND ROLL RATE DERIVATIVES NON-DIMENSIONALIZED BY R*LATREF/2*V *** END OF JOB ***

Figure 2.35: for006.dat File Example (Case 2 Output, Page 13) 70

VARIABLES=ALPHA,CN,CM,CA,CY,CLN,CLL,DELTA,CL,CD ZONE T="NO TRIM MACH= 0.40" -8.0000 -1.1995 1.3602 0.0325 0.0000 0.0000 0.0000 0.4000 -1.1833 0.1991 -4.0000 -0.5855 0.6819 0.0907 0.0000 0.0000 0.0000 0.4000 -0.5777 0.1313 0.0000 0.0000 0.0000 0.1127 0.0000 0.0000 0.0000 0.4000 0.0000 0.1127 4.0000 0.5855 -0.6819 0.0907 0.0000 0.0000 0.0000 0.4000 0.5777 0.1313 8.0000 1.1995 -1.3602 0.0325 0.0000 0.0000 0.0000 0.4000 1.1833 0.1991 ZONE T="NO TRIM MACH= 0.80" -8.0000 -1.2331 1.4210 0.0530 0.0000 0.0000 0.0000 0.8000 -1.2137 0.2241 -4.0000 -0.6044 0.7206 0.1099 0.0000 0.0000 0.0000 0.8000 -0.5953 0.1518 0.0000 0.0000 0.0000 0.1317 0.0000 0.0000 0.0000 0.8000 0.0000 0.1317 4.0000 0.6044 -0.7206 0.1099 0.0000 0.0000 0.0000 0.8000 0.5953 0.1518 8.0000 1.2331 -1.4210 0.0530 0.0000 0.0000 0.0000 0.8000 1.2137 0.2241 ZONE T="NO TRIM MACH= 2.00" -8.0000 -1.1082 1.0487 0.7708 0.0000 0.0000 0.0000 2.0000 -0.9902 0.9175 -4.0000 -0.5334 0.5691 0.7819 0.0000 0.0000 0.0000 2.0000 -0.4775 0.8172 0.0000 0.0000 0.0000 0.7856 0.0000 0.0000 0.0000 2.0000 0.0000 0.7856 4.0000 0.5334 -0.5691 0.7819 0.0000 0.0000 0.0000 2.0000 0.4775 0.8172 8.0000 1.1082 -1.0487 0.7708 0.0000 0.0000 0.0000 2.0000 0.9902 0.9175 ZONE T="NO TRIM MACH= 0.40" -8.0000 -1.1995 1.3602 0.0325 0.0000 0.0000 0.0000 0.4000 -1.1833 0.1991 -4.0000 -0.5855 0.6819 0.0907 0.0000 0.0000 0.0000 0.4000 -0.5777 0.1313 0.0000 0.0000 0.0000 0.1127 0.0000 0.0000 0.0000 0.4000 0.0000 0.1127 4.0000 0.5855 -0.6819 0.0907 0.0000 0.0000 0.0000 0.4000 0.5777 0.1313 8.0000 1.1995 -1.3602 0.0325 0.0000 0.0000 0.0000 0.4000 1.1833 0.1991 ZONE T="NO TRIM MACH= 0.80" -8.0000 -1.2331 1.4210 0.0530 0.0000 0.0000 0.0000 0.8000 -1.2137 0.2241 -4.0000 -0.6044 0.7206 0.1099 0.0000 0.0000 0.0000 0.8000 -0.5953 0.1518 0.0000 0.0000 0.0000 0.1317 0.0000 0.0000 0.0000 0.8000 0.0000 0.1317 4.0000 0.6044 -0.7206 0.1099 0.0000 0.0000 0.0000 0.8000 0.5953 0.1518 8.0000 1.2331 -1.4210 0.0530 0.0000 0.0000 0.0000 0.8000 1.2137 0.2241 ZONE T="NO TRIM MACH= 2.00" -8.0000 -1.1082 1.0487 0.7708 0.0000 0.0000 0.0000 2.0000 -0.9902 0.9175 -4.0000 -0.5334 0.5691 0.7819 0.0000 0.0000 0.0000 2.0000 -0.4775 0.8172 0.0000 0.0000 0.0000 0.7856 0.0000 0.0000 0.0000 2.0000 0.0000 0.7856 4.0000 0.5334 -0.5691 0.7819 0.0000 0.0000 0.0000 2.0000 0.4775 0.8172 8.0000 1.1082 -1.0487 0.7708 0.0000 0.0000 0.0000 2.0000 0.9902 0.9175

Figure 2.36: for003.dat File Example 71

VARIABLES: MACH,ALTITUDE,SIDESLIP,DEL1,DEL2,DEL3,DEL4 ROWS, TOTAL COLUMNS, COLUMNS OF DERIVATIVES DATA: ALPHA,CN,CM,CA,CY,CLN,CLL,CNQ,CMQ,CAQ,CYR,CLNR,CLLR,CYP,CLNP,CLLP 0.40 0.0 0.00 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 5.0 16.0 9.0 -0.800E+01 -0.1200E+01 0.1360E+01 0.3248E-01 0.2529E-07 -0.6334E-07 0.1074E-07 0.4265E+02 -0.9363E+02 0.3424E+01 0.4364E+02 -0.9656E+02 0.5687E-01 -0.3588E-01 0.8987E-01 -0.1351E+02 -0.400E+01 -0.5855E+00 0.6819E+00 0.9071E-01 -0.2940E-08 0.7365E-08 0.1541E-07 0.4073E+02 -0.8883E+02 0.8202E+00 0.4254E+02 -0.9380E+02 0.2176E-01 -0.3388E-02 0.8487E-02 -0.1256E+02 0.000E+00 0.0000E+00 -0.2934E-07 0.1127E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.4226E+02 -0.9266E+02 -0.2175E+01 0.4208E+02 -0.9266E+02 0.1212E-05 -0.9743E-06 0.2441E-05 -0.1150E+02 0.400E+01 0.5855E+00 -0.6819E+00 0.9071E-01 -0.1879E-07 0.4707E-07 0.1620E-07 0.4470E+02 -0.9878E+02 -0.4956E+01 0.4254E+02 -0.9380E+02 -0.2176E-01 0.3388E-02 -0.8487E-02 -0.1256E+02 0.800E+01 0.1200E+01 -0.1360E+01 0.3248E-01 -0.5080E-07 0.1273E-06 0.2556E-07 0.4499E+02 -0.9950E+02 -0.7183E+01 0.4364E+02 -0.9656E+02 -0.5687E-01 0.3588E-01 -0.8987E-01 -0.1351E+02 0.80 0.0 0.00 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 5.0 16.0 9.0 -0.800E+01 -0.1233E+01 0.1421E+01 0.5301E-01 0.2503E-07 -0.6227E-07 0.1141E-07 0.4354E+02 -0.1025E+03 0.3557E+01 0.4415E+02 -0.1045E+03 0.7166E-02 -0.3155E-01 0.7849E-01 -0.1399E+02 -0.400E+01 -0.6044E+00 0.7206E+00 0.1099E+00 0.1385E-07 -0.3445E-07 0.2393E-07 0.4189E+02 -0.9842E+02 0.8646E+00 0.4343E+02 -0.1027E+03 0.2009E-01 -0.3073E-02 0.7646E-02 -0.1317E+02 0.000E+00 0.0000E+00 0.9313E-09 0.1317E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.4330E+02 -0.1019E+03 -0.2236E+01 0.4312E+02 -0.1019E+03 0.1178E-05 -0.9279E-06 0.2309E-05 -0.1219E+02 0.400E+01 0.6044E+00 -0.7206E+00 0.1099E+00 -0.5782E-08 0.1438E-07 0.2264E-08 0.4535E+02 -0.1070E+03 -0.5121E+01 0.4343E+02 -0.1027E+03 -0.2009E-01 0.3070E-02 -0.7639E-02 -0.1317E+02 0.800E+01 0.1233E+01 -0.1421E+01 0.5301E-01 -0.4800E-07 0.1194E-06 0.2343E-07 0.4514E+02 -0.1065E+03 -0.7429E+01 0.4415E+02 -0.1045E+03 -0.7165E-02 0.3155E-01 -0.7849E-01 -0.1399E+02 2.00 0.0 0.00 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 5.0 16.0 9.0 -0.800E+01 -0.1108E+01 0.1049E+01 0.7708E+00 0.9829E-08 -0.2690E-07 -0.1791E-07 0.3934E+02 -0.1081E+03 0.0000E+00 0.3905E+02 -0.1074E+03 0.1684E-01 -0.3002E-02 0.8216E-02 -0.9839E+01 -0.400E+01 -0.5334E+00 0.5691E+00 0.7819E+00 0.7282E-10 -0.1993E-09 -0.8745E-08 0.3902E+02 -0.1073E+03 0.0000E+00 0.3948E+02 -0.1086E+03 0.2910E-02 -0.1501E-02 0.4109E-02 -0.9678E+01 0.000E+00 0.0000E+00 0.5588E-08 0.7856E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.4029E+02 -0.1107E+03 0.0000E+00 0.3988E+02 -0.1097E+03 0.2254E-05 -0.7973E-06 0.2183E-05 -0.9191E+01 0.400E+01 0.5334E+00 -0.5691E+00 0.7819E+00 -0.3837E-08 0.1050E-07 0.7883E-08 0.4076E+02 -0.1120E+03 0.0000E+00 0.3948E+02 -0.1086E+03 -0.2907E-02 0.1501E-02 -0.4108E-02 -0.9678E+01 0.800E+01 0.1108E+01 -0.1049E+01 0.7708E+00 -0.4747E-07 0.1299E-06 0.2443E-07 0.3958E+02 -0.1088E+03 0.0000E+00 0.3905E+02 -0.1074E+03 -0.1684E-01 0.3004E-02 -0.8223E-02 -0.9839E+01 0.40 0.0 0.00 5.0 0.0 0.0 -5.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 5.0 7.0 0.0 -0.800E+01 -0.1200E+01 0.1360E+01 0.3248E-01 0.2529E-07 -0.6334E-07 0.1074E-07 -0.400E+01 -0.5855E+00 0.6819E+00 0.9071E-01 -0.2940E-08 0.7365E-08 0.1541E-07 0.000E+00 0.0000E+00 -0.2934E-07 0.1127E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.400E+01 0.5855E+00 -0.6819E+00 0.9071E-01 -0.1879E-07 0.4707E-07 0.1620E-07 0.800E+01 0.1200E+01 -0.1360E+01 0.3248E-01 -0.5080E-07 0.1273E-06 0.2556E-07 0.80 0.0 0.00 5.0 0.0 0.0 -5.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 5.0 7.0 0.0 -0.800E+01 -0.1233E+01 0.1421E+01 0.5301E-01 0.2503E-07 -0.6227E-07 0.1141E-07 -0.400E+01 -0.6044E+00 0.7206E+00 0.1099E+00 0.1385E-07 -0.3445E-07 0.2393E-07 0.000E+00 0.0000E+00 0.9313E-09 0.1317E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.400E+01 0.6044E+00 -0.7206E+00 0.1099E+00 -0.5782E-08 0.1438E-07 0.2264E-08 0.800E+01 0.1233E+01 -0.1421E+01 0.5301E-01 -0.4800E-07 0.1194E-06 0.2343E-07 2.00 0.0 0.00 5.0 0.0 0.0 -5.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 5.0 7.0 0.0 -0.800E+01 -0.1108E+01 0.1049E+01 0.7708E+00 0.9829E-08 -0.2690E-07 -0.1791E-07 -0.400E+01 -0.5334E+00 0.5691E+00 0.7819E+00 0.7282E-10 -0.1993E-09 -0.8745E-08 0.000E+00 0.0000E+00 0.5588E-08 0.7856E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.400E+01 0.5334E+00 -0.5691E+00 0.7819E+00 -0.3837E-08 0.1050E-07 0.7883E-08 0.800E+01 0.1108E+01 -0.1049E+01 0.7708E+00 -0.4747E-07 0.1299E-06 0.2443E-07

Figure 2.37: for021.dat File Example 72

Chapter 3: Missile DATCOM Model of the Brumby

Unmanned Aerial Vehicle

The Brumby UAV is an ideal aerospace vehicle for using Missile DATCOM to create an aerodynamic model. The Brumby UAV has many characteristics similar to a missile such as its geometry and having a constant diameter body cross-section. The

Brumby UAV model will benefit from the very broad flight envelope allowed by Missile

DATCOM. A broad flight envelope makes the model useful for studying many different actual flight maneuvers.

3.1 Flight Conditions

The Brumby UAV was to be modeled under expected flight conditions. Data was generated for the Brumby UAV over a range of -5.0 to 35.0 degrees of angle-of-attack(

 ). The  values vary from -5.0 to 0.0 in 5.0 degree increments and from 0.0 to

20.0 degrees in 2.0 degree increments, to allow for nonlinearities in the aerodynamic coefficients. The  values from 20.0 to 35.0 degrees were taken in 5.0 degree increments. The maximum velocity for the Brumby UAV is approximately 100 miles per hour. The sea level speed of sound is approximately 1117 feet per second. The maximum

Brumby UAV velocity would be approximately 146.67 feet per second. This would correspond to a Mach value of approximately 0.13. The smallest Mach value that

Missile DATCOM will calculate is 0.01. This creates a lower velocity boundary of approximately 7.6 miles per hour, assuming sea level speed of sound. Table 3.1 includes the values input for each variable in namelist FLTCON. 73

Table 3.1: Brumby UAV Flight Conditions (FLTCON)

Variable Brumby Values Units Name NALPHA 15.0 N/A ALPHA -5.00,0.00,2.00,4.00,6.00,8.00,10.00,12.00,14.00,16.00,18.00,20.00, Degrees 25.00,30.00,35.00 BETA 0.0 Degrees PHI 0.0 Degrees NMACH 4.0 N/A MACH 0.05,0.08,0.10,0.15 N/A ALT 0.0,0.0 Meters

Missile DATCOM uses reference values in order to scale the aerodynamic coefficients. The reference values for longitudinal length, lateral length, and area are

LREF, LATREF, SREF. The reference values used for the Brumby UAV are the surface area of the wing planform area, the mean aerodynamic chord length, and the wing span length and are stored in SREF, LREF, LATREF. Missile DATCOM calculates the position of the aerodynamic center of pressure with respect to the center of gravity along the x-axis and is represented as the variable Xcp value. The aerodynamic center of pressure is defined as the point on the infinitely thin airfoil section where the aerodynamic moment is zero.[6] The aerodynamic center of pressure tends to be at approximately 0.25c aft from the leading edge of the airfoil section for commonly used airfoil sections at subsonic speeds.[5] The actual aerodynamic center of pressure changes as a function of the angle-of-attack[6], however the 0.25c value of the aerodynamic center of pressure is a commonly used approximation at subsonic speeds. Missile 74

DATCOM calculates the center of pressure for the Brumby UAV to be 0.90 meters aft from the tip of the nose. Therefore, the center of gravity must be approximately 0.90 meters or less aft from X0, where X0 is at the tip of the nose cone, in order for the

Brumby UAV to be longitudinally statically stable. For longitudinal static stability the center of pressure resides slightly aft of the center of gravity, such that the vehicle can be trimmed to be statically stable by use of the horizontal control surface or elevator. This also causes the vehicle to pitch nose over in the event that the free stream velocity is zero during flight. Typically, this is considered during the design of the airframe and includes the placement of equipment about the airframe in order to maintain the desired center of gravity location. The values used in the REFQ namelist variables are presented in Table

3.2. The center-of-gravity was chosen to be at 0.85 meters from the tip of the nose.

Missile DATCOM shows the center-of-pressure to be located at 0.90 meters from the tip of the nose. A center of gravity location of 0.85 meters from the tip of the nose will create static stability in the dynamics of the aircraft. Aircraft designers consider the location of the center of gravity during the design of the airframe. The aircraft designer locates the components of the aircraft, such as the airframe, power plant, and instrumentation, such that the location of the center of gravity forward of the center-of-pressure. 75

Table 3.2: Brumby UAV Reference Values (REFQ)

Variable Name Brumby Values Units SREF 1.251700 Meters^2 LREF 0.634700 Meters LATREF 2.324000 Meters XCG 0.85 Meters ZCG -0.04 Meters

3.2 Fuselage The fuselage of the Brumby UAV is a cylinder with a blunted ogive nose cone.

The body of the Brumby UAV will be modeled in Missile DATCOM using the Axially

Symmetric namelist (ASYM). This will allow the body geometry to be defined using the least number of variables. The body could have been defined at longitudinal station points, which would have required additional measurements and should yield similar results.

Figure 3.1: Brumby UAV Fuselage 76

Table 3.3: Brumby UAV Body Definition (ASYM)

Variable Name Brumby UAV Value Units XO 0.0 Meters TNOSE OGIVE N/A LNOSE 0.1970 Meters DNOSE 0.1524 Meters BNOSE 0.0 Meters LCENTR 1.7730 Meters DCENTR 0.1524 Meters

The longitudinal point of reference, X0, was set to zero. This sets the origin of the

Missile DATCOM coordinate system at the tip of the nose on the Brumby UAV.[1]

Setting the origin of the coordinate system at the tip of the nose simplifies the measurements along the longitudinal axis.

3.3 Wing Planform

The main lifting surface planform of the Brumby UAV is a delta wing that is located on the xy-plane. This planform is the foremost finset and is therefore labeled

FINSET1. The leading edge is located aft from the nose at a length of 0.97 meters. The wing is composed of two panels located at 90.0 degrees and -90.0 degrees from the positive z-axis. The wing had no measurable dihedral and therefore the GAM values are

0.0. The airfoil section was determined to be symmetric with an approximate chord thickness of 10% located 30% aft from the leading edge, which is an airfoil section of

NACA 0310. The Brumby UAV wing planform airfoil section is modeled in Missile

DATCOM as NACA-1-4-0310. A detailed explanation of how to implement the control 77 surfaces for this planform will be discussed in Section 3.5.

Figure 3.2: Brumby UAV Wing Planform

3.4 Vertical Stabilizer

The Brumby UAV has twin vertical stabilizers located on the wing planform approximately 0.2660 meters from the body mold line. The twin vertical stabilizers are aft of the wing planform and are defined as FINSET2 in Missile DATCOM. In order to place the panels perpendicular to the wing panel the vertical stabilizers are located at a

PHIF angle of 90.0 and -90.0. Then the vertical stabilizers will be given dihedral angles

GAM of -90.0 and 90.0, which will roll the panels into a vertical orientation. In order to 78 place the panels away from the body and onto the wing the first SSPAN value will be half the body diameter plus the distance of the panel from the body mold line. The second value for SSPAN is the distance from the first SSPAN length to the end of the panel. The values used for SSPAN are 0.2660 and 0.6790 for the starboard and port panels. The panels are 1.57 meters from X0 along the body x-axis. The twin vertical tails are swept from the root chord aft of the aircraft toward the tip chord. Missile DATCOM allows the user to define the sweep angle with reference to either the leading edge or the trailing edge. For the twin vertical stabilizers on the Brumby UAV the sweep angle is 13.99 degrees and is assigned to the variable SWEEP. To assign the reference edge to be the trailing edge, assign STA the value of 1.0. The Brumby UAV vertical stabilizers use a symmetric airfoil section with an airfoil thickness of 20% , located 30% aft from the leading edge. This is represented using a NACA 4 series airfoil as NACA 0320. To assign FINSET2, the user inputs the following control card NACA 2-4-0320. 79

Table 3.4: Brumby UAV Twin Vertical Tail Planform Definition (FINSET2)

Variable Name Default Value Units XLE 1.57 Meters CHORD 0.4000,0.1952 Meters SSPAN 0.2660,0.6790 Meters CFOC 0.2600,0.5328 N/A NPANEL 2.0 N/A PHIF 90.00,270.00 Degrees GAM -90.00,90.00 Degrees SECTYPE NACA N/A STA 1.0 N/A SWEEP 13.99 Degrees

Figure 3.3: Brumby UAV Vertical Planform 80

3.5 Control Surfaces

Traditional aircraft have control surfaces located on the planforms. The wing planform typically contains ailerons that create rolling moments about the x-axis. The

Horizontal stabilizer either has a movable section or is completely movable where the control surface is defined as the elevator. The elevator is used to create pitching moments about the y-axis, as well as to trim the vehicle longitudinally. The vertical stabilizer has a control surface defined as the rudder, which creates yawing moments about the aircraft z-axis.

The Brumby UAV has a delta wing planform with two control surfaces on each panel, as well as a control surface on each of the twin vertical tails. The delta wing has ailerons that create rolling moments as well as elevators that create pitching moments.

Missile DATCOM does not allow for multiple control surfaces on a fin. In order to accomplish a similar control scheme using a single control surface, the elevator and aileron displacement inputs must be geometrically summed together. This configuration of control surfaces that can create both rolling and pitching moments are defined as

Elevons. When the wing planform control surfaces are deflected equally up or down, then the control surfaces contribute to the pitching moment, similar to elevators. When the control surfaces on the wing panels are deflected an equal distance in opposite directions, the control surfaces contribute to the rolling moment, similar to ailerons. By combining the elevator and aileron deflection angles for each wing panel control surface we can accomplish simultaneous aileron and elevator control.

The size of the control surfaces are defined in Missile DATCOM using the flap 81 chord length to chord length ratio CFOC.

Flap Chord Length CFOC = CHORD (3.1)

The wing planform is defined as FINSET1, and the the twin vertical stabilizers will be defined as FINSET2. Notice that the control surface of the twin vertical stabilizers extends over the length of the panel span, while the control surface on the wing planform extends only over a portion of the span length.

In order to create aerodynamic data over the range of control surface deflection values, the model must be run for each deflection angle of each control surface. This is accomplished by saving the geometric model definition from previous cases and only changing the deflection values for the control surface under consideration, and by setting the control surface deflection angles to 0.0 for the base case. Invoking the SAVE control card retains the previous variable definitions so the values can be used during the execution of the next case. As an example, consider a base case where the SAVE control card is included before the NEXT CASE control card. The namelist DEFLCT contains a variable for each finset. Since the wing has been defined as FINSET1, the deflection angle for the starboard panel is contained in the first element of the array DELTA1. The

Second element is the port panel deflection angle, which will be set to zero for this case, because the base model values will be used in the preceding case, the SAVE control card must be used before the NEXT CASE control card. In subsequent cases the aerodynamic data will be computed for the same model with different values of side-slip angle, altitude, and control surface deflection angles. 82

CASEID Brumby DAMP PLOT DIM M DERIV RAD $FLTCON NMACH=1.0,ALT=12*0.,NALPHA=15.0, MACH = 0.05,0.08,0.10,0.15, ALT = 0.00,0.00,0.00,0.00, ALPHA = -5.00,0.00,2.00,4.00,6.00, ALPHA(6)=8.00,10.00,12.00,14.00,16.00, ALPHA(11)=18.00,20.00,25.00,30.00,35.00, BETA=0.,$ $REFQ SREF=1.251700,LREF=0.634700,LATREF=2.324000,XCG=0.85,ZCG=-0.04,$ $AXIBOD TNOSE=OGIVE,LNOSE=0.1970,DNOSE=0.1524,LCENTR=1.7730,DCENTR=0.1524,$ $FINSET1 SECTYP=NACA, SSPAN=0.0000,0.2660,1.0096, CHORD=1.0033,0.8048,0.2500, XLE=0.97, NPANEL=2., PHIF=90.00,270.00, GAM=0.00,0.00, CFOC=0.0000,0.1553,0.5000,$ NACA-1-4-1310 $FINSET2 SECTYP=NACA, SSPAN=0.2660,0.6790, CHORD=0.1040,0.1040, XLE=1.57, CFOC=1.0000,1.0000, STA=1., SWEEP=13.99, NPANEL=2., PHIF=90.00,-90.00, GAM=-90.00,90.00,$ NACA-2-4-0020 SAVE NEXT CASE

Figure 3.4: Brumby UAV Vehicle Description Case for005.dat File 83

CASEID WING FLAPS $DEFLCT DELTA1=-45.00,0.,$ SAVE NEXT CASE $DEFLCT DELTA1=-35.00,0.,$ SAVE NEXT CASE $DEFLCT DELTA1=-25.00,0.,$ SAVE NEXT CASE $DEFLCT DELTA1=-15.00,0.,$ SAVE NEXT CASE $DEFLCT DELTA1=-5.00,0.,$ SAVE NEXT CASE $DEFLCT DELTA1=5.00,0.,$ SAVE NEXT CASE $DEFLCT DELTA1=15.00,0.,$ SAVE NEXT CASE $DEFLCT DELTA1=25.00,0.,$ SAVE NEXT CASE $DEFLCT DELTA1=35.00,0.,$ SAVE NEXT CASE $DEFLCT DELTA1=45.00,0.,$ SAVE NEXT CASE $DEFLCT DELTA1=0.0,45.00,$ SAVE NEXT CASE $DEFLCT DELTA1=0.0,35.00,$ SAVE NEXT CASE $DEFLCT DELTA1=0.0,25.00,$ SAVE NEXT CASE $DEFLCT DELTA1=0.0,15.00,$ SAVE NEXT CASE $DEFLCT DELTA1=0.0,5.00,$ SAVE NEXT CASE $DEFLCT DELTA1=0.0,-5.00,$ SAVE NEXT CASE $DEFLCT DELTA1=0.0,-15.00,$ SAVE NEXT CASE $DEFLCT DELTA1=0.0,-25.00,$ SAVE NEXT CASE $DEFLCT DELTA1=0.0,-35.00,$ SAVE NEXT CASE $DEFLCT DELTA1=0.0,-45.00,$ SAVE NEXT CASE

Figure 3.5: Brumby UAV Wing Control Deflection Cases for005.dat File 84

CASEID RIGHT RUDDER $DEFLCT DELTA1=0.,0.,$ $DEFLCT DELTA2=-25.00,0.0,$ SAVE NEXT CASE $DEFLCT DELTA2=-15.00,0.0,$ SAVE NEXT CASE $DEFLCT DELTA2=-5.00,0.0,$ SAVE NEXT CASE $DEFLCT DELTA2=5.00,0.0,$ SAVE NEXT CASE $DEFLCT DELTA2=15.00,0.0,$ SAVE NEXT CASE $DEFLCT DELTA2=25.00,0.0,$ SAVE NEXT CASE $DEFLCT DELTA2=0.0,-25.00,$ SAVE NEXT CASE $DEFLCT DELTA2=0.0,-15.00,$ SAVE NEXT CASE $DEFLCT DELTA2=0.0,-5.00,$ SAVE NEXT CASE $DEFLCT DELTA2=0.0,5.00,$ SAVE NEXT CASE $DEFLCT DELTA2=0.0,15.00,$ SAVE NEXT CASE $DEFLCT DELTA2=0.0,25.00,$ SAVE NEXT CASE

Figure 3.6: Brumby UAV Twin Vertical Tail Control Deflection Cases for005.dat

File 85

$DEFLCT DELTA2=0.,0.,$ $FLTCON BETA=-20.,$ SAVE NEXT CASE $FLTCON BETA=-10.,$ SAVE NEXT CASE $FLTCON BETA=10.,$ SAVE NEXT CASE $FLTCON BETA=20.,$ SAVE NEXT CASE $FLTCON BETA=0.,ALT=5*100.,$ SAVE NEXT CASE $FLTCON ALT=10*100.,$ SAVE NEXT CASE

Figure 3.7: Brumby UAV Side-Slip Angle and Altitude Cases for005.dat File 86

Chapter 4: Equations of Motion and Rigid Body Modeling

This chapter will discuss the mathematical model of a rigid body. The nonlinear equations of motion for a non-rotating flat earth reference frame will be presented. The equations will be presented in a manner that they can be integrated into a simulation environment. For a derivation of the equations presented here the reader is directed to

Reference [7].

4.1 Equations of Motion for A Rigid Body

The movement of an object can be described with respect to an inertial reference frame. For three-dimensional motion this is done by defining a coordinate system in a reference frame. Multiple coordinate frames exist, however, only coordinate systems that obey the right hand rule for vector orientation will be considered.

The first coordinate system of interest is one that is defined on an “Earth Fixed

Inertial Frame” located on the earth's surface.[7] This coordinate system aligns the positive x-axis increasing in the East direction and the positive y-axis increasing in the

North direction. The positive z-axis increasing along the equatorial plane and is often abbreviated using: East, North, Up or ENU.[7]

The next coordinate system of interest is the frame with respect to the vehicle navigation. The coordinate system is defined as having the origin located on the surface of the earth. The x-axis is positive increasing toward the North direction. The y-axis is positive increasing toward the East direction. The z-axis is positive increasing Down toward the center of the Earth, in accordance with the right hand rule. This coordinate system is known as the “vehicle navigation” frame and is abbreviated as: North, East, 87

Down, or NED.[7]

The next coordinate system of interest is the coordinate frame with respect to which the vehicle's stability is defined. The coordinate system is defined as having the origin located at the center of mass of the vehicle. The positive x-axis is increasing toward the nose of the aircraft. The positive y-axis is increasing toward the starboard wing tip. The positive z-axis is increasing in accordance with the right hand rule. This is known as the “body fixed” coordinate system and is defined as the body axes.[7]

The final coordinate systems of interest are the stability axes coordinate system and the wind axes coordinate system. These coordinate systems relate the aerodynamic forces acting on the vehicle and each has its origin at the center of mass of the vehicle.

The angular difference between the body x-axis and the stability x-axis along the x-z plane, is known as the angle-of-attack (  ). The angular difference between the body x-axis and the wind x-axis along the x-y plane, is known as the side-slip angle (  ).

The z-axis always obeys the right hand rule. Figure 4.1 illustrates the stability and wind frames. 88

Figure 4.1: Aerodynamic Angles

Table 4.5: Brumby UAV Wing Planform Definition (FINSET1)

Variable Name Default Value Units XLE 0.97 Meters CHORD 1.0033,0.8048,0.2500 Meters SSPAN 0.0000,0.2660,1.0096 Meters CFOC 0.0000,0.1553,0.5000 N/A NPANEL 2.0 N/A PHIF 90.00,270.00 Degrees GAM 0.0,0.0 Degrees SECTYPE NACA N/A

Assume that there are two coordinate systems that are related by one coordinate system being rotated with respect to the other one about a parallel axis. Then let one of 89 the coordinate systems be rotated about one of the axes with respect to the other axis. Let us denote the angular difference between the two y-axes in the y-z plane as phi (  ), the angular difference between the two x-axes in the x-z plane as theta (  ), and finally the angle between the two x-axes in the x-z plane as psi (  ). These angles are the

Euler Angles.

A vector in one coordinate system (a) can be converted to another coordinate

system (b) by multiplying the vector by a Direction Cosine Matrix ( C a/ b ).

Equation 4.1 is a Direction Cosine Matrix that converts a vector from the navigation frame to the body frame.

coscos cossin  −sin  = [−              ] C b/n cos sin sin sin cos cos cos sin sin sin sin cos (4.2) sin sin cos sin cos −sin coscossin sin  cos cos

The fundamental equations of motion for the initial simulation of a vehicle can be as simple as the flat earth equations of motion such as Equations 4.3-7. Equation 4.3 shows that the Direction Cosine Matrix is a function of the Euler Angles. The derivative of position in the navigation frame is the velocity vector in the body frame converted into the position frame, Equation 4.4. The differential equation of the Euler angles is shown in

Equation 4.5. The translational accelerations are given by Equation 4.6. The rotational accelerations are given in Equation 4.7.

=  C b/n fn (4.3)

Poisson's Kinematic Equation

˙ ne = b (4.4) p C b/ n vCM/ e 90

Euler Kinematic Equation

˙ =  b (4.5) H b / e

Translational Acceleration

˙ bb =  /  b  n − b b (4.6) vCM / e 1 m F A,T C b/n g b/ e vCM / e

Rotational Acceleration

˙bb =  b −1 [ b − b b b ] (4.7) b / e J M A,T b/ e J b /e

Where:

1 tansin  tancos  = [  −  ] H 0 cos sin (4.8) 0 sin /cos cos/cos

b b = b × b b/e vCM/ e b / e vCM / e (4.9)

− J x 0 J x z b=[ ] J 0 J y 0 (4.10) − J xz 0 J z

F b =[ F F F ][F F F ] (4.11) A,T Ax A y Az T x T y T z

M b =[M M M ][M M M ] (4.12) A,T Ax A y Az T x T y T z

g n=[0 0 g] (4.13)

The vectors of interest for control and navigation purposes are the Navigation

Position Vector (Equation 4.11), the Euler Angles Vector (Equation 4.12), the

Translational Velocity Vector(Equation 4.13), and the Angular Velocity Vector

(Equation 4.14). 91

ne = [ ]T p p N pe pD (4.14)

T (4.15)  = [    ]

b = [ ]T (4.16) vCM / e UVW

b = [ ]T (4.17) b/e PQR

The matrix given in Equation 4.9 contains the moments of inertia for the vehicle = = in question. Due to symmetry in the xz-plane J XY J YX 0 , moments of inertia

about J XX , J YY , J ZZ , and J XZ are non-zero. In situations where the inertia tensor is difficult to obtain analytically, there exist experimental methods such as the pendulum method outlined by M. P. Miller .[8]

4.2 Aerodynamic Coefficients

The forces and moments of interest are taken about the aircraft's center of mass.

Drag is friction caused by the aircraft moving through the air. The air molecules move around the aircraft as it moves through the atmosphere. The molecules that cling to the surface of the aircraft create skin friction.[6] The natural texture of the surface of the aircraft is aerodynamically rough and is specified using the Roughness Height Rating

(RHR). The RHR is the arithmetic mean of the surface variation in millionths of an inch.

[4] Missile DATCOM allows the user to input the roughness factor of the vehicle surface.

The Lift force is created by both Bernoulli Lift and Vortex Lift.[6] Typically, the side force is very small in an aircraft flying in wings level steady flight with side-slip angle at or near zero. The aerodynamic forces of Lift and Drag are defined in the Stability frame. 92

The total moment acting on the aircraft is considered about the principle axes of the coordinate system. The moment about the body x-axis is known as the Rolling Moment, the moment about the body y-axis is known as the Pitching Moment, and the moment about the body z-axis is known as the Yawing Moment. The sense of the moments are defined using the right hand rule and are defined as follows. A positive Rolling Moment is one in which the pilot experiences a clockwise rotation about the x-axis. The starboard wing would be moving toward the positive z-axis and the port wing would be moving toward the negative z-axis. A positive Pitching Moment is one in which the pilot experiences the nose of the aircraft moving toward the positive z-axis and the tail of the aircraft moving toward the negative z-axis. The positive Yawing Moment is one in which the pilot experiences a clockwise rotation about the z-axis. The sense of the moments is illustrated in Figure 4.4. Aircraft moving through fluid will experience certain restoring forces, such as the vehicle to returning to a straight flight after experiencing a side-slip perturbation. This is caused by the vertical stabilizer and is known as weather veining.

These restoring forces are represented as damping derivatives. Equation 4.20 shows how to dimmensionalize the non-dimmensionalized aerodynamic coefficients and derivatives.

The rate value is the rotational rate with respect to the derivative, this is either Rolling

Rate p, the Pitching Rate q, or the Yawing Rate r. The constant k is either the wing span length b in the case of roll and yaw rates or the mean aerodynamic chord c with respect to the pitch rate. The damping derivative coefficients of interest are typically:

C Rolling Moment with respect to Roll Rate l p ,

C Pitching Moment with respect to Pitch Rate mq , 93

C Yawing Moment with respect to Yaw Rate nr ,

C Rolling Moment with respect to Yaw Rate l r ,

C Yawing Moment with respect to Roll Rate n p ,

C Lift Force with respect to Pitch Rate Lr ,

C Side Force with respect to Roll Rate Y P ,

C Side Force with respect to Yaw Rate Y r .

There are also derivatives of the force and moment coefficients with respect to the various control surfaces. These are typically:

C q Pitching Moment with respect to Elevator Deflection Angle ele ,

C Lift Force with respect to Elevator Deflection Angle L , ele

C l  Rolling Moment with respect to Aileron Deflection Angle ail ,

C n Yawing Moment with respect to Aileron Deflection Angle ail ,

C Rolling Moment with respect to Rudder Deflection Angle l  , rud

C Yawing Moment with respect to Rudder Deflection Angle n . rud

There also are derivatives for force and moment coefficients with respect to changes in

Mach, altitude, and thrust.

The aerodynamic forces, moments, and derivative coefficients are non- dimensionalized so that aerodynamic data for an aircraft is scaled from the coefficients.

This allows data taken from models in wind tunnels to be used on full scale aircraft. The equations used to create dimensionalized forces, moments, and derivatives are given in 94

Equations 4.17 – 23. Missile DATCOM provides force and moment coefficients for each

Mach and Alpha pair specified in the FLTCON namelist. Missile DATCOM only provides coefficients for the dynamic derivatives over the Alpha range specified. The

Drag, Lift, and Cross-Wind Forces are projected onto the body frame of the vehicle using the Direction Cosine Matrix given in Equation 4.36.

(a) (b)

Figure 4.2: Lift Coefficient (a) and Drag Coefficient (b)

The lift and drag force coefficients are plotted in Figure 4.2 and the force and moment coefficients in the body frame are plotted in Figure 4.3 for the Brumby UAV with zero control surface deflection angles. If there aircraft is in straight and level flight equilibrium then the longitudinal and lateral coefficients are be decoupled. For straight level flight the lateral force coefficients of Side-force coefficients, Rolling and Pitching moment coefficients will be of lower magnitude than the longitudinal force coefficients of Axial and Normal force coefficients, and pitching moment coefficient. 95

(a) (d)

(b) (e)

( c) (f)

Figure 4.3: Force(a, b, c) and Moment Coefficients(d, e, f) 96

Aerodynamic Forces

Drag Force

=  D q S C D (4.18)

Lift Force

=  L q S C L (4.19)

Side Force

=  Y q S C S (4.20)

Aerodynamic Moments

Rolling Moment

=  l W q S b C l (4.21)

Pitching Moment

=   mW q S c Cm (4.22)

Yawing Moment

=  nW q S b C n (4.23)

Dynamic Derivatives

∇ =     × k × C  C  , ,M , h , s rate (4.24) 2 V T

Where:

= 1  2  /  q V Dynamic Pressure units of force unit area (4.25) 2 T 97 b= Wing Span units of length (4.26) c= mean aerodynamic chord unitsof length (4.27)

S = Wing Area units of length2 (4.28)

 = massdensity mass /cubic volume (4.29)

=  /  V T Speed unit distance unit time (4.30)

k = dimensionless rate scale factor (4.31)

 = − −   Angle of Attack units of angle (4.32)

 = −   Side Slip Angle unitsof angle (4.33)

= M MACH (4.34) 98

=   (4.35) h Altittude units of length

 =     s Control Surface S deflection angle unitsof angle (4.36)

cos cos  sin sincos / = [−    −  ] C w b cos sin cos sin sin (4.37) −sin 0 cos

Figure 4.4: Brumby UAV Moment Definition

The total aerodynamic forces and moments acting on the vehicle are the sum of 99 the individual forces and moments. For example the total lift force acting on the vehicle is a function of Mach, Alpha, Beta, Altitude, and Control Surface deflection angles summed with the thrust forces.

The cumulative forces and moments enter into the equations of motion through

b b the vectors F A,T and M A,T . The aerodynamic force components

[ ]T F A , x F A , y F A, z are either the aerodynamic forces in the wind frame converted to

b = × [− − ]T the body frame F A,T C b /w Dw Y w Lw or already in the body frame

b = [ ]T F A,T Ab Y b N b . Figure 4.6 illustrates the Lift and Drag force vectors. Due to the coordinate frame that Missile DATCOM is using the forces in the body frame are

b = [− − ]T − defined F A,T Ab Y b N b . Where Ab is the axial force with respect to

the body and is positive increasing toward the nose along the positive body x-axis, Y b is the side force with respect to the free-stream and is positive increasing out the − starboard wing from the center of mass, N b is the normal force and is positive increasing from the center of mass along the positive body z-axis. Missile DATCOM provides these values for every Mach and Alpha point. Equations 4.37 and 4.38 are the dimensionalized Axial and Normal forces respectively, dimmensionalized Side force is listed in Equations 4.19. Figure 4.5 illustrates the Axial and Normal force vectors.

Axial Force

=  A q S C A (4.38)

Normal Force 100

=  N q S C N (4.39)

Figure 4.5: Axial and Normal Forces

Figure 4.6: Lift and Drag Forces 101

4.3 Six Degree-of-Freedom Aircraft Model

A nonlinear six-degrees of freedom model was created using the flat earth nonlinear equations of motion 4.3–7 . Since the variables of interest are only available through integration of the nonlinear equations, one could linearize the nonlinear model about an equilibrium point and represent the linearized system using a state transition matrix typically used in state-space control theory. This, however, can be very difficult and tedious, considering that the state transition matrix would have to be recalculated due to the changing nonlinear time-varying aerodynamic contribution beyond the allowable deviation from the equilibrium point. A better method would be to create a non-linear model in The Mathwork's Matlab and Simulink environments. This allows the nonlinear model to be created in the Simulink environment and programmed as an s-function. The benefit of using an s-function is that, by the use of flags Simulink will integrate and keep track of the state variables. The state variable vector given in Equation 4.39 is the position vector in the navigation frame pne T , Euler Angle Vector T , Translational

b T Velocity Vector in the body frame vCM / e , and Angular Velocity Vector in the body

b frame b/e .

T = [ ne T T b T b T ] X p vCM/ e b/ e (4.40)

The forces and moments acting on the vehicle enter the equations of motion as

b the Force Vector in the body frame F A,T and the Moment Vector in the body frame

b M A,T . The forces and moments are the sum of the aerodynamic contribution, denoted 102 with a subscript A, and the thrust contribution, denoted with a subscript T. The Thrust force vector is composed of the forces acting on the center of mass in the body frame.

The total force equation is given in Equation 4.40 and the total moments equation is given in Equation 4.41.

b = [ ]T  [ ]T F A,T F A , x F A , y F A , z F T , x F T , y F T , z (4.41)

b = [ ]T  [ ]T M A,T l A, b mA ,b nA, b l T ,b mT , b nT ,b (4.42) 103

Chapter 5: Simulation

In this chapter a nonlinear aircraft model is developed using The Mathwork's

Matlab and Simulink environments. The nonlinear model has the aerodynamics trimmed around an equilibrium point and then a linearized model is created. The linearized model is used to analyze the static and dynamic stability of the model.

5.1 Simulink Nonlinear Aircraft Model

The nonlinear aircraft model is implemented as a Simulink model. The model uses an s-function to perform the equations-of-motion calculations and Matlab functions execute DATCOMTableMex.dll to perform the linear interpolation on the Missile

DATCOM for0021.dat data file. The model, shown in Figure 5.1, allows the user to input the gravitational acceleration, inertia matrix, initial conditions, as well as input values for the control surfaces.

Figure 5.1: Simulink Nonlinear Aircraft Model 104

Table 5.1: Brumby UAV Mass Properties

Mass Properties of the Brumby UAV Values Units Mass 22.8543 Kg Moment of Inertia (Jxx) 2.41583571804 Kg*m^2 Moment of Inertia (Jyy) 21.973713217110 kg*m^2 Moment of Inertia (Jzz) 23.942135938328 kg*m^2 Moment of Inertial (Jxz) -0.16090180279 kg*m^2 Gravity Constant 9.81 m/s^2

The components of the Brumby UAV including the airframe, power plant, as well as the onboard instrumentation were treated as point masses and used to calculate the location of the center-of-mass and the inertia matrix values. Table 5.2 contains the mass properties of the Brumby UAV that were calculated by Sean Calhoun.[1]

The s-function requires the inertia matrix values, the gravity constant, current time step, and the initial state vector as function inputs. Execution of the s-function with the appropriate flags is controlled by the Simulink environment. The s-function provides the following functionality shown in Table 5.2. There are other flags which are not used in this simulation, and therefore will not be discussed.

Table 5.2: S-function Functionality

Flag Value Output 0 Initialization of state vector 1 Calculate derivatives at current time 3 Output current state values 105

The Matlab function that calculates the forces and moments acting on the aircraft perform several important tasks. The inputs to the function are the state vector and the control input values. The function tests the input values to see if the control surface deflections are within the physical tolerances of the full scale aircraft. After an aircraft specification has been created in the for005.dat file, the user must execute Missile

DATCOM to create the for021.dat file. The Matlab function is used to calculate the forces and moments is a wrapper function for DATCOMTableMex.dll.

DATCOMTableMex.dll requires the for021.dat file be read and the aerodynamic coefficients be stored in random access memory. Storing the aerodynamic data in memory is accomplished by executing DATCOMTableMEX with the inputs being a flag of 1 and the for021.dat filename. DATCOMTableMex.dll will return the table identification number that signifies the location of the data in random access memory.

DATCOMTableMex accesses the aerodynamic coefficients stored in memory when executed with a flag of 2. Executing DATCOMTableMex with inputs: a flag of 2, table identification number, angle-of-attack in degrees, Mach value, altitude in units of length, side slip value in degrees, control surface deflection values in degrees returns the following outputs: the incremental contribution to the aerodynamic coefficients, stability derivatives, and base aerodynamic coefficients for the aircraft model with zero control surface deflection. These coefficients must be dimensionalized by using the equations defined in Chapter 4. To overcome the complexity of calling DATCOMTableMex and then dimensionalizing the aerodynamic forces and moments from the Simlulink model environment a driver function was written. The Matlab driver function was defined as 106 forces_moments.m and outputs the force and moments in the body frame.

Forces_moments.m requires the state vector and the control input values as inputs. The function then proceeds to calculate the aerodynamic angle-of-attack, side slip, and Mach values which are inputs needed when datcomderive.m is called. The driver function datcomderive.m then executes DATCOMTableMex.dll with the appropriate inputs. The aerodynamic forces and moments returned by datcomderive.m are added to the thrust forces and moments to create the total forces and moments that are acting on the aircraft with respect to the body frame.

Table 5.3: DATCOMTableMex.dll Functionality

Flag Function Definition 1 tableID = DATCOMTableMex(flag,filename) 2 [DepDeltaIncrements, Derivatives_Stab, DepBaseIncrements] =DATCOMTableMex(flag,tableID,IndVariables) 4 DATCOMTableMex(flag)

Where,

filename - 'for021.dat' tableID - pointer to data table in memory deltadeg - [Starboard Ailevon Deflection Angle,Port Ailevon Deflection Angle,Starboard Rudder Deflection Angle,Port Rudder Deflection Angle,0,0] IndVariables - [Angle-of-Attack  , MACH, altitude (-Z), SideSlip Angle  , deltadeg] DepDeltaIncrements - Incremental Control Surface Forces and Moments Contributions Derivatives_Stab - Stability Derivatives DepBaseIncrements - Vehicle with zero control surfaces deflection angles Force and Monents Contributions

Execution of datcomderive.m requires the user to input the angle-of-attack in degrees, the side slip value in degrees, the altitude, control surface deflection angle vector, Mach, the angular velocity vector, table identification number, the lateral reference length, the longitudinal reference length, the reference area, the speed of sound, and the fluid density. Both forces_moments.m and datcomderive.m are included 107 in the Appendix .

5.2 Trimmed Aircraft Flight

The simulation was trimmed for straight and level flight using the Simulink Trim command. The trimmed control input values are given in Table 5.4. The values for the trimmed initial conditions are given in Table 5.5. The control surfaces on the aircraft are deflected such that the forces and moments on the aircraft are in equilibrium. The translational and rotational accelerations on the aircraft are zero. This condition is known as trimming the aircraft. Typically, this is performed for wings level straight and steady flight. For a trimmed aircraft the translational velocity derivatives, rotational velocity derivatives, and the derivatives of roll and pitch Euler angles are zero. The velocity component along the Body x-axis velocity (U) and the velocity component along the

 Body z-axis (W), the Euler Angle Theta ( ), and the Down position ( P Z ) are

allowed to have non zero constant values. East position ( P X ) and North position (

P Y ) are allowed to vary with time, while all other state variables must maintain values of zero.

Table 5.4: Brumby UAV Control Input Trimmed Values (Case 1)

Control Inputs Values Units Elevator -20.9891 Degrees Aileron 0.0000 Degrees Rudder 0.0000 Degrees Thrust Force 80.4064 Newtons 108

Table 5.5: Brumby UAV State Variables Initial Condition Values (Case 1)

State Vector Initial Conditions Values Units

Navigation East Position ( P X ) 0.0000 meters

Navigation East Position ( PY ) 0.0000 meters

Navigation East Position ( PZ ) 0.0000 meters Euler Angle (  ) 0.0000 radians Euler Angle (  ) 0.1087 radians Euler Angle (  ) 0.0000 radians Translational Velocity (U) 90.3775 meters/second Translational Velocity (V) 0.0000 meters/second Translational Velocity (W) 9.8658 meters/second Angular Velocity (p) 0.0000 radians/second Angular Velocity (q) 0.0000 radians/second Angular Velocity (r) 0.0000 radians/second

Table 5.6: Brumby UAV Trimmed Aerodynamic Values (Case 1)

Aerodynamic Values of Trimmed Values Units Condition angle-of-attack (  ) 6.2299 degrees Side-slip Angle (  ) 0.0000 degrees Speed (S) 90.9144 meters / second

5.3 Linearized Aircraft Model

The model was linearized around this equilibrium point using the Simulink command linmod. The state-space equation is defined in Equation 5.1 with state vector x defined in Equation 5.2, and input vector given in Equation 5.3, and the output vector 109 given in Equation 5.4. The state-space representation coefficient matrices A, B, C and D are listed as Equations 5.5-8. ˙ X = Ax  Bu (5.1) Y = Cx  Du

T = [ ∇ ne T ∇ T ∇ b T ∇ b T ] x p vCM /e b/ e (5.2)

= [ ∇ ∇  ∇  ∇  ]T u ail ele rud thrust (5.3)

T = [ ∇ ne T ∇ T ∇ b T ∇ b T ] Y p vCM / e b /e (5.4)

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.9941 0.0000 0.1085 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 −9.8658 0.0000 90.9144 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 −90.9144 0.0000 −0.1085 0.0000 0.9941 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.1092 0.0000 0.0000 0.0000 −0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 A = 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0059 [0.0000 0.0000 0.0000 0.0000 −9.7521 0.0000 −0.2003 0.0000 1.3375 0.0000 −9.8104 0.0000 ] (5.5) 0.0000 0.0000 0.0000 9.7521 0.0000 0.0000 −0.0000 −2.5025 0.0000 12.4164 0.0000 −79.6670 0.0000 0.0000 0.0000 0.0000 −1.0646 0.0000 0.9197 0.0000 −10.4017 0.0000 89.8141 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 −0.0000 −5.0595 0.0000 −20.1594 0.0000 19.4863 0.0000 0.0000 −0.0000 0.0000 0.0000 0.0000 0.0733 −0.0000 −0.6712 0.0000 −1.5359 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 11.0800 −0.0000 −12.0662 0.0000 −52.4232

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 B = 0.0000 0.0000 0.0000 0.0000 [ 0.0000 0.1407 0.0296 0.0438] (5.6) −0.9344 0.0000 −0.8710 0.0000 0.0000 −1.7621 0.0000 0.0000 4.4417 0.0000 −1.5911 0.0000 0.0000 −0.1178 −0.0011 0.0000 4.4318 −0.0000 4.1784 0.0000

1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 C = 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 [0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000] (5.7) 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 110

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 = 0.0000 0.0000 0.0000 0.0000 D 0.0000 0.0000 0.0000 0.0000 (5.8) [0.0000 0.0000 0.0000 0.0000] 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

Eigenvalues of the state differential equation A matrix are given in Equation 5.9.

The position vector in the navigation frame pne T , was removed from the A coefficient matrix whose eigenvalues are listed in Equation 5.9. The position vector in the navigation frame is not needed for the stability analysis that is being performed in this chapter.

0 −27.388323.5888i −27.3883−23.5888i  = −6.03156.3905i [ −6.0315−6.3905i ] −20.3019 (5.9) −0.03750.1280i −0.0375−0.1280i −0.0067

−0.03750.1280i  = [−0.0375−0.1280i] Longitudinal −6.03156.3905i (5.10) −6.0315−6.3905i

0  = −0.0067 [−27.388323.5888i] Lateral (5.11) −27.3883−23.5888i −20.3019

For Bounded-Input Bounded-Output (BIBO) stability the non-zero eigenvalues of the A Coefficient Matrix must contain only negative real parts. [9] Bounded-Input

Bounded-Output stability represents longitudinal and lateral stability in aircraft.

Equation 5.10 contains the longitudinal eigenvalues for the state variables

 ,U ,W ,q , and Equation 5.12 relates the eigenvalues to the longitudinal dynamics of   the aircraft. The lateral eigenvalues for the state variables , ,V , p ,q , associated 111 with the lateral dynamics are listed in Equation 5.11 and an explanation of the eigenvalues effect on the lateral dynamics is listed in Equation 5.13. The explanation of the eigenvalues includes the period of natural oscillation T  and the damping ratio

  for complex conjugate pairs and the time constant   for real and distinct eigenvalues.

−0.0375±0.1280i Phugoid Mode , T = 47.1060 s , =0.2811 (5.12) −6.0315±6.3905i Short−Period Mode , T = 0.7150 s , =0.6864

The short-period mode is the natural mode of the aircraft and is the transient response in the longitudinal direction. Once the short-period mode has decayed the aircraft experiences a very lightly damped oscillation known as the phugoid mode.[7]

−27.3883±23.5888i  Dutch Roll Mode , T = 0.1738 s , =0.7577 −20.3019 Roll Subsidence Mode ,  = 0.0493 s (5.13)

−0.0067 Spiral Mode ,  = 148.8067 s

The dutch roll mode of the aircraft consists of rolling and yawing motion with some side-slip and is similar to the motion of a drunken ice skater. The Brumby has a dutch roll mode period of 0.1738 seconds and a damping ratio of 0.7577. The dutch roll mode period for the Brumby is very short but highly damped. The roll subsidence mode gives an indication of the time required for the rolling moment control inputs create the rolling moment. The Brumby has a quick roll response at 0.0493 seconds. The spiral mode of the aircraft is the time lapse before the aircraft to go into a downward spiral with no control input correction. [7]

The Brumby was also trimmed for a coordinated turn with a constant rate of climb. The turn rate used for this trim condition is 0.1 radians per second, and the rate of 112 climb is 0.5 meters per second. The turn rate was chosen so that the centripetal acceleration on the aircraft would be less than 0.5 times the force of gravity during the turn.

Table 5.7: Brumby UAV Control Input Trimmed Values (Case 2)

Control Inputs Values Units Elevator -42.0978 Degrees Aileron -0.0174 Degrees Rudder 0.2862 Degrees Thrust Force 92.9212 Newtons

5.4 Nonlinear Simulation Results

Nonlinear simulation was performed on the trimmed aircraft model, and the state variables are plotted in this section. Figures 5.2–9 show the Brumby UAV trimmed for straight and level flight (SLF). The reader should note that the Euler angle  is obscured by the Euler angle  in Figure 5.3. This demonstrates the aircraft in a cruise maneuver, such as when flying from one way point to another . Nonlinear simulation results for the Brumby UAV trimmed for a coordinated turn with a constant rate of climb

(CTROC) are shown in Figures 5.10–17. 113

Figure 5.2: Navigation Position Output (SLF)

Figure 5.3: Euler Angles Output (SLF) 114

Figure 5.4: Translational Velocities Output (SLF)

Figure 5.5: Angular Velocities Output (SLF) 115

Figure 5.6: Velocity Magnitude Output (SLF)

Figure 5.7: Aerodynamic Angles Output (SLF) 116

Figure 5.8: Flight-Path Angle Output (SLF)

Figure 5.9: Rate-of-Climb Output (SLF) 117

Figure 5.10: Navigation Position Ground Track Output (SLF)

Figure 5.11: Navigation Position 3-Dimensional Output (SLF) 118

Figure 5.12: Navigation Position Output (CTROC)

Figure 5.13: Euler Angles Output (CTROC) 119

Figure 5.14: Translational Velocities Output (CTROC)

Figure 5.15: Angular Velocities Output (CTROC) 120

Figure 5.16: Velocity Magnitude Output (CTROC)

Figure 5.17: Aerodynamic Angles Output (CTROC) 121

Figure 5.18: Flight-Path Angle Output (CTROC)

Figure 5.19: Rate-of-Climb Output (CTROC) 122

Figure 5.20: Navigation Position Ground Track Output (CTROC)

Figure 5.21: Navigation Position 3-Dimensional Output (CTROC) 123

5.4 Control Surface Doublet Simulation Results

The aircraft model will now be subjected to perturbations about the trimmed equilibrium point. A doublet is composed of a positive displacement immediately followed by a negative displacement with equal magnitude. The doublet differs from a step input in that, a doublet has a finite duration and returns to the initial value. The positive displacement must be identical to the negative displacement in both magnitude and duration. Because the input is returned to the trimmed input value the net effect of the doublet on the steady-state output is zero. The first trim condition is that of straight and level flight and the input values are listed in Table 5.4. The second flight condition is that of the coordinated turn with a constant rate of climb and the input values are listed in

Table 5.7 The Brumby UAV model will be subjected to the similar control surface doublets as those presented in Reference [1]. Figures 5.18–26 shows input perturbations for the Brumby UAV trimmed for straight and level flight (SLF). Results for the input perturbations to the Brumby UAV trimmed for a coordinated turn with a constant rate of climb (CTROC) are shown in Figures 5.27–35. 124

Table 5.8: Brumby UAV Control Effector Doublet Values

Control Effector Values Units Time (s) Elevator Positive Displacement Trim Value + 0.01 Radians 7 Elevator Negative Displacement Trim Value - 0.01 Radians 9 Elevator Return to Trim Trim Value Radians 11 Aileron Positive Displacement Trim Value + 0.1 Radians 133 Aileron Negative Displacement Trim Value - 0.1 Radians 135 Aileron Return to Trim Trim Value Radians 137 Rudder Positive Displacement Trim Value + 0.01 Radians 261 Rudder Positive Displacement Trim Value - 0.01 Radians 263 Rudder Return to Trim Trim Value Radians 265 Thrust Force Positive Trim Value + 5.0 newtons 433 Displacement Thrust Force Negative Trim Value – 5.0 newtons 435 Displacement Thrust Force Return to Trim Trim Value newtons 437 125

Figure 5.22: Doublet Response Navigation Position Output (SLF)

Figure 5.23: Doublet Response Euler Angles Output(SLF) 126

Figure 5.24: Doublet Response Translational Velocities Output(SLF)

Figure 5.25: Doublet Response Angular Velocities Output (SLF) 127

Figure 5.26: Doublet Response Velocity Magnitude Output (SLF)

Figure 5.27: Doublet Response Aerodynamic Angles Output (SLF) 128

Figure 5.28: Flight-Path Angle Output (SLF)

Figure 5.29: Rate-of-Climb Output (SLF) 129

Figure 5.30: Doublet Response Navigation Ground Track Output (SLF)

Figure 5.31: Doublet Response Navigation 3-Dimensional Output (SLF) 130

Figure 5.32: Control Surface Deflection Input Angles (SLF)

Figure 5.33: Aerodynamic Control Surface Deflection Input Angles (SLF) 131

Figure 5.34: Doublet Response Navigation Position Output (CTROC)

Figure 5.35: Doublet Response Euler Angles Output (CTROC) 132

Figure 5.36: Doublet Response Translational Velocities Output (CTROC)

Figure 5.37: Doublet Response Angular Velocities Output (CTROC) 133

Figure 5.38: Doublet Response Velocity Magnitude Output (CTROC)

Figure 5.39: Doublet Response Aerodynamic Angles Output (CTROC) 134

Figure 5.40: Flight-Path Angle Output (CTROC)

Figure 5.41: Rate-of-Climb Output (CTROC) 135

Figure 5.42: Doublet Response Navigation Ground Track Output (CTROC)

Figure 5.43: Doublet Response Navigation 3-Dimensional Output (CTROC) 136

Figure 5.44: Control Surface Deflection Input Angles (CTROC)

Figure 5.45: Aerodynamic Control Surface Deflection Input Angles (CTROC) 137

In Figures 5.23-26 and Figures 5.27-35 the Brumby UAV returns to the trimmed equilibrium point after the doublet perturbation is applied. The model's ability to return to the equilibrium point illustrates that the model is statically stable as well as dynamically stable. Dynamic stability is defined as the time-dependent behavior of the aircraft being stable in response to an impulsive input.[7] Once perturbed from the equilibrium point the aircraft will return to the equilibrium point some time after the perturbation is applied.

The model presented in this thesis has been shown to be stable. In Calhoun's Thesis the aerodynamic model created from time sampled data was shown to be unstable. The model presented here has had the center of mass chosen so that it creates a longitudinal statically stable aircraft. The benefit of using computational fluid dynamic prediction codes is that the center-of-pressure of the lifting surface can be determined and the point masses located in such a manner as to induce static stability. 138

Chapter 6: Conclusions and Future Work

This research has presented a six degree-of-freedom model of the Brumby UAV using a computational fluid dynamic prediction code. The time history simulations show that the trimmed nonlinear model is both longitudinally and laterally stable. The Brumby

UAV returns to the trimmed condition after the perturbation. This is just the first step in creating a flight control system to be implemented on the physical vehicle.

The model must first be validated against flight test data to ensure that the model is an adequate approximation of the physical model. The center of gravity of the aircraft and the Inertia Tensor need to be recalculated. The Brumby UAV elevator control surfaces are located on the lifting planform. If the center of gravity does not lie forward of the center of pressure, then the center of gravity will create a negative moment about the center of pressure. In order to cancel out this moment traditional aircraft use the elevator control surface located on the horizontal stabilizer. If the center of gravity is forward of the center of pressure then the elevator would need to apply a positive moment. This acts as a spoiler on the lifting planform of the Brumby UAV, decreasing the lift coefficient, increasing the drag coefficient, and inducing a positive pitching moment. If the center of gravity is aft of the center of pressure then the elevators must provide negative pitching moment. This would act as an additional lifting surface on the

Brumby UAV which would increase the amount of lift as well as increasing the amount of drag, and inducing a positive pitching moment. The Brumby UAV may not have enough control authority to correct for extreme misalignment between the center of gravity and the center of pressure without introducing instability in the aerodynamic 139 forces and moments. The placement of components in the Brumby UAV should be performed with consideration of the center of pressure. It may be possible for a human pilot to counter act the natural instability of the aircraft induced by misaligned center of gravity and center of pressure. The inertia matrix can be determined using the method outlined by Miller in Reference [8].

The compensation scheme should include a state feedback loop as well as an observer. The state feedback gains as well as the observer gains should be gain scheduled. Gain Scheduling requires a finite set of feedback gains whose values are valid only over a defined flight condition. This type of controller requires the least amount of processor time or memory. [9]

The model and associated compensation scheme must be validated by simulation of disturbance inputs. Disturbances should include responses to control surface failures, wind gusts, and the power plant perturbations. The Simulink model described in Chapter

5 should be considered as a starting point in the simulation of perturbations. Failure modes that should be explored are effectors that are seized or that have become disconnected from the drive mechanism and are free to move, or a combination thereof.

140

References

[1] Calhoun, S.M., Six Degree-of-Freedom Modeling on an Uninhabited Aerial

Vehicle, Thesis: Ohio University, 2006.

[2] McDonnell Douglas Corporation, USAF Stability and Control DATCOM, 1960.

[3] McDonnell Douglas Corporation, The USAF Stability and Control Digital

DATCOM, 1979.

[4] United States Air Force, Missile DATCOM User's Manual, 1997.

[5] Abbott, I. H., A. E. Doenhoff, Theory of Wing Sections, New York: Dover, 1958.

[6] Anderson, J. D. , Fundamentals of Aerodynamics, New York: McGraw-Hill, 2001.

[7] Stevens, B. L., F. L. Lewis, Aircraft Control and Simulation, New York: Wiley,

2003.

[8] Miller, M. P., An Accurate Method of Measuring the Moments of Inertia of

Airplanes, 1930.

[9] Williams, R. L., D. A. Lawrence, Linear State-Space Control Systems, New York:

Wiley, 2007. 141

Appendix A.1: for005.dat File

CASEID Brumby DAMP PLOT DIM M DERIV RAD $FLTCON NMACH=1.0,ALT=12*0.,NALPHA=15.0, MACH = 0.05,0.08,0.10,0.15, ALT = 0.00,0.00,0.00,0.00, ALPHA = -5.00,0.00,2.00,4.00,6.00, ALPHA(6)=8.00,10.00,12.00,14.00,16.00, ALPHA(11)=18.00,20.00,25.00,30.00,35.00, BETA=0.,$ $REFQ SREF=1.251700,LREF=0.634700,LATREF=2.324000,XCG=0.85,ZCG=-0.04,$ $AXIBOD TNOSE=OGIVE,LNOSE=0.1970,DNOSE=0.1524,LCENTR=1.7730,DCENTR=0.1524, $ $FINSET1 SECTYP=NACA, SSPAN=0.0000,0.2660,1.0096, CHORD=1.0033,0.8048,0.2500, XLE=0.97, NPANEL=2., PHIF=90.00,270.00, GAM=0.00,0.00, CFOC=0.0000,0.1553,0.5000,$ NACA-1-4-1310 $FINSET2 SECTYP=NACA, SSPAN=0.2660,0.6790, CHORD=0.1040,0.1040, XLE=5.66, CFOC=1.0000,1.0000, STA=1., SWEEP=13.99, NPANEL=2., PHIF=90.00,-90.00, GAM=-90.00,90.00,$ NACA-2-4-0020 SAVE NEXT CASE CASEID WING FLAPS $DEFLCT DELTA1=-45.00,0.,$ SAVE NEXT CASE 142

$DEFLCT DELTA1=-35.00,0.,$ SAVE NEXT CASE $DEFLCT DELTA1=-25.00,0.,$ SAVE NEXT CASE $DEFLCT DELTA1=-15.00,0.,$ SAVE NEXT CASE $DEFLCT DELTA1=-5.00,0.,$ SAVE NEXT CASE $DEFLCT DELTA1=5.00,0.,$ SAVE NEXT CASE $DEFLCT DELTA1=15.00,0.,$ SAVE NEXT CASE $DEFLCT DELTA1=25.00,0.,$ SAVE NEXT CASE $DEFLCT DELTA1=35.00,0.,$ SAVE NEXT CASE $DEFLCT DELTA1=45.00,0.,$ SAVE NEXT CASE $DEFLCT DELTA1=0.0,45.00,$ SAVE NEXT CASE $DEFLCT DELTA1=0.0,35.00,$ SAVE NEXT CASE $DEFLCT DELTA1=0.0,25.00,$ SAVE NEXT CASE $DEFLCT DELTA1=0.0,15.00,$ SAVE NEXT CASE $DEFLCT DELTA1=0.0,5.00,$ SAVE NEXT CASE $DEFLCT DELTA1=0.0,-5.00,$ SAVE 143

NEXT CASE $DEFLCT DELTA1=0.0,-15.00,$ SAVE NEXT CASE $DEFLCT DELTA1=0.0,-25.00,$ SAVE NEXT CASE $DEFLCT DELTA1=0.0,-35.00,$ SAVE NEXT CASE $DEFLCT DELTA1=0.0,-45.00,$ SAVE NEXT CASE CASEID RIGHT RUDDER $DEFLCT DELTA1=0.,0.,$ $DEFLCT DELTA2=-25.00,0.0,$ SAVE NEXT CASE $DEFLCT DELTA2=-15.00,0.0,$ SAVE NEXT CASE $DEFLCT DELTA2=-5.00,0.0,$ SAVE NEXT CASE $DEFLCT DELTA2=5.00,0.0,$ SAVE NEXT CASE $DEFLCT DELTA2=15.00,0.0,$ SAVE NEXT CASE $DEFLCT DELTA2=25.00,0.0,$ SAVE NEXT CASE $DEFLCT DELTA2=0.0,-25.00,$ SAVE NEXT CASE $DEFLCT DELTA2=0.0,-15.00,$ SAVE NEXT CASE $DEFLCT DELTA2=0.0,-5.00,$ SAVE NEXT CASE $DEFLCT DELTA2=0.0,5.00,$ SAVE 144

NEXT CASE $DEFLCT DELTA2=0.0,15.00,$ SAVE NEXT CASE $DEFLCT DELTA2=0.0,25.00,$ SAVE NEXT CASE $DEFLCT DELTA2=0.,0.,$ $FLTCON BETA=-20.,$ SAVE NEXT CASE $FLTCON BETA=-10.,$ SAVE NEXT CASE $FLTCON BETA=10.,$ SAVE NEXT CASE $FLTCON BETA=20.,$ SAVE NEXT CASE $FLTCON BETA=0.,ALT=5*100.,$ SAVE NEXT CASE $FLTCON ALT=10*100.,$ SAVE NEXT CASE 145

Appendix A.2 : Truncated for006.dat File

1 ***** THE USAF AUTOMATED MISSILE DATCOM * REV 9/02 ***** AERODYNAMIC METHODS FOR MISSILE CONFIGURATIONS

CONERR - INPUT ERROR CHECKING

ERROR CODES - N* DENOTES THE NUMBER OF OCCURENCES OF EACH ERROR A - UNKNOWN VARIABLE NAME B - MISSING EQUAL SIGN FOLLOWING VARIABLE NAME C - NON-ARRAY VARIABLE HAS AN ARRAY ELEMENTDESIGNATION - (N) D - NON-ARRAY VARIABLE HAS MULTIPLE VALUES ASSIGNED E - ASSIGNED VALUES EXCEED ARRAY DIMENSION F - SYNTAX ERROR

************************* INPUT DATA CARDS *************************

1 CASEID Brumby 2 DAMP 3 PLOT 4 DIM M 5 DERIV RAD 6 $FLTCON NMACH=1.0,ALT=12*0.,NALPHA=15.0, 7 MACH = 0.05,0.08,0.10,0.15, 8 ALT = 0.00,0.00,0.00,0.00, 9 ALPHA = -5.00,0.00,2.00,4.00,6.00, 10 ALPHA(6)=8.00,10.00,12.00,14.00,16.00, 11 ALPHA(11)=18.00,20.00,25.00,30.00,35.00, 12 BETA=0.,$ 13 $REFQ SREF=1.251700,LREF=0.634700,LATREF=2.324000,XCG=0.85,ZCG=-0.04,$ 14 $AXIBOD TNOSE=OGIVE,LNOSE=0.1970,DNOSE=0.1524,LCENTR=1.7730,DCENTR=0.1524, $ ** SUBSTITUTING NUMERIC FOR NAME OGIVE 15 $FINSET1 SECTYP=NACA, ** SUBSTITUTING NUMERIC FOR NAME NACA 16 SSPAN=0.0000,0.2660,1.0096, 17 CHORD=1.0033,0.8048,0.2500, 18 XLE=0.97, 19 NPANEL=2., 20 PHIF=90.00,270.00, 146

21 GAM=0.00,0.00, 22 CFOC=0.0000,0.1553,0.5000,$ 23 NACA-1-4-0310 24 $FINSET2 SECTYP=NACA, ** SUBSTITUTING NUMERIC FOR NAME NACA 25 SSPAN=0.2660,0.6726, 26 CHORD=0.4000,0.1952, 27 XLE=5.66, 28 CFOC=0.2600,0.5328, 29 STA=1., 30 SWEEP=13.99, 31 NPANEL=2., 32 PHIF=90.00,-90.00, 33 GAM=-90.00,90.00,$ 34 NACA-2-4-0310 35 SAVE 36 NEXT CASE 37 CASEID WING FLAPS 38 $DEFLCT DELTA1=-45.00,0.,$ 39 SAVE 40 NEXT CASE 41 $DEFLCT DELTA1=-35.00,0.,$ 42 SAVE 43 NEXT CASE 44 $DEFLCT DELTA1=-25.00,0.,$ 45 SAVE 46 NEXT CASE 47 $DEFLCT DELTA1=-15.00,0.,$ 48 SAVE 49 NEXT CASE 50 $DEFLCT DELTA1=-5.00,0.,$ 51 SAVE 52 NEXT CASE 53 $DEFLCT DELTA1=5.00,0.,$ 54 SAVE 55 NEXT CASE 56 $DEFLCT DELTA1=15.00,0.,$ 57 SAVE 58 NEXT CASE 59 $DEFLCT DELTA1=25.00,0.,$ 60 SAVE 61 NEXT CASE 62 $DEFLCT DELTA1=35.00,0.,$ 63 SAVE 147

64 NEXT CASE 65 $DEFLCT DELTA1=45.00,0.,$ 66 SAVE 67 NEXT CASE 68 $DEFLCT DELTA1=0.0,45.00,$ 69 SAVE 70 NEXT CASE 71 $DEFLCT DELTA1=0.0,35.00,$ 72 SAVE 73 NEXT CASE 74 $DEFLCT DELTA1=0.0,25.00,$ 75 SAVE 76 NEXT CASE 77 $DEFLCT DELTA1=0.0,15.00,$ 78 SAVE 79 NEXT CASE 80 $DEFLCT DELTA1=0.0,5.00,$ 81 SAVE 82 NEXT CASE 83 $DEFLCT DELTA1=0.0,-5.00,$ 84 SAVE 85 NEXT CASE 86 $DEFLCT DELTA1=0.0,-15.00,$ 87 SAVE 88 NEXT CASE 89 $DEFLCT DELTA1=0.0,-25.00,$ 90 SAVE 91 NEXT CASE 92 $DEFLCT DELTA1=0.0,-35.00,$ 93 SAVE 94 NEXT CASE 95 $DEFLCT DELTA1=0.0,-45.00,$ 96 SAVE 97 NEXT CASE 98 CASEID RIGHT RUDDER 99 $DEFLCT DELTA1=0.,0.,$ 100 $DEFLCT DELTA2=-25.00,0.0,$ 101 SAVE 102 NEXT CASE 103 $DEFLCT DELTA2=-15.00,0.0,$ 104 SAVE 105 NEXT CASE 106 $DEFLCT DELTA2=-5.00,0.0,$ 107 SAVE 148

108 NEXT CASE 109 $DEFLCT DELTA2=5.00,0.0,$ 110 SAVE 111 NEXT CASE 112 $DEFLCT DELTA2=15.00,0.0,$ 113 SAVE 114 NEXT CASE 115 $DEFLCT DELTA2=25.00,0.0,$ 116 SAVE 117 NEXT CASE 118 $DEFLCT DELTA2=0.0,-25.00,$ 119 SAVE 120 NEXT CASE 121 $DEFLCT DELTA2=0.0,-15.00,$ 122 SAVE 123 NEXT CASE 124 $DEFLCT DELTA2=0.0,-5.00,$ 125 SAVE 126 NEXT CASE 127 $DEFLCT DELTA2=0.0,5.00,$ 128 SAVE 129 NEXT CASE 130 $DEFLCT DELTA2=0.0,15.00,$ 131 SAVE 132 NEXT CASE 133 $DEFLCT DELTA2=0.0,25.00,$ 134 SAVE 135 NEXT CASE 136 $DEFLCT DELTA2=0.,0.,$ 137 $FLTCON BETA=-20.,$ 138 SAVE 139 NEXT CASE 140 $FLTCON BETA=-10.,$ 141 SAVE 142 NEXT CASE 143 $FLTCON BETA=10.,$ 144 SAVE 145 NEXT CASE 146 $FLTCON BETA=20.,$ 147 SAVE 148 NEXT CASE 149 $FLTCON BETA=0.,ALT=5*100.,$ 150 SAVE 151 NEXT CASE 149

152 $FLTCON ALT=10*100.,$ 153 SAVE NEXT CASE ** MISSING NEXT CASE CARD ADDED ** 1 ***** THE USAF AUTOMATED MISSILE DATCOM * REV 9/02 ***** CASE 1 AERODYNAMIC METHODS FOR MISSILE CONFIGURATIONS PAGE 1 CASE INPUTS FOLLOWING ARE THE CARDS INPUT FOR THIS CASE

CASEID Brumby DAMP PLOT DIM M DERIV RAD $FLTCON NMACH=1.0,ALT=12*0.,NALPHA=15.0, MACH = 0.05,0.08,0.10,0.15, ALT = 0.00,0.00,0.00,0.00, ALPHA = -5.00,0.00,2.00,4.00,6.00, ALPHA(6)=8.00,10.00,12.00,14.00,16.00, ALPHA(11)=18.00,20.00,25.00,30.00,35.00, BETA=0.,$ $REFQ SREF=1.251700,LREF=0.634700,LATREF=2.324000,XCG=0.85,ZCG=-0.04,$ $AXIBOD TNOSE=1.,LNOSE=0.1970,DNOSE=0.1524,LCENTR=1.7730,DCENTR=0.1524,$ $FINSET1 SECTYP=1., SSPAN=0.0000,0.2660,1.0096, CHORD=1.0033,0.8048,0.2500, XLE=0.97, NPANEL=2., PHIF=90.00,270.00, GAM=0.00,0.00, CFOC=0.0000,0.1553,0.5000,$ NACA-1-4-0310 $FINSET2 SECTYP=1., SSPAN=0.2660,0.6726, CHORD=0.4000,0.1952, XLE=5.66, CFOC=0.2600,0.5328, STA=1., SWEEP=13.99, NPANEL=2., PHIF=90.00,-90.00, 150

GAM=-90.00,90.00,$ NACA-2-4-0310 SAVE NEXT CASE THE BOUNDARY LAYER IS ASSUMED TO BE TURBULENT THE INPUT UNITS ARE IN METERS, THE SCALE FACTOR IS 1.0000 1 ***** THE USAF AUTOMATED MISSILE DATCOM * REV 9/02 ***** CASE 1 AERODYNAMIC METHODS FOR MISSILE CONFIGURATIONS PAGE 2 Brumby STATIC AERODYNAMICS FOR BODY-FIN SET 1 AND 2

******* FLIGHT CONDITIONS AND REFERENCE QUANTITIES ******* MACH NO = 0.05 REYNOLDS NO = 1.159E+06 /M ALTITUDE = 0.0 M DYNAMIC PRESSURE = 177.32 N/M**2 SIDESLIP = 0.00 DEG ROLL = 0.00 DEG REF AREA = 1.252 M**2 MOMENT CENTER = 0.850 M REF LENGTH = 0.63 M LAT REF LENGTH = 2.32 M

----- LONGITUDINAL ------LATERAL DIRECTIONAL -- ALPHA CN CM CA CY CLN CLL

-5.00 -0.314 0.028 0.002 0.000 0.000 0.000 0.00 0.000 0.001 0.017 0.000 0.000 0.000 2.00 0.122 -0.010 0.014 0.000 0.000 0.000 4.00 0.249 -0.022 0.007 0.000 0.000 0.000 6.00 0.379 -0.034 -0.004 0.000 0.000 0.000 8.00 0.512 -0.047 -0.018 0.000 0.000 0.000 10.00 0.646 -0.060 -0.026 0.000 0.000 0.000 12.00 0.774 -0.071 -0.015 0.000 0.000 0.000 14.00 0.887 -0.080 0.006 0.000 0.000 0.000 16.00 0.989 -0.089 0.017 0.000 0.000 0.000 18.00 1.078 -0.097 0.016 0.000 0.000 0.000 20.00 1.150 -0.104 0.015 0.000 0.000 0.000 25.00 1.250 -0.114 0.015 0.000 0.000 0.000 30.00 1.161 -0.106 0.013 0.000 0.000 0.000 35.00 1.046 -0.095 0.016 0.000 0.000 0.000

ALPHA CL CD CL/CD X-C.P.

-5.00 -0.313 0.029 -10.775 -0.089 0.00 0.000 0.017 0.000 -0.088 2.00 0.122 0.019 6.539 -0.081 151

4.00 0.248 0.025 10.017 -0.087 6.00 0.378 0.036 10.579 -0.090 8.00 0.510 0.053 9.603 -0.092 10.00 0.641 0.087 7.360 -0.092 12.00 0.760 0.146 5.197 -0.092 14.00 0.859 0.220 3.906 -0.090 16.00 0.946 0.289 3.278 -0.090 18.00 1.020 0.348 2.933 -0.090 20.00 1.075 0.408 2.637 -0.091 25.00 1.126 0.541 2.080 -0.091 30.00 0.998 0.592 1.687 -0.091 35.00 0.848 0.613 1.384 -0.091

X-C.P. MEAS. FROM MOMENT CENTER IN REF. LENGTHS, NEG. AFT OF MOMENT CENTER 1 ***** THE USAF AUTOMATED MISSILE DATCOM * REV 9/02 ***** CASE 1 AERODYNAMIC METHODS FOR MISSILE CONFIGURATIONS PAGE 3 Brumby STATIC AERODYNAMICS FOR BODY-FIN SET 1 AND 2

------DERIVATIVES (PER RADIAN) ------ALPHA CNA CMA CYB CLNB CLLB -5.00 3.6478 -0.3067 -0.7278 1.4592 -0.0384 0.00 3.5494 -0.3106 -0.7165 1.4485 -0.0458 2.00 3.5692 -0.3245 -0.9691 1.9917 -0.0665 4.00 3.6835 -0.3461 -1.0636 2.1875 -0.0766 6.00 3.7641 -0.3635 -0.8770 1.7770 -0.0677 8.00 3.8196 -0.3681 -0.7706 1.5396 -0.0643 10.00 3.7468 -0.3418 -0.6814 1.3398 -0.0613 12.00 3.4472 -0.2911 -0.5935 1.1431 -0.0559 14.00 3.0888 -0.2591 -0.5019 0.9390 -0.0499 16.00 2.7384 -0.2478 -0.4146 0.7446 -0.0442 18.00 2.3028 -0.2185 -0.3337 0.5643 -0.0369 20.00 1.6032 -0.1534 -0.2625 0.4057 -0.0283 25.00 0.0607 -0.0087 -0.1120 0.0709 -0.0059 152

30.00 -1.1676 0.1066 -0.0437 -0.0828 0.0282 35.00 -1.4636 0.1327 0.1671 -0.5398 0.0212

PANEL DEFLECTION ANGLES (DEGREES) SET FIN 1 FIN 2 FIN 3 FIN 4 FIN 5 FIN 6 FIN 7 FIN 8 1 0.00 0.00 2 0.00 0.00 1 ***** THE USAF AUTOMATED MISSILE DATCOM * REV 9/02 ***** CASE 1 AERODYNAMIC METHODS FOR MISSILE CONFIGURATIONS PAGE 4 Brumby BODY + 2 FIN SETS DYNAMIC DERIVATIVES

------DYNAMIC DERIVATIVES (PER RADIAN) ------ALPHA CNQ CMQ CAQ CNAD CMAD -5.00 0.573 -2.389 0.045 6.838 1.555 0.00 0.536 -2.388 0.005 6.838 1.555 2.00 0.550 -2.391 -0.019 6.838 1.555 4.00 0.568 -2.394 -0.036 6.838 1.555 6.00 0.579 -2.396 -0.058 6.838 1.555 8.00 0.588 -2.397 -0.053 6.838 1.555 10.00 0.585 -2.393 0.011 6.838 1.555 12.00 0.527 -2.383 0.081 6.838 1.555 14.00 0.483 -2.380 0.067 6.838 1.555 16.00 0.431 -2.379 0.001 6.838 1.555 18.00 0.372 -2.374 0.000 6.838 1.555 20.00 0.295 -2.368 0.000 6.838 1.555 25.00 0.116 -2.353 0.000 6.838 1.555 30.00 -0.200 -2.327 0.000 6.838 1.555 35.00 0.034 -2.343 0.000 6.838 1.555

PITCH RATE DERIVATIVES NON-DIMENSIONALIZED BY Q*LREF/2*V 1 ***** THE USAF AUTOMATED MISSILE DATCOM * REV 9/02 ***** CASE 1 AERODYNAMIC METHODS FOR MISSILE CONFIGURATIONS PAGE 5 153

Brumby BODY + 2 FIN SETS DYNAMIC DERIVATIVES

------DYNAMIC DERIVATIVES (PER RADIAN) ------ALPHA CYR CLNR CLLR CYP CLNP CLLP -5.00 3.058 -6.727 0.208 0.798 -1.721 -0.260 0.00 3.024 -6.655 0.205 0.675 -1.456 -0.259 2.00 3.020 -6.646 0.205 0.678 -1.462 -0.263 4.00 3.017 -6.639 0.205 0.700 -1.511 -0.268 6.00 3.013 -6.631 0.205 0.716 -1.544 -0.268 8.00 3.008 -6.621 0.204 0.728 -1.570 -0.271 10.00 3.003 -6.609 0.203 0.722 -1.558 -0.266 12.00 2.997 -6.596 0.203 0.625 -1.347 -0.234 14.00 2.992 -6.584 0.202 0.543 -1.172 -0.210 16.00 2.989 -6.579 0.202 0.453 -0.976 -0.188 18.00 2.991 -6.583 0.202 0.348 -0.750 -0.157 20.00 2.998 -6.598 0.202 0.217 -0.468 -0.119 25.00 3.044 -6.698 0.204 -0.083 0.179 -0.031 30.00 3.140 -6.903 0.210 -0.573 1.235 0.118 35.00 3.293 -7.234 0.220 -0.212 0.458 0.010

YAW AND ROLL RATE DERIVATIVES NON-DIMENSIONALIZED BY R*LATREF/2*V 1 ***** THE USAF AUTOMATED MISSILE DATCOM * REV 9/02 ***** CASE 2 AERODYNAMIC METHODS FOR MISSILE CONFIGURATIONS PAGE 1 CASE INPUTS FOLLOWING ARE THE CARDS INPUT FOR THIS CASE 154

Appendix A.3 : for003.dat File

VARIABLES=ALPHA,CN,CM,CA,CY,CLN,CLL,DELTA,CL,CD ZONE T="NO TRIM MACH= 0.05" -5.0000 -0.3140 0.0280 0.0017 0.0000 0.0000 0.0000 0.0500 -0.3127 0.0290 0.0000 0.0000 0.0011 0.0167 0.0000 0.0000 0.0000 0.0500 0.0000 0.0167 2.0000 0.1222 -0.0099 0.0143 0.0000 0.0000 0.0000 0.0500 0.1216 0.0186 4.0000 0.2492 -0.0216 0.0074 0.0000 0.0000 0.0000 0.0500 0.2481 0.0248 6.0000 0.3793 -0.0340 -0.0040 0.0000 0.0000 0.0000 0.0500 0.3777 0.0357 8.0000 0.5120 -0.0470 -0.0184 0.0000 0.0000 0.0000 0.0500 0.5095 0.0531 10.0000 0.6460 -0.0597 -0.0255 0.0000 0.0000 0.0000 0.0500 0.6406 0.0870 12.0000 0.7735 -0.0708 -0.0150 0.0000 0.0000 0.0000 0.0500 0.7597 0.1462 14.0000 0.8867 -0.0800 0.0056 0.0000 0.0000 0.0000 0.0500 0.8590 0.2199 16.0000 0.9892 -0.0889 0.0167 0.0000 0.0000 0.0000 0.0500 0.9463 0.2887 18.0000 1.0778 -0.0973 0.0155 0.0000 0.0000 0.0000 0.0500 1.0203 0.3478 20.0000 1.1500 -0.1042 0.0154 0.0000 0.0000 0.0000 0.0500 1.0753 0.4078 25.0000 1.2495 -0.1139 0.0148 0.0000 0.0000 0.0000 0.0500 1.1262 0.5414 30.0000 1.1605 -0.1057 0.0134 0.0000 0.0000 0.0000 0.0500 0.9984 0.5919 35.0000 1.0457 -0.0953 0.0157 0.0000 0.0000 0.0000 0.0500 0.8476 0.6126 155

Appendix A.4 : for021.dat File

VARIABLES: MACH,ALTITUDE,SIDESLIP,DEL1,DEL2,DEL3,DEL4 ROWS, TOTAL COLUMNS, COLUMNS OF DERIVATIVES DATA: ALPHA,CN,CM,CA,CY,CLN,CLL,CNQ,CMQ,CAQ,CYR,CLNR,CLLR,CYP,CLNP,C LLP 0.05 0.0 0.00 0.0 0.0 0.0 0.0 0.0 0.0 15.0 16.0 9.0 -0.500E+01 -0.3140E+00 0.2799E-01 0.1657E-02 0.7704E-08 0.4366E-08 -0.9096E-10 0.5728E+00 -0.2389E+01 0.4518E-01 0.3058E+01 -0.6727E+01 0.2077E+00 0.7981E+00 -0.1721E+01 -0.2599E+00 0.000E+00 0.0000E+00 0.1053E-02 0.1671E-01 0.0000E+00 0.0000E+00 0.0000E+00 0.5362E+00 -0.2388E+01 0.4789E-02 0.3024E+01 -0.6655E+01 0.2054E+00 0.6749E+00 -0.1456E+01 -0.2592E+00 0.200E+01 0.1222E+00 -0.9857E-02 0.1434E-01 -0.2767E-08 -0.2208E-08 0.5058E-10 0.5503E+00 -0.2391E+01 -0.1890E-01 0.3020E+01 -0.6646E+01 0.2053E+00 0.6777E+00 -0.1462E+01 -0.2631E+00 0.400E+01 0.2492E+00 -0.2160E-01 0.7400E-02 -0.8889E-08 0.2516E-08 -0.1054E-09 0.5677E+00 -0.2394E+01 -0.3617E-01 0.3017E+01 -0.6639E+01 0.2053E+00 0.7004E+00 -0.1511E+01 -0.2675E+00 0.600E+01 0.3793E+00 -0.3402E-01 -0.3973E-02 -0.7599E-08 -0.8943E-08 0.3458E-09 0.5792E+00 -0.2396E+01 -0.5754E-01 0.3013E+01 -0.6631E+01 0.2047E+00 0.7160E+00 -0.1544E+01 -0.2678E+00 0.800E+01 0.5120E+00 -0.4698E-01 -0.1837E-01 -0.2033E-07 0.9691E-08 -0.3499E-09 0.5878E+00 -0.2397E+01 -0.5294E-01 0.3008E+01 -0.6621E+01 0.2040E+00 0.7280E+00 -0.1570E+01 -0.2711E+00 0.100E+02 0.6460E+00 -0.5972E-01 -0.2552E-01 -0.3312E-07 0.2838E-07 -0.7916E-09 0.5851E+00 -0.2393E+01 0.1105E-01 0.3003E+01 -0.6609E+01 0.2034E+00 0.7223E+00 -0.1558E+01 -0.2657E+00 0.120E+02 0.7735E+00 -0.7084E-01 -0.1496E-01 -0.2848E-07 0.9935E-08 -0.3882E-09 0.5272E+00 -0.2383E+01 0.8061E-01 0.2997E+01 -0.6596E+01 0.2027E+00 0.6247E+00 -0.1347E+01 -0.2340E+00 0.140E+02 0.8867E+00 -0.8004E-01 0.5581E-02 -0.4062E-07 0.2866E-07 -0.9540E-09 0.4830E+00 -0.2380E+01 0.6722E-01 0.2992E+01 -0.6584E+01 0.2020E+00 0.5433E+00 -0.1172E+01 -0.2103E+00 0.160E+02 0.9892E+00 -0.8893E-01 0.1670E-01 -0.2650E-07 -0.8508E-08 0.1298E-09 0.4311E+00 -0.2379E+01 0.9943E-03 0.2989E+01 -0.6579E+01 0.2016E+00 0.4527E+00 -0.9764E+00 -0.1885E+00 0.180E+02 0.1078E+01 -0.9734E-01 0.1553E-01 -0.3353E-07 0.8890E-09 -0.1576E-09 0.3715E+00 -0.2374E+01 -0.3844E-03 0.2991E+01 -0.6583E+01 0.2015E+00 0.3479E+00 -0.7504E+00 -0.1575E+00 0.200E+02 0.1150E+01 -0.1042E+00 0.1541E-01 -0.2487E-07 -0.2242E-07 0.6461E-09 0.2951E+00 -0.2368E+01 -0.2330E-03 0.2998E+01 -0.6598E+01 156

0.2018E+00 0.2169E+00 -0.4678E+00 -0.1190E+00 0.250E+02 0.1250E+01 -0.1139E+00 0.1476E-01 -0.2552E-07 -0.2709E-07 0.6431E-09 0.1160E+00 -0.2353E+01 -0.2254E-04 0.3044E+01 -0.6698E+01 0.2045E+00 -0.8301E-01 0.1790E+00 -0.3076E-01 0.300E+02 0.1161E+01 -0.1057E+00 0.1339E-01 -0.1689E-07 -0.3895E-07 0.1129E-08 -0.2002E+00 -0.2327E+01 0.8931E-04 0.3140E+01 -0.6903E+01 0.2104E+00 -0.5727E+00 0.1235E+01 0.1176E+00 0.350E+02 0.1046E+01 -0.9526E-01 0.1566E-01 -0.3030E-07 -0.1511E-08 -0.7884E-10 0.3410E-01 -0.2343E+01 0.8568E-05 0.3293E+01 -0.7234E+01 0.2204E+00 -0.2124E+00 0.4581E+00 0.9807E-02 0.05 0.0 0.00 -45.0 0.0 0.0 0.0 0.0 0.0 15.0 7.0 0.0 -0.500E+01 -0.5850E+00 0.5764E-01 0.5884E-01 -0.1234E+00 0.2662E+00 0.3176E-01 0.000E+00 -0.2620E+00 0.2735E-01 0.3523E-01 -0.1432E+00 0.3090E+00 0.3119E-01 0.200E+01 -0.1344E+00 0.1581E-01 0.3091E-01 -0.1421E+00 0.3066E+00 0.3087E-01 0.400E+01 -0.7230E-02 0.4018E-02 0.2301E-01 -0.1377E+00 0.2969E+00 0.3112E-01 0.600E+01 0.1188E+00 -0.7932E-02 0.1222E-01 -0.1355E+00 0.2922E+00 0.3156E-01 0.800E+01 0.2430E+00 -0.2003E-01 -0.2035E-02 -0.1366E+00 0.2947E+00 0.3209E-01 0.100E+02 0.3679E+00 -0.3234E-01 -0.1694E-01 -0.1390E+00 0.2998E+00 0.3271E-01 0.120E+02 0.4936E+00 -0.4449E-01 -0.2622E-01 -0.1403E+00 0.3026E+00 0.3290E-01 0.140E+02 0.6138E+00 -0.5589E-01 -0.3030E-01 -0.1377E+00 0.2970E+00 0.3195E-01 0.160E+02 0.7299E+00 -0.6726E-01 -0.3844E-01 -0.1320E+00 0.2848E+00 0.3031E-01 0.180E+02 0.8406E+00 -0.7856E-01 -0.5191E-01 -0.1225E+00 0.2643E+00 0.2785E-01 0.200E+02 0.9421E+00 -0.8859E-01 -0.5773E-01 -0.1093E+00 0.2357E+00 0.2477E-01 0.250E+02 0.1132E+01 -0.1070E+00 -0.5933E-01 -0.6329E-01 0.1365E+00 0.1522E-01 0.300E+02 0.1192E+01 -0.1142E+00 -0.7227E-01 0.5965E-02 -0.1286E-01 -0.5753E-04 0.350E+02 0.1154E+01 -0.1116E+00 -0.7500E-01 0.6543E-01 -0.1411E+00 -0.1318E-01 0.05 0.0 0.00 -35.0 0.0 0.0 0.0 0.0 0.0 15.0 7.0 0.0 157

-0.500E+01 -0.5509E+00 0.5319E-01 0.4019E-01 -0.1138E+00 0.2455E+00 0.2764E-01 0.000E+00 -0.2237E+00 0.2328E-01 0.2895E-01 -0.1261E+00 0.2721E+00 0.2651E-01 0.200E+01 -0.9688E-01 0.1184E-01 0.2507E-01 -0.1220E+00 0.2632E+00 0.2646E-01 0.400E+01 0.2944E-01 0.1780E-03 0.1798E-01 -0.1166E+00 0.2514E+00 0.2682E-01 0.600E+01 0.1542E+00 -0.1163E-01 0.7684E-02 -0.1164E+00 0.2511E+00 0.2723E-01 0.800E+01 0.2784E+00 -0.2375E-01 -0.6805E-02 -0.1190E+00 0.2566E+00 0.2765E-01 0.100E+02 0.4050E+00 -0.3623E-01 -0.2174E-01 -0.1215E+00 0.2621E+00 0.2810E-01 0.120E+02 0.5320E+00 -0.4848E-01 -0.3071E-01 -0.1225E+00 0.2642E+00 0.2815E-01 0.140E+02 0.6531E+00 -0.5999E-01 -0.3494E-01 -0.1193E+00 0.2573E+00 0.2712E-01 0.160E+02 0.7701E+00 -0.7136E-01 -0.4148E-01 -0.1127E+00 0.2431E+00 0.2534E-01 0.180E+02 0.8806E+00 -0.8227E-01 -0.4909E-01 -0.1023E+00 0.2206E+00 0.2297E-01 0.200E+02 0.9781E+00 -0.9149E-01 -0.4816E-01 -0.8993E-01 0.1940E+00 0.2046E-01 0.250E+02 0.1159E+01 -0.1092E+00 -0.5214E-01 -0.4844E-01 0.1045E+00 0.1170E-01 0.300E+02 0.1204E+01 -0.1146E+00 -0.6175E-01 0.1397E-01 -0.3012E-01 -0.2196E-02 0.350E+02 0.1147E+01 -0.1100E+00 -0.6110E-01 0.6016E-01 -0.1297E+00 -0.1245E-01 0.05 0.0 0.00 -25.0 0.0 0.0 0.0 0.0 0.0 15.0 7.0 0.0 -0.500E+01 -0.5158E+00 0.4891E-01 0.2578E-01 -0.1042E+00 0.2247E+00 0.2323E-01 0.000E+00 -0.1875E+00 0.1952E-01 0.2432E-01 -0.1073E+00 0.2315E+00 0.2228E-01 0.200E+01 -0.6153E-01 0.8209E-02 0.2127E-01 -0.1008E+00 0.2175E+00 0.2239E-01 0.400E+01 0.6374E-01 -0.3318E-02 0.1480E-01 -0.9714E-01 0.2095E+00 0.2272E-01 0.600E+01 0.1877E+00 -0.1506E-01 0.4312E-02 -0.9826E-01 0.2119E+00 0.2311E-01 0.800E+01 0.3130E+00 -0.2731E-01 -0.1049E-01 -0.1011E+00 0.2180E+00 0.2345E-01 158

0.100E+02 0.4412E+00 -0.3995E-01 -0.2543E-01 -0.1035E+00 0.2233E+00 0.2375E-01 0.120E+02 0.5692E+00 -0.5231E-01 -0.3433E-01 -0.1044E+00 0.2251E+00 0.2369E-01 0.140E+02 0.6912E+00 -0.6385E-01 -0.3761E-01 -0.1004E+00 0.2165E+00 0.2249E-01 0.160E+02 0.8087E+00 -0.7503E-01 -0.4036E-01 -0.9285E-01 0.2003E+00 0.2071E-01 0.180E+02 0.9168E+00 -0.8534E-01 -0.4196E-01 -0.8292E-01 0.1788E+00 0.1870E-01 0.200E+02 0.1011E+01 -0.9399E-01 -0.3767E-01 -0.7232E-01 0.1560E+00 0.1665E-01 0.250E+02 0.1182E+01 -0.1109E+00 -0.4377E-01 -0.3600E-01 0.7763E-01 0.8698E-02 0.300E+02 0.1212E+01 -0.1147E+00 -0.5101E-01 0.2002E-01 -0.4317E-01 -0.3837E-02 0.350E+02 0.1131E+01 -0.1075E+00 -0.4712E-01 0.4986E-01 -0.1075E+00 -0.1056E-01 0.05 0.0 0.00 -15.0 0.0 0.0 0.0 0.0 0.0 15.0 7.0 0.0 -0.500E+01 -0.4812E+00 0.4492E-01 0.1535E-01 -0.9107E-01 0.1964E+00 0.1900E-01 0.000E+00 -0.1541E+00 0.1611E-01 0.2115E-01 -0.8735E-01 0.1884E+00 0.1849E-01 0.200E+01 -0.2901E-01 0.4946E-02 0.1901E-01 -0.8168E-01 0.1762E+00 0.1859E-01 0.400E+01 0.9512E-01 -0.6467E-02 0.1267E-01 -0.7984E-01 0.1722E+00 0.1892E-01 0.600E+01 0.2192E+00 -0.1825E-01 0.1813E-02 -0.8153E-01 0.1759E+00 0.1925E-01 0.800E+01 0.3460E+00 -0.3066E-01 -0.1325E-01 -0.8380E-01 0.1807E+00 0.1948E-01 0.100E+02 0.4755E+00 -0.4343E-01 -0.2816E-01 -0.8591E-01 0.1853E+00 0.1971E-01 0.120E+02 0.6044E+00 -0.5587E-01 -0.3694E-01 -0.8628E-01 0.1861E+00 0.1957E-01 0.140E+02 0.7270E+00 -0.6735E-01 -0.3815E-01 -0.8164E-01 0.1761E+00 0.1823E-01 0.160E+02 0.8440E+00 -0.7814E-01 -0.3537E-01 -0.7409E-01 0.1598E+00 0.1657E-01 0.180E+02 0.9484E+00 -0.8774E-01 -0.3151E-01 -0.6623E-01 0.1428E+00 0.1504E-01 0.200E+02 0.1040E+01 -0.9616E-01 -0.2748E-01 -0.5695E-01 0.1228E+00 0.1314E-01 159

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0.350E+02 0.1047E+01 -0.9137E-01 0.7901E-01 -0.5669E-02 0.1223E-01 0.4460E-03 0.05 0.0 0.00 0.0 45.0 0.0 0.0 0.0 0.0 15.0 7.0 0.0 -0.500E+01 -0.5613E-01 0.4253E-02 0.1845E-01 -0.1459E+00 0.3147E+00 0.3068E-01 0.000E+00 0.2620E+00 -0.2291E-01 0.3518E-01 -0.1420E+00 0.3063E+00 0.3127E-01 0.200E+01 0.3925E+00 -0.3421E-01 0.3961E-01 -0.1382E+00 0.2980E+00 0.3199E-01 0.400E+01 0.5225E+00 -0.4512E-01 0.5088E-01 -0.1347E+00 0.2905E+00 0.3178E-01 0.600E+01 0.6461E+00 -0.5542E-01 0.6370E-01 -0.1293E+00 0.2788E+00 0.3022E-01 0.800E+01 0.7648E+00 -0.6579E-01 0.6964E-01 -0.1219E+00 0.2629E+00 0.2791E-01 0.100E+02 0.8758E+00 -0.7558E-01 0.7480E-01 -0.1112E+00 0.2398E+00 0.2459E-01 0.120E+02 0.9718E+00 -0.8355E-01 0.8745E-01 -0.9828E-01 0.2120E+00 0.2091E-01 0.140E+02 0.1051E+01 -0.9004E-01 0.9948E-01 -0.8198E-01 0.1768E+00 0.1681E-01 0.160E+02 0.1116E+01 -0.9570E-01 0.1039E+00 -0.6207E-01 0.1339E+00 0.1227E-01 0.180E+02 0.1161E+01 -0.9982E-01 0.1042E+00 -0.3894E-01 0.8397E-01 0.7142E-02 0.200E+02 0.1175E+01 -0.1012E+00 0.1015E+00 -0.8058E-02 0.1738E-01 0.3955E-03 0.250E+02 0.1139E+01 -0.9850E-01 0.8741E-01 0.6306E-01 -0.1360E+00 -0.1526E-01 0.300E+02 0.1073E+01 -0.9264E-01 0.8521E-01 0.3043E-01 -0.6563E-01 -0.8126E-02 0.350E+02 0.1050E+01 -0.9098E-01 0.9009E-01 -0.8845E-02 0.1907E-01 0.8644E-03 0.05 0.0 0.00 0.0 0.0 -25.0 0.0 0.0 0.0 15.0 7.0 0.0 -0.500E+01 -0.3140E+00 0.2832E-01 0.6961E-02 0.5399E-01 -0.1165E+00 0.3680E-02 0.000E+00 0.0000E+00 0.1600E-02 0.2538E-01 0.5550E-01 -0.1197E+00 0.3807E-02 0.200E+01 0.1222E+00 -0.9195E-02 0.2486E-01 0.5610E-01 -0.1210E+00 0.3835E-02 0.400E+01 0.2492E+00 -0.2084E-01 0.1938E-01 0.5554E-01 -0.1198E+00 0.3783E-02 171

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0.180E+02 0.1078E+01 -0.9675E-01 0.2490E-01 0.4722E-01 -0.1018E+00 0.3184E-02 0.200E+02 0.1150E+01 -0.1036E+00 0.2444E-01 0.4745E-01 -0.1023E+00 0.3200E-02 0.250E+02 0.1250E+01 -0.1133E+00 0.2286E-01 0.4772E-01 -0.1029E+00 0.3221E-02 0.300E+02 0.1161E+01 -0.1052E+00 0.2050E-01 0.4754E-01 -0.1025E+00 0.3212E-02 0.350E+02 0.1046E+01 -0.9485E-01 0.2215E-01 0.4728E-01 -0.1020E+00 0.3196E-02 0.05 0.0 0.00 0.0 0.0 -5.0 0.0 0.0 0.0 15.0 7.0 0.0 -0.500E+01 -0.3140E+00 0.2798E-01 0.1562E-02 0.1569E-01 -0.3385E-01 0.1076E-02 0.000E+00 0.0000E+00 0.1092E-02 0.1733E-01 0.1513E-01 -0.3263E-01 0.1028E-02 0.200E+01 0.1222E+00 -0.9803E-02 0.1520E-01 0.1554E-01 -0.3353E-01 0.1066E-02 0.400E+01 0.2492E+00 -0.2153E-01 0.8452E-02 0.1581E-01 -0.3410E-01 0.1090E-02 0.600E+01 0.3793E+00 -0.3395E-01 -0.2792E-02 0.1598E-01 -0.3447E-01 0.1105E-02 0.800E+01 0.5120E+00 -0.4690E-01 -0.1711E-01 0.1609E-01 -0.3471E-01 0.1115E-02 0.100E+02 0.6460E+00 -0.5964E-01 -0.2422E-01 0.1617E-01 -0.3489E-01 0.1121E-02 0.120E+02 0.7735E+00 -0.7075E-01 -0.1363E-01 0.1622E-01 -0.3499E-01 0.1123E-02 0.140E+02 0.8867E+00 -0.7996E-01 0.6899E-02 0.1624E-01 -0.3502E-01 0.1122E-02 0.160E+02 0.9892E+00 -0.8884E-01 0.1800E-01 0.1624E-01 -0.3502E-01 0.1120E-02 0.180E+02 0.1078E+01 -0.9726E-01 0.1680E-01 0.1622E-01 -0.3498E-01 0.1116E-02 0.200E+02 0.1150E+01 -0.1041E+00 0.1663E-01 0.1619E-01 -0.3491E-01 0.1111E-02 0.250E+02 0.1250E+01 -0.1138E+00 0.1585E-01 0.1608E-01 -0.3467E-01 0.1095E-02 0.300E+02 0.1161E+01 -0.1056E+00 0.1432E-01 0.1590E-01 -0.3429E-01 0.1075E-02 0.350E+02 0.1046E+01 -0.9520E-01 0.1646E-01 0.1575E-01 -0.3396E-01 0.1060E-02 0.05 0.0 0.00 0.0 0.0 5.0 0.0 0.0 0.0 15.0 7.0 0.0 173

-0.500E+01 -0.3140E+00 0.2808E-01 0.3086E-02 -0.1623E-01 0.3501E-01 -0.1127E-02 0.000E+00 0.0000E+00 0.1092E-02 0.1733E-01 -0.1513E-01 0.3263E-01 -0.1028E-02 0.200E+01 0.1222E+00 -0.9833E-02 0.1473E-01 -0.1496E-01 0.3228E-01 -0.1012E-02 0.400E+01 0.2492E+00 -0.2158E-01 0.7636E-02 -0.1510E-01 0.3257E-01 -0.1023E-02 0.600E+01 0.3793E+00 -0.3401E-01 -0.3838E-02 -0.1530E-01 0.3300E-01 -0.1039E-02 0.800E+01 0.5120E+00 -0.4697E-01 -0.1831E-01 -0.1546E-01 0.3334E-01 -0.1052E-02 0.100E+02 0.6460E+00 -0.5972E-01 -0.2551E-01 -0.1557E-01 0.3358E-01 -0.1060E-02 0.120E+02 0.7735E+00 -0.7084E-01 -0.1496E-01 -0.1563E-01 0.3371E-01 -0.1063E-02 0.140E+02 0.8867E+00 -0.8004E-01 0.5579E-02 -0.1564E-01 0.3373E-01 -0.1062E-02 0.160E+02 0.9892E+00 -0.8892E-01 0.1672E-01 -0.1562E-01 0.3369E-01 -0.1058E-02 0.180E+02 0.1078E+01 -0.9734E-01 0.1559E-01 -0.1558E-01 0.3361E-01 -0.1053E-02 0.200E+02 0.1150E+01 -0.1042E+00 0.1551E-01 -0.1552E-01 0.3348E-01 -0.1046E-02 0.250E+02 0.1250E+01 -0.1138E+00 0.1496E-01 -0.1536E-01 0.3312E-01 -0.1028E-02 0.300E+02 0.1161E+01 -0.1057E+00 0.1373E-01 -0.1524E-01 0.3286E-01 -0.1014E-02 0.350E+02 0.1046E+01 -0.9523E-01 0.1608E-01 -0.1524E-01 0.3286E-01 -0.1013E-02 0.05 0.0 0.00 0.0 0.0 15.0 0.0 0.0 0.0 15.0 7.0 0.0 -0.500E+01 -0.3140E+00 0.2866E-01 0.1225E-01 -0.4572E-01 0.9862E-01 -0.3096E-02 0.000E+00 0.0000E+00 0.1410E-02 0.2237E-01 -0.4574E-01 0.9867E-01 -0.3149E-02 0.200E+01 0.1222E+00 -0.9555E-02 0.1914E-01 -0.4499E-01 0.9704E-01 -0.3084E-02 0.400E+01 0.2492E+00 -0.2133E-01 0.1167E-01 -0.4466E-01 0.9633E-01 -0.3046E-02 0.600E+01 0.3793E+00 -0.3377E-01 -0.6767E-04 -0.4459E-01 0.9618E-01 -0.3032E-02 0.800E+01 0.5120E+00 -0.4674E-01 -0.1470E-01 -0.4463E-01 0.9627E-01 -0.3029E-02 174

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0.250E+02 0.1250E+01 -0.1134E+00 0.2124E-01 -0.5496E-01 0.1185E+00 -0.3715E-02 0.300E+02 0.1161E+01 -0.1052E+00 0.2049E-01 -0.5542E-01 0.1195E+00 -0.3743E-02 0.350E+02 0.1046E+01 -0.9478E-01 0.2322E-01 -0.5582E-01 0.1204E+00 -0.3765E-02 0.05 0.0 0.00 0.0 0.0 0.0 0.0 0.0 0.0 15.0 7.0 0.0 -0.500E+01 -0.3140E+00 0.2893E-01 0.1658E-01 0.5176E-01 -0.1116E+00 0.3491E-02 0.000E+00 0.0000E+00 0.1600E-02 0.2538E-01 0.5550E-01 -0.1197E+00 0.3807E-02 0.200E+01 0.1222E+00 -0.9394E-02 0.2170E-01 0.5467E-01 -0.1179E+00 0.3750E-02 0.400E+01 0.2492E+00 -0.2118E-01 0.1401E-01 0.5427E-01 -0.1171E+00 0.3719E-02 0.600E+01 0.3793E+00 -0.3363E-01 0.2193E-02 0.5414E-01 -0.1168E+00 0.3697E-02 0.800E+01 0.5120E+00 -0.4660E-01 -0.1250E-01 0.5414E-01 -0.1168E+00 0.3688E-02 0.100E+02 0.6460E+00 -0.5936E-01 -0.1983E-01 0.5420E-01 -0.1169E+00 0.3684E-02 0.120E+02 0.7735E+00 -0.7048E-01 -0.9335E-02 0.5427E-01 -0.1171E+00 0.3683E-02 0.140E+02 0.8867E+00 -0.7969E-01 0.1123E-01 0.5436E-01 -0.1172E+00 0.3684E-02 0.160E+02 0.9892E+00 -0.8856E-01 0.2243E-01 0.5444E-01 -0.1174E+00 0.3686E-02 0.180E+02 0.1078E+01 -0.9697E-01 0.2139E-01 0.5453E-01 -0.1176E+00 0.3689E-02 0.200E+02 0.1150E+01 -0.1038E+00 0.2142E-01 0.5464E-01 -0.1178E+00 0.3694E-02 0.250E+02 0.1250E+01 -0.1134E+00 0.2124E-01 0.5496E-01 -0.1185E+00 0.3715E-02 0.300E+02 0.1161E+01 -0.1052E+00 0.2049E-01 0.5542E-01 -0.1195E+00 0.3743E-02 0.350E+02 0.1046E+01 -0.9478E-01 0.2322E-01 0.5582E-01 -0.1204E+00 0.3765E-02 0.05 0.0 0.00 0.0 0.0 0.0 0.0 0.0 0.0 15.0 7.0 0.0 -0.500E+01 -0.3140E+00 0.2866E-01 0.1225E-01 0.4572E-01 -0.9862E-01 0.3096E-02 0.000E+00 0.0000E+00 0.1410E-02 0.2237E-01 0.4574E-01 -0.9867E-01 0.3149E-02 176

0.200E+01 0.1222E+00 -0.9555E-02 0.1914E-01 0.4499E-01 -0.9704E-01 0.3084E-02 0.400E+01 0.2492E+00 -0.2133E-01 0.1167E-01 0.4466E-01 -0.9633E-01 0.3046E-02 0.600E+01 0.3793E+00 -0.3377E-01 -0.6767E-04 0.4459E-01 -0.9618E-01 0.3032E-02 0.800E+01 0.5120E+00 -0.4674E-01 -0.1470E-01 0.4463E-01 -0.9627E-01 0.3029E-02 0.100E+02 0.6460E+00 -0.5950E-01 -0.2200E-01 0.4470E-01 -0.9642E-01 0.3029E-02 0.120E+02 0.7735E+00 -0.7062E-01 -0.1149E-01 0.4478E-01 -0.9658E-01 0.3030E-02 0.140E+02 0.8867E+00 -0.7982E-01 0.9069E-02 0.4484E-01 -0.9672E-01 0.3030E-02 0.160E+02 0.9892E+00 -0.8870E-01 0.2025E-01 0.4490E-01 -0.9685E-01 0.3030E-02 0.180E+02 0.1078E+01 -0.9711E-01 0.1919E-01 0.4496E-01 -0.9697E-01 0.3031E-02 0.200E+02 0.1150E+01 -0.1039E+00 0.1919E-01 0.4502E-01 -0.9710E-01 0.3032E-02 0.250E+02 0.1250E+01 -0.1136E+00 0.1892E-01 0.4524E-01 -0.9756E-01 0.3044E-02 0.300E+02 0.1161E+01 -0.1054E+00 0.1802E-01 0.4560E-01 -0.9834E-01 0.3072E-02 0.350E+02 0.1046E+01 -0.9494E-01 0.2059E-01 0.4594E-01 -0.9907E-01 0.3101E-02 0.05 0.0 0.00 0.0 0.0 0.0 0.0 0.0 0.0 15.0 7.0 0.0 -0.500E+01 -0.3140E+00 0.2808E-01 0.3086E-02 0.1623E-01 -0.3501E-01 0.1127E-02 0.000E+00 0.0000E+00 0.1092E-02 0.1733E-01 0.1513E-01 -0.3263E-01 0.1028E-02 0.200E+01 0.1222E+00 -0.9833E-02 0.1473E-01 0.1496E-01 -0.3228E-01 0.1012E-02 0.400E+01 0.2492E+00 -0.2158E-01 0.7636E-02 0.1510E-01 -0.3257E-01 0.1023E-02 0.600E+01 0.3793E+00 -0.3401E-01 -0.3838E-02 0.1530E-01 -0.3300E-01 0.1039E-02 0.800E+01 0.5120E+00 -0.4697E-01 -0.1831E-01 0.1546E-01 -0.3334E-01 0.1052E-02 0.100E+02 0.6460E+00 -0.5972E-01 -0.2551E-01 0.1557E-01 -0.3358E-01 0.1060E-02 0.120E+02 0.7735E+00 -0.7084E-01 -0.1496E-01 0.1563E-01 -0.3371E-01 0.1063E-02 177

0.140E+02 0.8867E+00 -0.8004E-01 0.5579E-02 0.1564E-01 -0.3373E-01 0.1062E-02 0.160E+02 0.9892E+00 -0.8892E-01 0.1672E-01 0.1562E-01 -0.3369E-01 0.1058E-02 0.180E+02 0.1078E+01 -0.9734E-01 0.1559E-01 0.1558E-01 -0.3361E-01 0.1053E-02 0.200E+02 0.1150E+01 -0.1042E+00 0.1551E-01 0.1552E-01 -0.3348E-01 0.1046E-02 0.250E+02 0.1250E+01 -0.1138E+00 0.1496E-01 0.1536E-01 -0.3312E-01 0.1028E-02 0.300E+02 0.1161E+01 -0.1057E+00 0.1373E-01 0.1524E-01 -0.3286E-01 0.1014E-02 0.350E+02 0.1046E+01 -0.9523E-01 0.1608E-01 0.1524E-01 -0.3286E-01 0.1013E-02 0.05 0.0 0.00 0.0 0.0 0.0 0.0 0.0 0.0 15.0 7.0 0.0 -0.500E+01 -0.3140E+00 0.2798E-01 0.1562E-02 -0.1569E-01 0.3385E-01 -0.1076E-02 0.000E+00 0.0000E+00 0.1092E-02 0.1733E-01 -0.1513E-01 0.3263E-01 -0.1028E-02 0.200E+01 0.1222E+00 -0.9803E-02 0.1520E-01 -0.1554E-01 0.3353E-01 -0.1066E-02 0.400E+01 0.2492E+00 -0.2153E-01 0.8452E-02 -0.1581E-01 0.3410E-01 -0.1090E-02 0.600E+01 0.3793E+00 -0.3395E-01 -0.2792E-02 -0.1598E-01 0.3447E-01 -0.1105E-02 0.800E+01 0.5120E+00 -0.4690E-01 -0.1711E-01 -0.1609E-01 0.3471E-01 -0.1115E-02 0.100E+02 0.6460E+00 -0.5964E-01 -0.2422E-01 -0.1617E-01 0.3489E-01 -0.1121E-02 0.120E+02 0.7735E+00 -0.7075E-01 -0.1363E-01 -0.1622E-01 0.3499E-01 -0.1123E-02 0.140E+02 0.8867E+00 -0.7996E-01 0.6899E-02 -0.1624E-01 0.3502E-01 -0.1122E-02 0.160E+02 0.9892E+00 -0.8884E-01 0.1800E-01 -0.1624E-01 0.3502E-01 -0.1120E-02 0.180E+02 0.1078E+01 -0.9726E-01 0.1680E-01 -0.1622E-01 0.3498E-01 -0.1116E-02 0.200E+02 0.1150E+01 -0.1041E+00 0.1663E-01 -0.1619E-01 0.3491E-01 -0.1111E-02 0.250E+02 0.1250E+01 -0.1138E+00 0.1585E-01 -0.1608E-01 0.3467E-01 -0.1095E-02 0.300E+02 0.1161E+01 -0.1056E+00 0.1432E-01 -0.1590E-01 0.3429E-01 -0.1075E-02 178

0.350E+02 0.1046E+01 -0.9520E-01 0.1646E-01 -0.1575E-01 0.3396E-01 -0.1060E-02 0.05 0.0 0.00 0.0 0.0 0.0 0.0 0.0 0.0 15.0 7.0 0.0 -0.500E+01 -0.3140E+00 0.2819E-01 0.4874E-02 -0.4460E-01 0.9621E-01 -0.3033E-02 0.000E+00 0.0000E+00 0.1410E-02 0.2237E-01 -0.4574E-01 0.9867E-01 -0.3149E-02 0.200E+01 0.1222E+00 -0.9432E-02 0.2110E-01 -0.4658E-01 0.1005E+00 -0.3200E-02 0.400E+01 0.2492E+00 -0.2110E-01 0.1526E-01 -0.4704E-01 0.1015E+00 -0.3218E-02 0.600E+01 0.3793E+00 -0.3347E-01 0.4752E-02 -0.4708E-01 0.1015E+00 -0.3208E-02 0.800E+01 0.5120E+00 -0.4639E-01 -0.9045E-02 -0.4690E-01 0.1012E+00 -0.3184E-02 0.100E+02 0.6460E+00 -0.5911E-01 -0.1580E-01 -0.4671E-01 0.1007E+00 -0.3162E-02 0.120E+02 0.7735E+00 -0.7022E-01 -0.5075E-02 -0.4665E-01 0.1006E+00 -0.3152E-02 0.140E+02 0.8867E+00 -0.7943E-01 0.1540E-01 -0.4677E-01 0.1009E+00 -0.3157E-02 0.160E+02 0.9892E+00 -0.8832E-01 0.2634E-01 -0.4697E-01 0.1013E+00 -0.3168E-02 0.180E+02 0.1078E+01 -0.9675E-01 0.2490E-01 -0.4722E-01 0.1018E+00 -0.3184E-02 0.200E+02 0.1150E+01 -0.1036E+00 0.2444E-01 -0.4745E-01 0.1023E+00 -0.3200E-02 0.250E+02 0.1250E+01 -0.1133E+00 0.2286E-01 -0.4772E-01 0.1029E+00 -0.3221E-02 0.300E+02 0.1161E+01 -0.1052E+00 0.2050E-01 -0.4754E-01 0.1025E+00 -0.3212E-02 0.350E+02 0.1046E+01 -0.9485E-01 0.2215E-01 -0.4728E-01 0.1020E+00 -0.3196E-02 0.05 0.0 0.00 0.0 0.0 0.0 0.0 0.0 0.0 15.0 7.0 0.0 -0.500E+01 -0.3140E+00 0.2832E-01 0.6961E-02 -0.5399E-01 0.1165E+00 -0.3680E-02 0.000E+00 0.0000E+00 0.1600E-02 0.2538E-01 -0.5550E-01 0.1197E+00 -0.3807E-02 0.200E+01 0.1222E+00 -0.9195E-02 0.2486E-01 -0.5610E-01 0.1210E+00 -0.3835E-02 0.400E+01 0.2492E+00 -0.2084E-01 0.1938E-01 -0.5554E-01 0.1198E+00 -0.3783E-02 179

0.600E+01 0.3793E+00 -0.3320E-01 0.9053E-02 -0.5464E-01 0.1179E+00 -0.3707E-02 0.800E+01 0.5120E+00 -0.4611E-01 -0.4682E-02 -0.5387E-01 0.1162E+00 -0.3643E-02 0.100E+02 0.6460E+00 -0.5883E-01 -0.1144E-01 -0.5337E-01 0.1151E+00 -0.3599E-02 0.120E+02 0.7735E+00 -0.6994E-01 -0.7155E-03 -0.5321E-01 0.1148E+00 -0.3581E-02 0.140E+02 0.8867E+00 -0.7915E-01 0.1976E-01 -0.5338E-01 0.1151E+00 -0.3588E-02 0.160E+02 0.9892E+00 -0.8804E-01 0.3070E-01 -0.5373E-01 0.1159E+00 -0.3609E-02 0.180E+02 0.1078E+01 -0.9648E-01 0.2926E-01 -0.5421E-01 0.1169E+00 -0.3640E-02 0.200E+02 0.1150E+01 -0.1033E+00 0.2879E-01 -0.5476E-01 0.1181E+00 -0.3677E-02 0.250E+02 0.1250E+01 -0.1131E+00 0.2703E-01 -0.5608E-01 0.1209E+00 -0.3768E-02 0.300E+02 0.1161E+01 -0.1050E+00 0.2439E-01 -0.5696E-01 0.1228E+00 -0.3831E-02 0.350E+02 0.1046E+01 -0.9462E-01 0.2579E-01 -0.5711E-01 0.1232E+00 -0.3842E-02 0.05 0.0 -20.00 0.0 0.0 0.0 0.0 0.0 0.0 15.0 7.0 0.0 -0.500E+01 -0.2989E+00 0.2024E+00 0.2775E+01 0.2243E+00 -0.4249E+00 0.9436E-02 0.000E+00 -0.6004E-07 0.3013E+00 0.4781E+01 0.2208E+00 -0.4178E+00 0.1311E-01 0.200E+01 0.1161E+00 0.2191E+00 0.3645E+01 0.2192E+00 -0.4143E+00 0.1451E-01 0.400E+01 0.2369E+00 0.1677E+00 0.3007E+01 0.2176E+00 -0.4107E+00 0.1598E-01 0.600E+01 0.3616E+00 0.1305E+00 0.2599E+01 0.2143E+00 -0.4031E+00 0.1733E-01 0.800E+01 0.4912E+00 0.9761E-01 0.2268E+01 0.2113E+00 -0.3962E+00 0.1849E-01 0.100E+02 0.6240E+00 0.4780E-01 0.1675E+01 0.2077E+00 -0.3878E+00 0.1930E-01 0.120E+02 0.7528E+00 -0.1500E-01 0.8697E+00 0.2041E+00 -0.3796E+00 0.1937E-01 0.140E+02 0.8723E+00 -0.6805E-01 0.2060E+00 0.2010E+00 -0.3722E+00 0.1876E-01 0.160E+02 0.9769E+00 -0.9653E-01 -0.8973E-01 0.1959E+00 -0.3605E+00 0.1855E-01 180

0.180E+02 0.1068E+01 -0.1107E+00 -0.1776E+00 0.1907E+00 -0.3486E+00 0.1763E-01 0.200E+02 0.1143E+01 -0.1213E+00 -0.2334E+00 0.1858E+00 -0.3375E+00 0.1613E-01 0.250E+02 0.1246E+01 -0.1418E+00 -0.4033E+00 0.1732E+00 -0.3089E+00 0.1115E-01 0.300E+02 0.1168E+01 -0.1331E+00 -0.3826E+00 0.1754E+00 -0.3130E+00 0.1121E-02 0.350E+02 0.1051E+01 -0.1658E+00 -0.1072E+01 0.1141E+00 -0.1810E+00 0.3807E-02 0.05 0.0 -10.00 0.0 0.0 0.0 0.0 0.0 0.0 15.0 7.0 0.0 -0.500E+01 -0.3080E+00 0.5418E-01 0.4201E+00 0.1199E+00 -0.2350E+00 0.5684E-02 0.000E+00 -0.2454E-07 0.2696E-01 0.4277E+00 0.1260E+00 -0.2489E+00 0.7946E-02 0.200E+01 0.1194E+00 0.1448E-01 0.3994E+00 0.1306E+00 -0.2587E+00 0.9009E-02 0.400E+01 0.2438E+00 0.2381E-02 0.3852E+00 0.1445E+00 -0.2883E+00 0.1070E-01 0.600E+01 0.3729E+00 -0.8274E-02 0.4019E+00 0.1433E+00 -0.2853E+00 0.1118E-01 0.800E+01 0.5049E+00 -0.1914E-01 0.4208E+00 0.1385E+00 -0.2742E+00 0.1146E-01 0.100E+02 0.6387E+00 -0.3394E-01 0.3815E+00 0.1307E+00 -0.2565E+00 0.1147E-01 0.120E+02 0.7670E+00 -0.5281E-01 0.2707E+00 0.1221E+00 -0.2373E+00 0.1099E-01 0.140E+02 0.8812E+00 -0.7007E-01 0.1653E+00 0.1122E+00 -0.2151E+00 0.1035E-01 0.160E+02 0.9846E+00 -0.8211E-01 0.1277E+00 0.1017E+00 -0.1917E+00 0.9685E-02 0.180E+02 0.1074E+01 -0.9010E-01 0.1347E+00 0.9138E-01 -0.1687E+00 0.8680E-02 0.200E+02 0.1147E+01 -0.9686E-01 0.1373E+00 0.8176E-01 -0.1473E+00 0.7398E-02 0.250E+02 0.1249E+01 -0.1067E+00 0.1349E+00 0.5844E-01 -0.9555E-01 0.3662E-02 0.300E+02 0.1163E+01 -0.9876E-01 0.1343E+00 0.4341E-01 -0.6225E-01 -0.2498E-02 0.350E+02 0.1047E+01 -0.8826E-01 0.1348E+00 -0.2885E-02 0.3786E-01 -0.1993E-02 0.05 0.0 10.00 0.0 0.0 0.0 0.0 0.0 0.0 15.0 7.0 0.0 181

-0.500E+01 -0.3080E+00 0.5370E-01 0.4126E+00 -0.9783E-01 0.1874E+00 -0.4158E-02 0.000E+00 0.4239E-07 0.2696E-01 0.4277E+00 -0.1260E+00 0.2489E+00 -0.7946E-02 0.200E+01 0.1194E+00 0.1448E-01 0.3994E+00 -0.1306E+00 0.2587E+00 -0.9009E-02 0.400E+01 0.2438E+00 0.2381E-02 0.3852E+00 -0.1445E+00 0.2883E+00 -0.1070E-01 0.600E+01 0.3729E+00 -0.8274E-02 0.4019E+00 -0.1433E+00 0.2853E+00 -0.1118E-01 0.800E+01 0.5049E+00 -0.1914E-01 0.4208E+00 -0.1385E+00 0.2742E+00 -0.1146E-01 0.100E+02 0.6387E+00 -0.3394E-01 0.3815E+00 -0.1307E+00 0.2565E+00 -0.1147E-01 0.120E+02 0.7670E+00 -0.5281E-01 0.2707E+00 -0.1221E+00 0.2373E+00 -0.1099E-01 0.140E+02 0.8812E+00 -0.7007E-01 0.1653E+00 -0.1122E+00 0.2151E+00 -0.1035E-01 0.160E+02 0.9846E+00 -0.8211E-01 0.1277E+00 -0.1017E+00 0.1917E+00 -0.9685E-02 0.180E+02 0.1074E+01 -0.9010E-01 0.1347E+00 -0.9138E-01 0.1687E+00 -0.8680E-02 0.200E+02 0.1147E+01 -0.9686E-01 0.1373E+00 -0.8176E-01 0.1473E+00 -0.7398E-02 0.250E+02 0.1249E+01 -0.1067E+00 0.1349E+00 -0.5844E-01 0.9555E-01 -0.3662E-02 0.300E+02 0.1163E+01 -0.9876E-01 0.1343E+00 -0.4341E-01 0.6225E-01 0.2498E-02 0.350E+02 0.1047E+01 -0.8826E-01 0.1348E+00 0.2885E-02 -0.3786E-01 0.1993E-02 0.05 0.0 20.00 0.0 0.0 0.0 0.0 0.0 0.0 15.0 7.0 0.0 -0.500E+01 -0.2989E+00 0.2025E+00 0.2777E+01 -0.2218E+00 0.4196E+00 -0.9284E-02 0.000E+00 0.9035E-07 0.3013E+00 0.4781E+01 -0.2208E+00 0.4178E+00 -0.1311E-01 0.200E+01 0.1161E+00 0.2191E+00 0.3645E+01 -0.2192E+00 0.4143E+00 -0.1451E-01 0.400E+01 0.2369E+00 0.1677E+00 0.3007E+01 -0.2176E+00 0.4107E+00 -0.1598E-01 0.600E+01 0.3616E+00 0.1305E+00 0.2599E+01 -0.2143E+00 0.4031E+00 -0.1733E-01 0.800E+01 0.4912E+00 0.9761E-01 0.2268E+01 -0.2113E+00 0.3962E+00 -0.1849E-01 182

0.100E+02 0.6240E+00 0.4780E-01 0.1675E+01 -0.2077E+00 0.3878E+00 -0.1930E-01 0.120E+02 0.7528E+00 -0.1500E-01 0.8697E+00 -0.2041E+00 0.3796E+00 -0.1937E-01 0.140E+02 0.8723E+00 -0.6805E-01 0.2060E+00 -0.2010E+00 0.3722E+00 -0.1876E-01 0.160E+02 0.9769E+00 -0.9653E-01 -0.8973E-01 -0.1959E+00 0.3605E+00 -0.1855E-01 0.180E+02 0.1068E+01 -0.1107E+00 -0.1776E+00 -0.1907E+00 0.3486E+00 -0.1763E-01 0.200E+02 0.1143E+01 -0.1213E+00 -0.2334E+00 -0.1858E+00 0.3375E+00 -0.1613E-01 0.250E+02 0.1246E+01 -0.1418E+00 -0.4033E+00 -0.1732E+00 0.3089E+00 -0.1115E-01 0.300E+02 0.1168E+01 -0.1331E+00 -0.3826E+00 -0.1754E+00 0.3130E+00 -0.1121E-02 0.350E+02 0.1051E+01 -0.1658E+00 -0.1072E+01 -0.1141E+00 0.1810E+00 -0.3807E-02 0.05 328.0 0.00 0.0 0.0 0.0 0.0 0.0 0.0 15.0 7.0 0.0 -0.500E+01 -0.3140E+00 0.2799E-01 0.1728E-02 0.7704E-08 0.4366E-08 -0.9096E-10 0.000E+00 0.0000E+00 0.1055E-02 0.1674E-01 0.0000E+00 0.0000E+00 0.0000E+00 0.200E+01 0.1222E+00 -0.9855E-02 0.1438E-01 -0.2767E-08 -0.2208E-08 0.5058E-10 0.400E+01 0.2492E+00 -0.2160E-01 0.7454E-02 -0.8889E-08 0.2516E-08 -0.1054E-09 0.600E+01 0.3793E+00 -0.3402E-01 -0.3889E-02 -0.7599E-08 -0.8943E-08 0.3458E-09 0.800E+01 0.5120E+00 -0.4697E-01 -0.1825E-01 -0.2033E-07 0.9691E-08 -0.3499E-09 0.100E+02 0.6460E+00 -0.5971E-01 -0.2538E-01 -0.3312E-07 0.2838E-07 -0.7916E-09 0.120E+02 0.7735E+00 -0.7083E-01 -0.1484E-01 -0.2848E-07 0.9935E-08 -0.3882E-09 0.140E+02 0.8867E+00 -0.8004E-01 0.5642E-02 -0.4062E-07 0.2866E-07 -0.9540E-09 0.160E+02 0.9892E+00 -0.8892E-01 0.1673E-01 -0.2650E-07 -0.8508E-08 0.1298E-09 0.180E+02 0.1078E+01 -0.9734E-01 0.1557E-01 -0.3353E-07 0.8890E-09 -0.1576E-09 0.200E+02 0.1150E+01 -0.1042E+00 0.1544E-01 -0.2487E-07 -0.2242E-07 0.6461E-09 183

0.250E+02 0.1250E+01 -0.1139E+00 0.1479E-01 -0.2552E-07 -0.2709E-07 0.6431E-09 0.300E+02 0.1161E+01 -0.1057E+00 0.1342E-01 -0.1689E-07 -0.3895E-07 0.1129E-08 0.350E+02 0.1046E+01 -0.9525E-01 0.1568E-01 -0.3030E-07 -0.1511E-08 -0.7884E-10 184

Appendix B.1: Equations of Motion s-function function [sys,x0,str,ts] = EOM(t,x,u,flag,J_inertial,mass,g,x0) % %Parameters % J(1) = Jxx; % J(2) = Jyy; % J(3) = Jzz; % J(4) = Jxz; % mass = ; %slugs or kg % g = ; gravity %States % Xe(N) = x0(1); % Ye(E) = x0(2); % Ze(D) = x0(3); % Phi = x0(4); % Theta = x0(5); % Psi = x0(6); % U = x0(7); % V = x0(8); % W = x0(9); % P = x0(10); % Q = x0(11); % R = x0(12);

%#define GD=32.17; % ft/s

% % The following outlines the general structure of an S-function. % switch flag,

%%%%%%%%%%%%%%%%%% % Initialization % %%%%%%%%%%%%%%%%%% case 0, % % call simsizes for a sizes structure, fill it in and convert it to a % sizes array. % % Note that in this example, the values are hard coded. This is not a % recommended practice as the characteristics of the block are typically % defined by the S-function parameters. % 185 sizes = simsizes; sizes.NumContStates = 12; sizes.NumDiscStates = 0; sizes.NumOutputs = 12; sizes.NumInputs = 6; sizes.DirFeedthrough = 0; sizes.NumSampleTimes = 1; % at least one sample time is needed sys = simsizes(sizes);

% % str is always an empty matrix % str = [];

% % initialize the array of sample times % ts = [0 0]; % % initialize the initial conditions % Xe = x0(1); %units of linear distance Ye = x0(2); %units of linear distance Ze = x0(3); %units of linear distance Phi = x0(4); %radians Theta = x0(5); %radians Psi = x0(6); %radians U = x0(7); %units of linear distance per second V = x0(8); %units of linear distance per second W = x0(9); %units of linear distance per second P = x0(10); %units of radians per second Q = x0(11); %units of radians per second R = x0(12); %units of radians per second nav_dot=[0 0 0]'; euler_dot=[0 0 0]'; vel_dot=[0 0 0]'; omega_dot=[0 0 0]';

% end mdlInitializeSizes

186

%%%%%%%%%%%%%%% % Derivatives % %%%%%%%%%%%%%%% case 1,

%Inputs Fx = u(1); Fy = u(2); Fz = u(3); l = u(4); m = u(5); n = u(6); %states Xe = x(1); %units of linear distance Ye = x(2); %units of linear distance Ze = x(3); %units of linear distance Phi = x(4); %radians Theta = x(5); %radians Psi = x(6); %radians U = x(7); %units of linear distance per second V = x(8); %units of linear distance per second W = x(9); %units of linear distance per second P = x(10); %units of radians per second Q = x(11); %units of radians per second R = x(12); %units of radians per second v_cm_e = [U V W]'; %Velocity components Force = [Fx Fy Fz]'; %Foce Input Moment = [l m n]'; %Moment Input omega = [P Q R]'; %Body Rates

%Inertial Matrix - %From Stevens and Lewis pg.45 equation 1.5-7 Jxx = J_inertial(1); Jyy = J_inertial(2); Jzz = J_inertial(3); Jxz = J_inertial(4); %Inertial Matrix J = [ Jxx 0 -Jxz;... 0 Jyy 0; ... -Jxz 0 Jzz];

%Direction Cosine Matrix - Navigation Frame with respect to Body Frame 187

%From Stevens and Lewis pg.26 equation 1.3-20 % Direction Cosine Matrix from navigation frame to body frame DCM_b_n = [ cos(Theta)*cos(Psi) cos(Theta)*sin(Psi) -sin(Theta);... (-cos(Phi)*sin(Psi)+sin(Phi)*sin(Theta)*cos(Psi)) (cos(Phi)*cos(Psi) + sin(Phi)*sin(Theta)*sin(Psi)) sin(Phi)*cos(Theta);... (sin(Phi)*sin(Psi)+cos(Phi)*sin(Theta)*cos(Psi)) (-sin(Phi)*cos(Psi) + cos(Phi)*sin(Theta)*sin(Psi)) cos(Phi)*cos(Theta)];

% Cx=[1 0 0; 0 cos(Phi) -sin(Phi); 0 sin(Phi) cos(Phi)]; %roll % Cy=[cos(Theta) 0 sin(Theta); 0 1 0; -sin(Theta) 0 cos(Theta)]; %pitch % Cz=[cos(Psi) -sin(Psi) 0;sin(Psi) cos(Psi) 0; 0 0 1]; %yaw % % % DCM_b_n=Cz*Cy*Cx; %yaw,pitch,roll

% Direction Cosine Matrix from body frame to navigation frame DCM_n_b = inv(DCM_b_n);

%Specific Force Equ - Translational Velocity (Body Frame) %From Stevens and Lewis pg.52 equation 1.5-22d vel_dot = (1/mass)*Force - (cross(omega,v_cm_e)) + (DCM_b_n*[0; 0; g]);

%From Stevens and Lewis pg.27 equation 1.3-22a h_dot = [1 tan(Theta)*sin(Phi) tan(Theta)*cos(Phi);... 0 cos(Phi) -sin(Phi);... 0 (sin(Phi)/cos(Theta)) cos(Phi)/cos(Theta)];

%Kinematic Equ - Euler Angle Rates (Body Frame) %From Stevens and Lewis pg.52 equation 1.5-22c euler_dot = h_dot * omega;

%Moment Equ - Angular Acceleration (Body Frame) %From Stevens and Lewis pg.52 equation 1.5-22e %omega_dot = J\ (Moment - cross(omega,J*omega)); omega_dot = inv(J)*(Moment - cross(omega,J*omega)); %Navigation Euqations - Inertial Velocity (Navigation Frame) %From Stevens and Lewis pg.52 equation 1.5-22b nav_dot = DCM_n_b * v_cm_e;

%Form the state rate vector 188

%------%sys = [Udot;Vdot,Wdot;Phidot;Thetadot;Psidot;Pdot;Qdot;Rdot;Xedot;Yedot;Hdot]; sys = [nav_dot;euler_dot;vel_dot;omega_dot]; %------%------% end mdlDerivatives

%%%%%%%%%%% % Outputs % %%%%%%%%%%% case 3,

sys = x;

%%%%%%%%%%%%% % Terminate % %%%%%%%%%%%%% case {2, 4, 9}, % sys=mdlTerminate(t,x,u); sys=[]; %do nothing

%%%%%%%%%%%%%%%%%%%% % Unexpected flags % %%%%%%%%%%%%%%%%%%%% otherwise error(['Unhandled flag = ',num2str(flag)]); end 189

Appendix B.2: forces_moments.m function [ output_args ] = forces_moments( input_args )

x_e = input_args(1); y_e = input_args(2); z_e = input_args(3);

phi = input_args(4); theta = input_args(5); psi = input_args(6);

u = input_args(7); v = input_args(8); w = input_args(9);

p = input_args(10); q = input_args(11); r = input_args(12);

% delta_right_ail = input_args(13); % delta_left_ail = input_args(14); % delta_rud = input_args(15); % thrust = input_args(16); delta_ail = input_args(13); delta_elv = input_args(14); delta_rud = input_args(15); thrust = input_args(16);

delta_right_ail = delta_elv - delta_ail; delta_left_ail = delta_elv + delta_ail;

if delta_right_ail < -45 delta_right_ail = -45; else if delta_right_ail > 45 delta_right_ail = 45; end end

if delta_left_ail < -45 delta_left_ail = -45; else if delta_left_ail > 45 delta_left_ail = 45; 190

end end if delta_rud < -25 delta_rud = -25; else if delta_rud > 25 delta_rud = 25; end end

% if thrust < 0 % thrust = 0; % else if thrust > 25*4.448222 %4.448222 lbf = newtons % thrust = 25*4.448222; %4.448222 lbf = newtons % end % end

%delta_rud=0;

% delta_ail = input_args(13); % delta_ele = input_args(14); % delta_rud = input_args(15); % delta_thrust= input_args(16);

%************************************ Vt = sqrt(u^2+v^2+w^2); if u~=0 alpha = atan2(w,u); beta = asin(v/Vt); else alpha = pi/2; beta = 0; end tableID=1; %create thrust force vector F_thrust_b = [thrust 0 0]'; %altitude = - H = -z_e; deltadeg = [delta_right_ail,delta_left_ail,delta_rud,delta_rud,0,0]; omega = [p q r]'; rho = 1.229; %kg/m^3 1.229 bref = 2.3240; %m Sref = 1.2517; %m^2 191 cbar = .6347; %m

%this has been changed because the for005 file is setup in VINF m/s sos = 340.29;% 340.29 m/s at sea level alphadeg = alpha*180/pi; betadeg = beta*180/pi;

%aerodynamic forces [tau, f] = datcomderive(alphadeg, betadeg, H, deltadeg, Vt, omega, tableID,bref,cbar,Sref,sos,rho); F = f + F_thrust_b; T = tau; output_args = [F(1) F(2) F(3) T(1) T(2) T(3)]; return 192

Appendix B.3: datcomderive.m

%calculate the Forces and Moments function [tau, f] = datcomderive(alphadeg, betadeg, H, deltadeg, Vt, omega, tableID,bref,cbar,Sref,sos,rho)

% mach number mach = Vt / sos;

% dynamic pressure qbar = 0.5 * rho * Vt^2;

% Scale dynamic derivatives lat_scale = 0.5 * (bref) / Vt; long_scale = 0.5 * (cbar) / Vt; omega_scale = omega.* [lat_scale; long_scale; lat_scale];

% Call DATCOMTableMex IndVariables = [alphadeg, mach, H, betadeg, deltadeg]; % Vector for input into Action 2 [DepDeltaIncrements, Derivatives_Stab, DepBaseIncrements] = DATCOMTableMex(2,tableID,IndVariables); % calculate coefficients % alpha and mach effects for nominal C_norm = DepBaseIncrements; %components: [N, M, A, Y, ln, ll] % Scaling the Stability derivative coefficients C_scale = Derivatives_Stab .* omega_scale([2 2 2 3 3 3 1 1 1])'; %component: [CNQ, CMQ, CAQ, CYR, ClnR, CllR, CYP, ClnP, CllP] % Form base contributions to moments and forces C_BAE = C_norm + C_scale(1:6) + [zeros(1,3), C_scale(7:9)]; %components: [N, M, A, Y, ln, ll] % delta contributions to moments and forces C_delta = DepDeltaIncrements; %components: [N, M, A, Y, ln, ll] % % output moments and forces %MFoutput = qbar * (Sref) * [p.bref; p.cbar; p.bref; -1; 1; -1] .* (C_BAE([6 2 5 3 4 1])' + C_delta([6 2 5 3 4 1])'); %components: [L, M, N, X, Y, Z] tau = qbar * (Sref) * [bref; cbar; bref] .* (C_BAE([6 2 5])' + C_delta([6 2 5])'); %components: [L, M, N] f = qbar * (Sref) * [-1; 1; -1] .* (C_BAE([3 4 1])' + C_delta([3 4 1])'); %components: [X, Y, Z]