NASA cor~TRACTOR REPORT CR-166536
A MATHEMATICAL MODEL FOR REAL TIME FLIGHT SIMULATION OF A GENERIC TILT-ROTOR AIRCRAFT
S. W. FERGUSON
•
CONTRACT NAS2- 11317
SEPTEMBER 19881 REV. A
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NASA CONTRACTOR REPORT

CR-166536

A MATHEMATICAL MODEL FOR REAL TIME FLIGHT SIMULATION OF A GENERIC TILT-ROTOR AIRCRAFT

S. W. FERGUSON
• SYSTEMS TECHNOLOGY, INC• MOUNTAIN VIEW, CALIFORNIA

PREPARED FOR AMES RESEARCH CENTER
UNDER CONTRACT NAS2-11317

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National Aeronautics and Space Administration
Ames Research Center
Moffett Field, California 94035

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2672 BAYSHORE PARKWAY, SUITE 505 • M~J-f@!}\YJ~~.fALIFORNIA 94043-1011 • PHONE (415) 961-4674

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Technical Report No. 1195-2

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A MATHEMATICAL MODEL FOR REAL.TIME FLIGHT SIMULATION OF A GENERIC TILT ROTOR AIRCRAFT
Samuel W. Ferguson
October 1983 March 1988, Rev. A September 1988, Rev. A (Final)
National Aeronautics and Space Administration Ames Research Center
Moffett Field, California 94035 Contract NAS2-11317

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ABSTRACT
The objective of this report is to document a mathematical model for the real time flight simulation of a generic tilt-rotor aircraft which can be used in support of aircraft design, pilot training, and flight testing. The mathematical model was originally developed by Bell Helicopter Textron (BHT) under NASA Contract NAS2-6599 for the XV-15 tilt-rotor research aircraft. A real-time version of this model was implemented by Computer Sciences Corporation (CSC) on the NASA Ames Research Center (ARC) Flight Simulator for Advanced Aircraft (FSAA). Systems Technology, Inc., (STI) was given the task under NASA Contract NAS2-11317 to develop, document, and validate a generic tilt-rotor mathematical model version of the BHT mathematical model for XV-15 and generic tilt-rotor simulation on the NASA ARC Vertical Motion Simulator (VMS).
The generic tilt-rotor mathematical model development and documentation effort required that the following specific tasks be completed: (1) restructuring of the original BHT report by (a) updating the list of symbols, (b) rewriting the input/output format, (c) developing a cross reference between the VAX 11/780 and Sigma 8 versions of the generic model, and (d) modifying or adding equations to the mathematical model in several deficient areas; (2) programming, checkout, and validation of the generic tilt-rotor mathematical model; and (3) simulation support .

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TR-1195-2 (Rev. A)

iii

FOREWORD
STI wishes to acknowledge the help of several groups of people involved in developing and validating the GTRS mathematical model. Mr. Gary Churchill of the NASA ARC XV-15 Project Office was extremely helpful in directing the overall NASA generic tilt-rotor validation effort and in supporting STI from a technical standpoint. Messrs. Steve Belsley and Mike Weinstein of CSC implemented all of the STI-requested modifications to the Sigma 8/VMS version of the GTRS program and helped to check out the modifications. They also provided STI with information for development of the mathematical model/Sigma 8 cross reference (Appendix C) and spent a significant amount of time helping to review documentation in order to help insure accuracy. STI would like to acknowledge the assistance provided by BHT in providing some of the computer source code used in the VAX 11 /780 GTRS programming effort and in developing and writing the IFHC80 program (the program from which the GTRS program is derived) and its associated documentation. Without this assistance, it would have been impossible to develop the GTRS program in its present form. Messrs. Roger Marr, Narendra Batra, and Bradford Roberts of BHT were also very helpful in improving and updating the mathematical model for Revision A of this document, and their efforts are greatly appreciated.

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TR-1195-2 (Rev. A)

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TABLE OF CONTENTS

Section

Page

I INTRODUCTION AND BACKGROUND ............................... 1

A. Restructuring of the Report ••••••••••••••••••••••••••• 1

B. Implementation of the Generic Tilt-Rotor Mathematical Model on the VAX 11/780 and Sigma 8/VMS Computers ••••• 2

c. Validation of the Generic Tilt-Rotor Mathematical

Mode 1 ••••.•.•.•.••••.••••••••..••••••••..•••......•.•.

4

D. Simulation Support .................................... 5

II STRUCTURE OF THE MATHEMATICAL MODEL ....................... 7

............................... III A GENERAL DESCRIPTION OF THE MATHEMATICAL MODEL

AND INPUT DATA REQUIREMENTS

11

A• Subsystem 1: Rotor Aerodynamics •••••••••••••••••••••• 11
1. Rotor Forces and Moments .......................... 11

2. Rotor-Induced Velocity •••••••••••••••••••••••••••• 13

3. General Input Data Requirements

15

B. Subsystem 2: Rotor-Induced Velocities

(Also Parts of Subsystems 4, 5, 6, and 14)

18

1. Model Structure ................................... 18

2. Input Data Requirements ••••••••••••••••••••••••••• 20
c. Subsystem 3: Fuselage Aerodynamics ••••••••••••••••••• 24
1. Model Structure ................................... 24

2. Input Data Requirements ••••••••••••••••••••••••••• 24

D. Subsystem 4: Wing-Pylon Aerodynamics ••••••••••••••••• 24
1. Model Structure ................................... 24

2. Input Data Requirements ••••••••••••••••••••••••••• 26

TR-1195-2 (Rev. A)

V

TABLE OF CONTENTS (Continued)

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Section

Page

E. Subsystem 5: Horizontal Stabilizer Aerodynamics

27

1. Model Structure ................................... 27

2. Input Data Requirements ••••••••••••••••••••••••••• 35

F. Subsystem 6: Vertical Stabilizer Aerodynamics

35

1. Model Structure ................................... 35

2. Input Data Requirements ••••••••••••••••••••••••••• 38
G. Subsystem 7: Landing Gear ............................ 41 1. Model Structure ................................... 41 2. Input Data Requirements ........................... 41

H. Subsystem 8: Control System•••••••••••••••••••••~•••• 41
1. Model Structure ................................... 41

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2. Input Data Requirements ••••••••••••••••••••••••••• 42

I. Subsystem 9: CG and Inertia•••••••••••••••••••••••••• 42

J. Subsystems 10 Through 14: Coordinate Transformations

and Equations of Motion••••••••••••••••••••••••••••••• 43

.................... . K. Subsystem 15: Flight Environment

43

............... L. Subsystem 16: Pilot's Instrument Panel

43

............. . M. Subsystem 17: Rotor Collective Governor

44

1. Model Structure ................................... 44

2. Input Data Requirements ........................... 44

N. Subsystems 18 and 19: Engines, Fuel, Controls,

and Drive System Dynamics ••••••••••••••••••••••••••••• 45

1. Model Structure ................................... 45

2. Input Data Requirements ••••••••••••••••••••••••••• 47

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TR-1195-2 (Rev. A)

vi

LIST OF FIGURES

Humber

Page

1 Generic Tilt-Rotor Mathematical Model Structure and

Input/Output Summary••••••••••••••••••••••••••••••••••••••

8

2 Side-by-Side Rotor Effect on Induced Velocity••••••••••••• 16

3 Sideward Flight Rotor Effects on Induced Velocity••••••••• 16

4 Effect of Ground Proximity on Hover Power Required •••••••• 17

5 Rotor Wake on the Horizontal Stabilizer as a Function of Airspeed at a Nacelle Incidence of 90 Degrees •••••••••• 21

6 Representation of In-Ground Effect Rolling Moment ••••••••• 22

7 The Effect of Hover Height on Longitudinal Stick Position••••••••••••••••~••••••••••••••••••••••••••• 23

....... 8 Wing-Pylon Lift Coefficient Versus Angle of Attack

for Flap/Flaperon Settings of D/0 and 40/25 Degrees

28

9 Wing-Pylon Lift Coefficient Corrections Due to Compress! bili ty ........................................... . 29

.... Wing-Pylon Lift Coefficient at Large Negative Angles

of Attack for a Flap/Flaperon Setting of 40/25 Degrees

30

11 Wing-Pylon Drag Coefficient Versus Angle of Attack for Flap/Flaperon Settings of 0/0 and 40/25 Degrees ••••••••••• 31

12 Wing-Pylon Drag Coefficient Corrections Due to C~mpressibility ••••••••••••••••••••••••••••••••••••••••••• 32
13 Wing-Pylon Drag Coefficient at Large Negative Angles of Attack for a Flap/Flaperon Setting of 40/25 Degrees ••••••• 33

14 Wing-Pylon Wake Deflection (Downwash) at the Horizontal Stabilizer for Flap/Flaperon Settings of 0/0
and 40/25 Degrees ••••••••••••••••••••••••••••••••••••••••• 34

15 Horizontal Stabilizer Lift Coefficient Versus
Angle of Attack••••••••••••••••••••••••••••••••••••••••••• 36

16 Horizontal Stabilizer Drag Coefficient Versus Angle of Attack••••••••••••••••••••••••••••••••••••••••••• 37

TR-1195-2 (Rev. A)

viii

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LIST OF FIGURES (Concluded)

Number 17
18

Page
Vertical Stabilizer Side Force Coefficient Versus Sideslip Angle•••••••••••••••••••••••••••••••••••••••••••• 39
Vertical Stabilizer Drag Coefficient Versus Sideslip Angle•••••••••••••••••••••••••••••••••••••••••••• 40

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TR-1195-2 (Rev. A)

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TR-1195-2 (Rev. A)

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SECTION I INTRODUCTION AND BACKGROUND

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The objective of this report is to document a mathematical model for the real time flight simulation of a generic tilt-rotor aircraft which can be used in support of aircraft design, pilot training, and flight testing. The mathematical model was originally developed by Bell Helicopter Textron (BHT) under NASA Contract NAS2-6599 for the XV-15 tilt-rotor research aircraft (Ref. 1). A real-time version of this model was implemented by Computer Sciences Corporation ( CSC) on the NASA Ames Research Center (ARC) Flight Simulator for Advanced Aircraft (FSAA). Systems Technology, Inc., (STI) was given the task under NASA Contract NAS2-11317 to develop, document, and validate a generic tilt-rotor mathematical model version of the BHT mathematical model for XV-15 and generic tilt-rotor simulation on the NASA ARC Vertical Motion Simulator (VMS) . The first release of this development effort was completed in October 1983.
The generic tilt-rotor mathematical model development and documentation effort required that the following specific tasks be completed: ( 1) restructuring of the original BHT report by (a) updating the list of symbols, (b) rewriting the input/output format, (c) developing a cross reference between the VAX 11 /780 and Sigma 8 versions of the generic model, and (d) modifying or adding equations to the mathematical model in several deficient areas; (2) programming, checkout, and validation of the generic tilt-rotor mathematical model; and (3) simulation support.

A. RESTRUCTURING OF THE REPORT

The tilt-rotor mathematical model equations, as originally derived, represented the kinematic, dynamic, and aerodynamic characteristics of the XV-15 rotor, airframe, and flight control system. A description of the development of the mathematical model, in its original form, is presented in Ref. 1. The equations presented in that report are, in many instances,

TR-1195-2 (Rev. A)

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revised in this report to provide an improved generic model as based on XV-15 flight test data. The equations of this improved generic tilt-rotor mathematical model are provided in Appendix A of this report. The XV-15 input data array taken from Ref. 1 has also been significantly updated and restructured to the generic mathematical model input format and is presented in Appendix B of this report.
All pages from the original BHT mathematical model report (Ref. 1) which remain unchanged are presented in this report with the Bell report number, 301-099-001, located in the lower left-hand corner. New or corrected pages are identified by the STI report number, TR-1195-2. Pages that have been revised for this edition of the STI report are labeled TR-1195-2 (Rev. A).
Appendix C of this report contains a cross reference, developed by STI and CSC, of the mathematical model input data array and the associated computer variable names used in the Sigma 8/VMS version of the program.
B. IMPLEMENTATION OF THE GENERIC TILT-ROTOR MATHEMATICAL MODEL ON THE VAX 11/780 AND SIGMA 8/VMS COMPUTERS
The initial version of what is now the generic tilt-rotor program was developed in the 1970s by BHT for use as an offline XV-15 tilt-rotor analysis tool. A version of this program, IFHC80, was delivered in 1980 to NASA ARC in a non-generic form for use with the XV-15 only. This program is based on the XV-15 tilt-rotor mathematical model of Ref. 1 and was used as a checkout tool prior to BHT XV-15 simulations. A user's guide and programmer's guide, Refs. 2 and 3, were delivered for use with this program.
STI used the IFHC80 program, as requested by the XV-15 Tilt Rotor Project Office, as a basis for development of the generic tilt-rotor simulat ion program ( GTRS) described in this document. The GTRS program has been implemented on the NASA ARC VAX 11/780 computer, and the effort has involved an extensive reformatting and recoding of the IFHC80 program's complete input/output structure and format. In addition, several computer

TR-1195-2 (Rev. A)

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programming errors were corrected during the creation of the GTRS program. During development of both the original version and Revision A of GTRS, informal discussions were held between STI and BHT in an effort to define areas of similarity which might be maintained between STI's GTRS program and versions of a generic tilt-rotor program that have been developed by BHT for their internal use. As a result of these discussions, STI has adopted some FORTRAN coding supplied by BHT for use with the GTRS program. Almost all of this code is related to the input and internal storage of aerodynamic data so as to maintain commonality between BHT and NASA in the way in which tilt-rotor aerodynamic data is described for use in the program. During the debugging and checkout of the STI GTRS program, BHT was notified of the coding modifications and changes that would be required for any future use of the BHT-supplied FORTRAN code. This was because some of the STI code was developed before some of the BHT code.
A user's guide and a programmer's guide have been written for the VAX 11 /780 version of the GTRS program and were originally available as Refs. 4 and 5, respectively. Both of these reports are now superseded by Revision A versions with the same titles ( their release date is the same as the release date of this document). Appendices I and J of Ref. 4 provide a cross reference between the input data and computer variable names and the equations in the original version of this document for both the
VAX 11/780 and Sigma 8/VMS* computer versions of the generic tilt-rotor
mathematical model. The Sigma 8/VMS version has not been released in a Revision A upgrade. All information contained in Refs. 4 and 5, other than that supplied in Appendices I and J, is intended to apply to the STIdeveloped VAX 11 /780 version of the GTRS program only, unless otherwise specified, even though there are many similarities among the STI-, CSC-, and BHT-developed versions of the mathematical models and the associated versions of computer code.

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*The Sigma 8 version of the GTRS program was developed by CSC under a separate contract and is used presently for real-time simulation of the XV-15. GTRS is also an off-line version developed by STI for use on the VAX 11/780 computer.

TR-1195-2 (Rev. A)

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C. VALIDATION OF THE GENERIC TILT-ROTOR MATHEMATICAL MODEL
The original XV-15 mathematical model (Ref. 1) was validated by BHT through the use of wind tunnel tests, other computer programs, and limited flight tests. Work accomplished by STI has been directed toward validation of the GTRS program using the earlier XV-15 data base as well as the extensive flight test data base which is presently being developed with the XV-15. Both the VAX 11/780 and Sigma 8/VMS versions of the GTRS program have been used in the validation effort. Output from both of the simulation programs has also been compared for numerous flight conditions in order to ensure that both programs yield the same calculated results. While conducting the validation study with flight test data, the following limitations/deficiencies were identified by STI.
1. The prediction of hover performance was originally found to be clearly overly optimistic (helicopter and airplane forward flight performance was only slightly over predicted).
2. In-ground effect rotor modeling was unacceptable for rotor power calculation.
3. In-ground effect pitching moments were not predicted as observed in flight test.
4. The calculated hover in-ground effect rolling moment instability was excessive and of too high a frequency in comparison with flight data.
S. Spinner drag modeling was discovered to be implemented incorrectly.
6. Pylon drag modeling (including wing-pylon interference drag) was determined to be inadequate.
7. A static B1 rigging offset term was not included in the control system model so that the rotor controls could be rigged like the XV-15.
8. The XV-15 20-degree flap position (and associated aerodynamic tables) was not available for selection by the pilot with the model (this flap position is one of the three normal XV-15 flap positions)

TR-1195-2 (Rev. A)

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9. Simulated trimmed sideward flight data did not correlate well with XV-15 flight test data.
10. Short takeoffs and landings were found to require too much distance (possibly due to the lack of a wing inground effect model and inaccurate rotor/wing flow field modeling while in ground effect).
11. Questionable input data values were identified for elevator, rudder, and aileron effectiveness as well as the Q-loss value at the respective control surfaces (as observed through correlation of aircraft simulation response to flight test response for the same control input).
12. Values for the XV-15 inertias were demonstrated to be out of date ( airframe modifications and flight test instrumentation weights and locations were not included in the calculated inertias).

BHT was notified of each these model limitations/deficiencies. Modifications were made to the GTRS program or input data values which resolved all of the limitations/deficiencies except for the deficiencies involved with short takeoffs and landings. An investigation into the STOL deficiency was beyond the scope of effort STI was tasked to accomplish at that time. Interim results from the mathematical model validation effort are presented in Ref. 6. The final report (Ref. 7) for the contract provides a more detailed discussion of the results from the validation effort.

D. SIMULATION SUPPORT

STI provided engineering support to NASA and CSC for the initial generic tilt-rotor simulation validation effort that was conducted at NASA ARC from January to April 1983. The support to NASA was provided in order to aid in the evaluation of the XV-15 data input configuration (in the generic mathematical model format) and to modify the model as required. Both open- and closed-loop evaluations of the model were conducted using NASA and military XV-15 pilots. CSC support was provided to aid in implementation and checkout of the generic model on the Sigma 8 computer
and the VMS. Major off-line simulation efforts were conducted in 1983 and

TR-1195-2 (Rev. A)

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1984 to investigate improvements to the mathematical model and to correlate results with flight test data taken specifically for simulation validation purposes. Other off-line validation efforts have been conducted using the VAX 11/780 version of the program beginning in 1983 and continuing to the release of this report. Some of these efforts have also involved work with tilt-rotor configurations other than the XV-15.

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TR-1195-2 (Rev. A)

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SECTION II STRUCTURE OF THE MATHEMATICAL MODEL

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The generic tilt-rotor mathematical model structure is presented in the block diagram shown in Fig. 1. The mathematical model differs from that of a conventional fixed-wing aircraft in that there are added requirements to represent the dynamics and aerodynamics of the rotors, the interaction of the rotor wake with the airframe, and the rotor control and drive systems. The rigorousness of the mathematical model of the tiltrotor aircraft was constrained by two factors. One factor was the requirement to keep the computational loop time to less than 70 ms in order to maintain a real time simulation. In order to achieve this, it was necessary· to limit the rotor representation to steady, linearized aerodynamics having a uniform inflow and to approximate the rotor following time. Rotor stall and compressibility effects were used only to define a limit for the maximum rotor thrust coefficient as a function of advance ratio. This rotor mathematical model is satisfactory for most handling qualities studies but may be inadequate to evaluate flight conditions or maneuvers where stall, compressibility, or rotor dynamics are significant.
A second factor constraining the rigorousness of the mathematical model was the lack of sufficient experimental data on rotor wake-airframe aerodynamic interactions, such as the downwash (or upwash) of the rotors at the horizontal tail. The model of the rotor wake-airframe interaction was initially based on a limited amount of data from tests of a powered model of a tilt-rotor aircraft similar to the XV-15. Tests were subsequently completed using a powered model of the XV-15 to obtain detailed information on the rotor wake-airframe aerodynamic interactions. This data was used to update the simulation and refine the model for this important characteristic of a tilt-rotor. Other revisions were made to the mathematical model during the aircraft development in order to reflect design changes in the aircraft, corrections to the mathematical model, and

TR-1195-2 (Rev. A)

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1-"3 :::0
....I......
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CP
X Module numbe1, T Module title I Inputs from
other modules 0 Outputs to
other modules

I ROTOR
AEJRUODYNAHICS
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2 ROTOR Il'IDUCF:D
VELOCITIES

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' FUSELAGE AERODYNAMICS

INPtlT
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OUTPUT
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6,7 9, !Ob
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Control

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(8d)

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FORCE TRIM
SYSTEM

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OUTPUT 8b

JOe EULER ANGLE
TRANSiFORMATIONS
__T I E.!!!!!!!
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14 IHOHF.NT
SUUMMATION

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INPUT
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INPUT
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18

10. GROUND VELOCITY
SUMMATION

INPUT
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OUTPUT
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16 PILOT'S
INSTRUMENT
PAKEL

INPUT
7,8d IOe JOf 12 15 17 18 19

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' WING-PYLON AEROOTIMHICS

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•• PILOT'S CONTROL FUNCTION

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10, GROUND
REFF.RF.NC! DISTANCES

INPUT
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17 ROTOR
COLLECTIVE GOVERNOR

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' HORIZONTAL STABILIZER AERODYNAMICS

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6 10•

6 VERTICAL STABil,IZEJt
AERODYNAMICS

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OUTPUT 10•

1 LANDING CF.AR

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8d
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10• tJ 14 16

' C.G. AND INERTIA

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10,

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11 AlRCRAFT ANGlfl.AR
ACCELERATION AND VELOCITIES

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9 II 13 14

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18 ENGINES AND FUEL
CONTROLS

..INPUT

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16

12

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tJ

19

!Oa AIRFRAHF.: lllND TO BODY AXIS
TRANSFORMATION

~
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12 BODY AXIS l,INF.AR
ACCELf.RATIONS AND
VELOCITIES

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19 DRIVE
SYSTEM
DYNAMICS

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!Ob ROTOR WIND
TO BOOT AXIS TRANSFORMATION

INPUT
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"

tJ FORCE
StlMKATION

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7 10, !Ob 18

~
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20 SCAS
SYSTEH
.. .. lf!m ~ 8d toe 11 12

Figure I• Generic Tilt -Rotor Mathematical Model Structure and Input/Output Summary

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additions or improvements to the mathematical model. This latest revision provides the most recently updated documentation of the mathematical model in its generic tilt-rotor form. Many of the changes to the Revision A version of the mathematical model involve improvements that are incorporated as a result of correlation with XV-15 flight test data •

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TR-1195-2 (Rev. A)

9

• • •
TR-1195-2

•

SECTION III
A GENERAL DESCRIPTION OF THE MATHEMATICAL MODEL AND INPUT DATA REQUIREMENTS

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This section describes the mathematical models of the generic tiltrotor aircraft components--the rotors, the airframe, the control system, the engines and drive system, and the automatic flight control systems (Subsystems 1 through 9, and 17 through 20 in Fig. 1)--and the input data requirements for those components. The equations of motion used with the mathematical model (Subsystems 10 through 14 in Fig. 1) are the same as those found in Ref. 8.
Earth-, body-, wind-, and mast-axes systems are used in the generic tilt-rotor mathematical model. The rotor flapping, forces, and moments are calculated in a "wind-mast" axis system, while the airframe aerodynamic forces and moments are calculated in a wind-axis system. Forces and moments from the rotor and airframe are then resolved into the body-axis system for solution of the aircraft equations of motion. The flight path of the tilt-rotor is described with reference to earth-fixed axes with the orientation given by the Euler angles~, 0, and~, in that order of rotation. Details on individual subsystem sign conventions are provided in the following sections.
A. SUBSYSTEM 1: ROTOR AERODYNAMICS
l. Rotor Forces and Moments
The mathematical model of the rotor is similar to that described in Refs. 9 and 10, except that it is derived in a mast-axis system (the theory in Ref. 9 is based on an axis system perpendicular to the axis of no flapping, i.e., the tip-path-plane, and that of Ref. 10 is based on the

TR-1195-2 (Rev. A)

11

axis of no feathering) and contains provisions for prop-rotor character-
!sties such as nonlinear twist, flapping restraint, and pitch-flap
coupling. The mast-axis system and sign convention used for the rotor are
shown in Fig. Al-1 (in Appendix A). The rotor flapping, forces, and mo-
ments are calculated in the "wind-mast" axis system (a 1 , b1 , T, H, and Y) and are then transformed into the mast-axis system (a1, b1, T, H, and Y).
Major assumptions that are made in the rotor mathematical model
include:
1. Average values for the lift-curve slope and profile-drag coefficient are used over the entire span of the blade. These are adjusted to approximate the rotor thrust- and power-required characteristics.
2. The blade angle of attack, ar, is approximated by sin ar• Substitution of sin a for ar in the blade element equations makes it possible to develop equations for rotor forces with-
out restricting blade pitch, a, and inflow angle, ~, to small
angles.
3. Blade flapping with respect to the mast is considered to be small so that the small angle assumption can be made, and harmonics of flapping greater than one-per-revolution are ignored.
4. The blade flapping due to cyclic inputs is assumed to occur instantaneously, i.e., the flapping equations assume that the rotor is in an equilibrium condition. This assumption was made because of limits imposed by the computation time of the simulation computer. Differential equations for blade flapping that would properly account for the rotor following time were determined to require a solution time in excess of that allowable for real time simulation. Furthermore, there is a transport lag, between the time that a control input is made at the simulator cab and the time that an aircraft response is updated at the cab (by the motion and visual systems), of from one to two frame times. By neglecting the rotor following time in the equation of motion, this transport lag is approximated by the cab-control input to computer time lag; for example, in hover, the rotor following time is 0. 08 sec compared to an average computational lag time of at least as much as 0.075 sec using the Sigma 8 computer with the VMS located at NASA ARC.

TR-1195-2 (Rev. A)

12

•
• •

• •

S. Blade stall and compressibility effects are approximated by limiting the maximum rotor thrust coefficient as a function of advance ratio and by arbitrarily modifying coefficients in the rotor power required equation (i.e., rotor profile drag is increased as a function of the cubes of the rotor inflow and advance ratios multiplied by empirically adjusted coefficients).
2. Rotor-Induced Velocity
The rotor-induced velocity is computed by calculating the induced velocity of an isolated, out-of-ground effect rotor and then modifying the induced velocity to account for the side-by-side rotor effect, the tandem rotor effect (in sideward flight), and for operation in ground effect.
The mean value of the isolated, out-of-ground effect rotor-induced velocity is approximated using a modified expression from Ref. 11.

=

(QR)C

o.6lcTl 1"5 (lcTI - 8/3xlxl)

+ ------.,...-------,--

( Ic I+ 8µ 2 ) (lcl + sx 2 )

•

where C = CT/2B2 (the 0.866 factor on x2 has been added to improve power
correlation in hover).
The major assumption made with regard to induced velocity is that it is uniform over the rotor disk. The main effect of this assumption is that lateral flapping is underpredicted in the low-speed helicopter regime ( µ = 0. OS to O. 2). However, lateral flapping has only a second-order effect on stability and control characteristics in the helicopter mode, so this is not a serious limitation.
The side-by-side rotor effect on the rotor-induced velocity is approximated using an expression derived in Ref. 12 .

TR-1195-2 (Rev. A)

13

(QR)CT

=

xss

2B

2 µ

The factor x88 is called the mutual induction coefficient, and it is obtained from Fig. 3.7 of Ref. 12. In the determination of x88 , the
increased mass flow of the side-by-side configuration is taken into account, and the rotor wakes are assumed to remain separate if the distance between the rotor centers is greater than the rotor diameter. The value
of x88 depends on the direction of rotation, the distance between the
rotors, the advance ratio, and the rotor angle of attack. The value of
x88 given in Ref. 12 is valid forµ greater than 0.15. In this analysis, x the value of 88 forµ less than 0.15 has been approximated by providing a x smooth transition between a value of 88 equal to zero atµ= 0.06 and the value at µ = 0 .15. The term ~vi ss is added to the induced velocity for
the isolated rotor during the induced-velocity solution process.
The added induced-velocity component at the trailing rotor of the tilt rotor in sideward flight ( the tandem rotor effect) is approximated as a
function of the normalized sideward flight velocity (V). This component,
tNi , is then added to the induced velocity for the isolated trailing SF
rotor, along with the value for ~V. during the induced-velocity solution
1 SS process.
The reduction in induced velocity caused by ground effect is computed using an exponential expression

• •

where

G =

1-GEC0Nl (eGEC0N2(h8 /2R) )

and

W =

2 GEWASH(u

2 1/2
+v) •

> 0.001 or G 1, then G is set equal to one. This form of ground effect

equation is a variation of an equation derived by Hayden in Ref. 13 and

shown in Ref. 6 to provide excellent correlation of the mathematical model

with XV-15 flight test data. The factor ew washes out exponentially the

effect of ground proximity with forward speed. At 30 ft/sec and greater,

the effect is completely washed out.

TR-1195-2 (Rev. A)

14

•

•
•
•

3. General Input Data Requirements
The input data requirements for the rotor are described in an organized format on Pages A-5 through A-12 of Appendix A. The majority of the required rotor input data values are geometric constants which are selfexplanatory or are rotor- or blade-specific parameters which are configuration dependent [e.g., o3 , blade inertia (lb), flapping spring
rate (KR)]. The values for average rotor blade lift-curve slope and drag coefficient, a0 , 1 , 2 and oo,l,Z respectively, should be iteratively deter-
mined using rotor test stand data or other rotor performance programs via correlation with the generic tilt-rotor program output. If this type of approach is not possible, or if data does not exist, then input data values for these parameters should not be input without careful consideration, because it is highly unlikely that any prop-rotor configuration will have average rotor blade aerodynamic characteristics similar to the low twist and usually single airfoil section characteristics of untapered helicopter rotor blades.
Input data requirements for determining side-by-side (x88 ), tandem rotor (X8F), and ground effects are obtained using sources such as those discussed in the previous section. In most cases it would be expected that the input data used for the XV-15 would be appropriate for most tilt-
rotor investigations. The values for x88 and XsF are obtained from data
tables in the simulation computer program (plotted in Figs. 2 and 3); whereas, the coefficients for the ground effect equation (GECONl and GECON2) were iteratively determined by curve fitting data (originally presented in Ref. 14) and then correlating with XV-15 flight test data
(Fig. 4 from Ref. 6).
Input data values for Mach number effects and induced-velocity coefficients have been determined from experience and correlation with XV-15 wind-tunnel and flight-test data. Unless specific knowledge about rotor characteristics unquestionably indicates that a change is needed in one of these parameters, it is recommended that XV-15 values be used .

TR-1195-2 (Rev. A)

15

t,

0.02

•r-l C) I
OA r-l Z
(]) I
t> 'O ,,....."._'
(]) tr.!
~ :x:Ci'.l
'COl ----H .µ
C:
(]) (])
'O •r-1
•r-1 C) Ci'.l •r-1
I 4-i i>.,lt--i
f ~
~u

0 -0.02 -0.04 -0.06 -0.08

\
\
\
i ~

•r-1

Ci'.l

-0.10

0.0

0. 1

0.2

0.3

o.4

Advance Ratio Cu), -ND-

Figure 2. Side-by-Side Rotor Effect on Induced Velocity

i>., .µ
•r-1 C)
0 I
r(]-) lAz
t> I

'O(]) ,,....."._' CJ lz.i
~ Ci'.l 'O :x:

HC:

----.µ

.µ C:

,Q (])

bO •r-1

•r-l C)

r-l •r-l

1z.it 'O (]) ~ 0

~u

(])
'O
•r-1 Ci'.l

Lateral Velocity Advance Ratio (V), -ND-

Figure 3. Sideward Flight Rotor Effects
on Induced Velocity

TR-1195-2

16

• • •

• •

0 XV-15 Hover Data
GTRS Program

~ 3.0

,--."..'
p:j
~..__,
H 2.0
Q)
.p
Q)
@
•r-1
§ .o ...i.=...l... 1
•r-1
Q)
p:l

H

s+0'

0

1 .o

0.9

o.8

0.7

0.6

Total Power/Total Power OGE (P/P) 00

Figure 4 . Effect of Ground Proximity on Hover Power Required

•

TR-1195-2

17

The tables provided for setting an upper bound limit on usable rotor thrust coefficient CT are defined as a function ofµ and am. These tables can be modified from the XV-15 values based upon either analytical or rotor test data from the rotor which is to be simulated. For simulated flight conditions not requiring high thrust, e.g., high-g maneuvers, these tables have no effect on the calculated results and would not be in need of modification.
B. SUBSYSTEM 2: ROTOR-INDUCED VELOCITIES (ALSO PARTS OF SUBSYSTEMS 4, 5, 6, AND 14)
The rotor wake-airframe aerodynamic interferences (or rotor-induced velocities) represented in the generic tilt-rotor mathematical model consist of three parts:
1. The effect of the rotor wakes on the wing lift and drag.
2. The effect of the rotor wakes on the horizontal stabilizer and vertical fin lift and drag.
3. The effect of the rotor wake-airframe-ground interaction in producing net rolling moment and pitching moment effects when hovering near the ground.
1. Model Structure
The calculation of the wing aerodynamic forces and moments due to rotor wake effects is made separately from the forces and moments generated by the freestream flow. The calculation of the rotor wake effect involves calculating the area, angle of attack, and dynamic pressure of the portion of the wing immersed in the wake. Figure A4-l (in Appendix A) illustrates the representation of this effect.
The area of the wing immersed in the rotor wake, SiW ( shown in Fig. A4-l) is computed as a function of wake radius, conversion angle, wake angle of attack, and sideslip angle of the fuselage. The expression used to compute the wake radius of a hovering rotor as a function of vertical distance from the rotor disk is derived in Ref. 15. Experimental

TR-1195-2 (Rev. A)

18

• • •

•

data also show that the contracted wake remains stable as it reaches the wing and horizontal stabilizer surfaces. Therefore, the equation for the wake radius (Eq. 3 of Ref. 15) has been simplified, since the wing and stabilizer surfaces are located at approximately 0.4 R below the rotor disk.
= R{0.78 + 0.22 Exp [-(0.3 + 2Z ✓cRF + 60 CRF))}

The rotor-induced velocity at the wing varies with speed and mast tilt and is given by the following expression:

• •

w.,
1 R/W

where the constants K0_4 are determined from powered rotor test data • Wing loads at high negative incidences caused by the rotor wake at low speeds are determined using lift and drag coefficient data tables that are defined up to angles of attack of ± 90 deg. Asymmetric flight at low speeds, which causes unequal portions of the left and right wing to be affected by the left and right rotor wakes and which generates roll and yaw moments, is also taken into account.
The induced velocity at the horizontal stabilizer and the vertical fins (a function of airspeed and mast angle) is determined by first calculating the rotor-induced velocity for trimmed flight and then correcting it for angle of attack and sideslip from data tables based upon windtunnel data. The values calculated are assumed to be constant across the empennage for the analysis.
When hovering in ground effect (h/D < 2 .0), both an unstable rolling
moment and a pitching moment are generated by aerodynamic interaction between the rotor wake, fuselage, wing, horizontal stabilizer, and the ground. The rolling moment effect is represented in the mathematical model by a polynomial equation for the rolling moment as a function of h/d

TR-1195-2 (Rev. A)

19

and then applied at the aircraft center of gravity. The in-ground effect pitching moment is modeled as an exponential function of rotor thrust, rotor hub height above the ground, and airspeed; and the pitching moment is applied at the aircraft center of gravity. The decision to model this effect was made following an evaluation of pilot comments and flight test data first presented in Ref. 6 and later in Ref. 16.
2. Input Data Requirements
The details of the input data requirements are listed on Pages A-34
through A-37 of Appendix A. The coefficients K0 , 1 , 2 , 3 , 4 are determined
from powered-model wind-tunnel data. The rotor-induced velocity at the horizontal stabilizer and vertical fins is also based on powered-model wind-tunnel data. The velocity induced at the tail by the rotors was derived for the XV-15 by analysis of pitching moment data with the tail ON and OFF as well as with and without the rotors (Refs. 17 and 18). Data
generated by this method should look similar to the XV-15 data for Bm = 0
. deg presented in Fig. 5, which is plotted from Appendix B, Table 2-Ia on Page B-22. (Data for Bm values other than O deg are not plotted but are contained in the tables.) Further corrections to data from these tables (which are corrections for angle of attack) are made for sideslip from Table 2-II.
The data used to fit the polynomial equation for the rolling moment data was measured using a 0.2 scale powered XV-15 wind-tunnel model (Ref. 17). This data is shown plotted in Fig. 6. The data used to fit the in-ground effect pitching moment equation is based on flight test data from Refs. 14 and 16, which is presented in Fig. 7, and compared with the simulation results using the GTRS program.

• •

•

TR-1195-2 (Rev. A)

20

•

0.2

•

o.o

-0.2

-o.4

•r-1
-•r~-:1 : : :

-o.6

!3:

-o.8

-1 .o -1 .2

-1.4 0

Mast Angle, ~ = 9:) degrees (13m = 0 deg)
~ = -30
- - 16 deg -

, __ ,,
-16 deg

20

4o

6o

80

100

120

140

Airspeed, knots

Figure 5. Rotor Wake On the Horizontal Stabilizer as a Function of' Airspeed
at a Nacelle Incidence of' 90 Degrees

•

TR-1195-2

21

•
2.0

i

•r-1
~

::-i

Stable Unstable

0

+:>

0

P:.

0

0

X:V-15 Touchdown

'}JO

1000

Rolling Moment, ft-lb/deg

Figure 6. Representation of In-Ground Effect Rolling Moment

TR-1195-2

22

•

•
0 Flight Data (Ref. 14) • GTRS Data

•

p:j

40

~

.Cl

F-"t '

0)

.p

t: 30

I

i"'

•rl
A F-t

•rl
~

1 .5

0 ..µ

0

r-1
0)

20

§

~ ~
llll •rl

~

1-0

F-t 0

10

..µ 0

p:j

0.75

o.6
0 30

I I I I
• I I I
, I
I
•,,,,,I'
,
I
I
·•I I
I
I

40

50

Longitudinal Stick Position, percent forward

Figure 7. The Effect of Hover Height on Longitudinal Stick Position

•

TR-1195-2

23

C. SUBSYSTEM 3: FUSELAGE AERODYNAMICS

I. Model Structure

The fuselage, wing-pylon assembly, horizontal tail, and vertical fins are modeled separately in order to facilitate accounting for the influence of the rotor wake on the airframe aerodynamics. Equations for the fuselage lift, drag, side force, pitching moment, yawing moment, and rolling moment are referenced to the wind-axis system and defined at the input fuselage center of pressure. Aircraft angular rates as well as the rotor wakes are neglected in calculating the fuselage aerodynamic forces and moments.

2. Input Data Requirements

In general, the wind-axis airframe aerodynamics are extracted from wind tunnel test data. For the XV-15, this data is tabulated in Appendix B on Pages B-26 through B-30. Where wind tunnel data was not available for the XV-15, characteristics were estimated using Refs. 19, 20, and 21. [For the XV-15, the coefficients in the equations for angles of attack and sideslip less than or equal to 20 deg are based on windtunnel data. For angles of attack greater than 20 deg, the coefficients have been approximated.] The values for the constants LBFO, DBFO, and MBFO are the same values as those in the data tables at aF =SF= 0 and must be subtracted out. Otherwise, the equations would add the respective numbers together twice (once from each of the aF and BF tables), thereby resulting in double the actual value being used in calculations.

D. SUBSYSTEM 4: WING-PYLON AERODYNAMICS

1 • Model Structure

The wing-pylon aerodynamic forces and moments are defined in the local wind-axis system. Wing-body interference effects are included in the aerodynamic data.

TR-1195-2 (Rev. A)

24

• • •

• •

Calculation of the wing aerodynamic forces and moments is made up of two parts: the first part is composed of the part of the wing which is influenced by the rotor wakes, and the second part, that which is influenced by only the free stream flow. The mathematical model and all sign conventions are described and flow charted in Appendix A or in the previous section of text (Subsystem 2).
The wing-pylon lift and drag generated by the free stream flow are functions of angle of attack, conversion angle, flap setting, and Mach number. The pitching moment is a function of flap setting.
The wing lateral-directional aerodynamic forces and moments are calcu-
lated using equations for stability derivatives from Ref. 19.
Compressibility effects and the wing loading are included in the lateraldirectional characteristics.
Wing-pylon lift and drag coefficients are provided for mast angles of
0 deg and 90 deg and for four flap settings. Coefficients for interme-
diate mast angles and flap settings are obtained by interpolation. Mach number corrections are made only for the flaps-up airplane mode configuration.
The angle of attack of the wing is also modified in order to reflect the induction effect of the thrusting rotors. The expression for the wing angle of attack is:

CRFR + CRFL
= ~F - KXRW (xR/W ) [ - -2- - - ] ( 5 7 . 3 )
MAX (µ, 0.15)
where xR/W' the induction coefficient, is a function of the distance between the rotor and the wing and of mast angle; and CRFR,L are the nondimensionalized rotor force coefficients for the right and left rotors .

•

TR-1195-2 (Rev. A)

25

2. Input Data Requirements
The wing subsystem requires more data input than any other section of the GTRS model. A detailed listing of the input data requirements is provided on Pages A-45 through A-56 in Appendix A. Constants and many of the coefficients listed on Pages A-45 and A-46 are either wing geometric values or can be calculated using Ref. 19. (Other sources for calculation of wing lateral-directional stability derivatives should also be acceptable). Values for calculation of the constants in the equation for the rotor flow field effects on angle of attack are for the XV-15 and, in general, should be applicable for other tilt-rotor configurations similar to the XV-15. The constants in the rotor downwash/wing equations for flap effects are based on wind-tunnel or flight-test data and are used to adjust wing download as a function of flap setting. The spinner drag coefficients were determined for the XV-15 from wind-tunnel test data of the full scale XV-15 rotor and pylon (shown in Ref. 20). Values for the pylon interference drag were determined for the XV-15 from flight-test data and were a correction or addition to the model in order to account for extra drag due to wing-pylon interference. Significant differences exist between the "smooth and clean" skin surfaces of the wing tip and the inside surface of the pylon for the XV-15 wind tunnel model and the surfaces around the XV-15 wing-pylon interface. (These differences can easily be seen in a photograph of the XV-15 in helicopter flight.) This input variable will probably not be obtainable from wind-tunnel data, since the pylon drag will normally be included with the wing drag and input into the wing-pylon tables described in this model. However, in evaluating a tilt-rotor configuration using this program, it would nevertheless be advisable to use XV-15 input data as a minimum if flight-test data cannot be obtained. The effect of this parameter can be significant in the deceleration of the tilt rotor during reconversion to helicopter mode and is noticeable by pilots in a manned simulation environment.
Coefficients for wing lift, drag, and pitching moment should be obtained whenever possible through use of wind-tunnel testing. The XV-15 aerodynamic coefficients which are supplied in Appendix B (Pages B-31

TR-1195-2 (Rev. A)

26

• • •

•
•

through B-55) are based on wind-tunnel data for angles of attack up to stall. At angles of attack above stall, the coefficients are approximated based on the test data presented in Ref. 21. Examples of how data should look for wing lift and drag for the flap/flaperon settings of 0/0 and 40/25 deg are presented in Figs. 8 through 13. The dihedral effect of the wing-pylon is based on wind-tunnel test data and is a function of angle of attack and flap setting as well as sideslip. The aileron effectiveness and yawing moment coefficients are also based on wind-tunnel data (or in some cases may have to be calculated) and are a function of angle of attack, mast angle, and flaperon deflection.
The wake deflection or downwash at the empennage due to the wing-pylon for the XV-15 is determined from wind-tunnel data for angles of attack up to stall. Above wing stall, the downwash is approximated using data for the high wing-low tail configuration given in Ref. 22. Figure 14 presents example data for the XV-15 for two flap/flaperon positions at two mast angles (helicopter and airplane). The downwash at the empennage due to the rotor wake is discussed in a previous section.
E. SUBSYSTEM 5: HORIZONTAL STABILIZER AERODYNAMICS
I. Model Structure
Detailed input data requirements for the horizontal stabilizer model are described on Pages A-78 through A-82 in Appendix A. The dynamic pressure and angle of attack calculations for the horizontal stabilizer model, as shown in Fig. AS-1, take into account wing-body blockage, mast angle, the wing-pylon wake, the rotor wake, and the fuselage attitude and angular velocities •

•

TR-1195-2 (Rev. A)

27

8
..!.:.Ic....l

2.0

I I I I

\D \J1
I I\)
I\)
CX>

,......,,
J
..0___,
.µ Cl
Q)
•.-1
t)
•.-1 li-1 li-1
(!)
0 0
~
•.-1 ...:I
§
i I
~
•.-1 ~

1.6 1.2
o.8 0.4 0 -o.4 -o.8

-

iN = 0 deg (airplane)

/ ' ' - - iN = 90 deg (helicopte ~ ~

/

VI//v/j/ -", __ -- -- I ./ I

' Flap/Flaperon Settin1
/// / in degrees= x/y

/
'//

v-- ~

/

. ~

_/'
,.

--

., / /

/_,.,,-

.....

,/

.•

4o~o/~ )~v/

--- 1 -- ----- -

I
/J

, , f10

- J'. .,,, ~

~

'~

/:I
,;

I/

V ~ "'-' I

-1.2

-40 -32

-24 -16 -8

O

8

16

24

32

40

Wing-Pylon Angle of Attack (aw), degrees

Figure 8. Wing-Pylon Lif't Coefficient Versus Angle of Attack for Flap/Flaperon Settings of o/o and 40/25 Degrees

•

•

•

•
1-3
.!.:I.t.l
-'
\0 V1
I I\)
I\)
\0

•

•

1.6

I

I

I

it_= 0 deg (airplane)

~

ch Number

F p/Flaperon = o/o
1.2

"

~

o.4
0.5

0-0.2

.,,...._
3 o.8
..C__)...

~
Cl)

o.4

/'h -o.6
, I

•r-1

C)

•r-1
G-1 G-1
Cl)

I

0

0

C)

t:

•r-1

i-=I
s::

-o.4

V
/..,

0

~

P-1

I
sb:l: )

-o.8

•:r:-;1:

.hf
......... lb'

V

-1.2

-32 -24 -16 -8

o

8

16 24 32

Wing-Pylon Angle of Attack (°'w), degrees

Figure 9 . Wing-Pylon Lift Coefficient Corrections Due to Compresibility

~
_I . _.
'-0 \Jl
I I\)

~

0

,......_ P-i

-0.2

$

..u........

~ -o.4

<!)

•rl

()

•rl

G--! G--!
<!)

-o.6

u 0

- t

•rl
H

-o.8

\,,,

t1

\ '

/ ------,
/

"' , // .

/

'\•

- I' -.......--. .,.,,,,,.

"

"-,,

"'' V \ I~

iN = 0 deg (airplane)

§

- - - iN = 90 deg (helicopter)

~
I -1.0
bO i::: •rl ~

-1.2

-100 -90 -80 -70 -6o -1:f)

-40 -30 -20 -10

0

Wing-Pylon Angle of Attack (°'w"), degrees

Figure 10. Wing-Pylon Lift Coefficient at Large Negative Angles 0£ Attack
for a Flap/Flaperon Setting of 40/25 Degrees

•

•

•

•

•

..~..I....

\.0

VI
I f\)

1.4 --+----'-----'-----·--'·----- I

. I

I

I

I

I

iN = 0 deg (airplane)

- - - iN = 90 deg (helicopter)

,...._ 1.2

~

..0.......

~

1.0 r - - - t - - - + - - - ~ - - 1 - - -·I

I

I

I

I

I

(I)

•r-l t)

Flap/Flaperon Setting

t•r-l

o.8 - - - - - - - 1 1 - - - - - - - - -

in degrees= x/y

(I)

0

0

v...J.

bl)

ti!
~

o.6 I J

'.J

I

I

I

I

I

I ,,- A - -,r::

§

~
P-1

o.4 I P,

~

I "- I

I

I / -"f

y< '1 1

I

I

bl)

Cl
:•r:-;1:

0.2 I I

I ......__

•
I I
I I
I I
I I

0

-40 -32 -24 -16 -8

o

8 16 24 32 40

Wing-Pylon Angle of Attack (a-w-), degrees

Figure 11. Wing-Pylon Drag Coefficient Versus Angle of Attack for Flap/Flaperon
Settings of o/o and 40/25 Degrees

8

!::d

_I .

_.
\.0 \J1
I I\)

,-,..

l1 .2 iN = 0 deg (airplane Flap/flaperons = o/o

~ 1.0

0 '-'

~ o.8

(I)

•rl

CJ

•rl

~ ~ (I)

I
o.6

I

I

I

I

I

I Mach Number

0
u

ctiuO

~
A

o.4

s::

0

\.)J I\)

~

P-!

I
sti:O:

0.2

•rl

~

0
-32

-24 - 16 -8

O

8 16 24

Wing-Pylon Angle of Attack (aw), degrees

32

40

Figure 12. Wing-Pylon Drag Coefficient Corrections Due to Compressibility

•

•

•

•

•

•

~
-_.'

1.6

' \

\0 VI
I I\) \.>l \.>l

1.4

..........
~

1.2

..0........

+:> s:I

1.0

(I)

•r-1

()

iQ

Cf; (I) 0

o.8

0

bl)
(lj
~ o.6
§
~
o.4 I
bl)
s:I
•:rs-1:

\

-

\

iN = 0 deg (airplane)

\ - - - - iN = 90 deg (helicopter)

\

" \ \\

..--
i - / ,.. /

\' \ .

I/
~ /)

' '\
I\\

vf'/

/I

\

\ l\

//)

✓·;

\', t--..._.,,,

0.2

"'~- / V

0

-100 -80 -6o -40 -20 0

20

40

60

80

100

Wing-Pylon Angle of Attack (°w), degrees

Figure 13. Wing-Pylon Drag Coefficient at Large Negative Angles of Attack
for a Flap/Flaperon Setting of 40/25 Degrees

1-3 !:X:J
...I..

--'

\.0 \J1

16

I

I\)

Flap/Flaperon Setting

in degrees= x/y
12

r/.l

(I)

(I)
H

8

ttO

(I)

rd

"'

---~-- 4 I I

I

I

I

I /7' I 7-X I

I

"..: \

I

I

J=

'-"

(I)

r-1

0

\J,J

stt:O:

-i::-

~

..cl

rc/o.l
~

-4

6

A

-8

- - - i N = 0 deg (airplane)
- - -iN = 90 deg (helicopter)!

-12
-40 -32 -24 -16 -8

o

8 16 24 32

40

Wing-Pylon Angle of Attack (aw), degrees

Figure 14. Wing-Pylon Wake Deflection (Downwash) at the Horizontal Stabilizer For
Flap/Flaperon Settings of o/o and 40/25 Degrees

•

•

•

• • •

2. Input Data Requirements
The constants required for the horizontal stabilizer model on Page A-79 are geometric in nature and are a function of the empennage configuration of interest. The value(s) for elevator effectiveness (Te) can be measured both from a wind-tunnel model (Ref. 23) or from sources such as Ref. 19. Data table input allows for further correction due to Mach number effects. The values for change in horizontal stabilizer lift coefficient C1H with sideslip and pitching moment are best determined
B from sources such as Ref. 19. The horizontal stabilizer dynamic pressure loss multiplier (KHNU) is included in the model for the purpose of providing a simple term to provide the capability to account for the dynamic pressure loss if detailed wind-tunnel data is not available for mapping empennage dynamic pressure losses as a function of angle of attack, sideslip, and airspeed. If this type of data is available, it can be entered as data tables as described on Page A-79 and tabulated on Pages B-65 through B-68 •
The lift and drag coefficients for the horizontal stabilizer should be determined from wind-tunnel test data for angles of attack up to stall whenever possible. Examples of the data requirement, as measured for the XV-15 are presented in Figs. 15 and 16. Otherwise, sources such as Ref. 19 can be used to compute these coefficients. Above stall, the coefficients can be approximated using data from Ref. 21.
F. SUBSYSTEM 6: VERTICAL STABILIZER AERODYNAMICS
1. Model Structure
The GTRS model assumes an H-tail vertical fin configuration like the XV-15, and the forces and moments on the left and right fins are computed separately in order to account for the variation in rotor wake effects with sideslip. The dynamic pressure and angle of attack at the fins, as shown in Fig. A6-l, take into account the wing-body blockage, mast angle,

TR-1195-2 (Rev. A)

35

Figure From Ref. 1

1-3 !:d __I..
\0 \J1
I [\)
,.....,.
!:d
.(!)
<:
._:_i>_,,

,,-....
~
._0__,,

AIRFOIL SECTION: NACA 64A01'

1.2 ,___

ASPECT RATIO: AR= SH= 50.25 ft2

3.28

C1t.x, = 0.071/deg

o.8

o.4

~' ,.,.,.. - ,,-.

-;::::.

...

t::::="--:::-
1---'"'

VII/ '/ , , / . , , , . I,...........-

/

r,.,
,

/,,-

I
1/ ~I _ II 1

i,......'
- . ~ ~ i----

I j V.

i.-,

V /. '-.,,,.

--- .__.

-1-

II -- I .j,/v

i..--- -
,_ .........

\..N 0\

+:> s:::
(I) •r-l
CJ •r-1
G--t G--t
(I)
0 0
~
•r-1
...:I

0
-o.4 -o.8
-1 .2
-1.6

1//

//

,/, 'I /

/ / I '/

-- IV i.--
- ------- V ..- --------

I ; , 5e ( deg) -201/'/ / /

V/ / -15
I,,. .......,,v<-10

1/

, /

0

_...... v""-......i I O

/'" /

I 10 / /15 20

- - ,- .4

- ~ I I/

---.=.. -

- ,_.... -- ,

V.
I 'V

,.. .8

~)

/

~
'~

V

\\' ~. JI

't "' ~- ~ ..... ..... ~ I\. ' ..... ~ -

"~"
~'\

~

.._ r--

.~ .......

-100 -to -60 •i,..,rljJ

I/ Insert for CLH versus

V

a,H from

-40 to -100

degrees

-l.JD -32

-24 -16 -8

0

8

16 24 32 40

Angle of Attack (°'II), Degrees

Figure 15. Horizontal Stabilizer Lift Coefficient Versus Angle of Attack

•

•

•

•

•

•

~nsert for CDH versus aH from -40 to -100 degrees

1-3

!:d

_I_.

__.

'y'igure from Ref. 1

\.0

\Jl
I I\)

!~

,......_ !:d
Cl)
:::
:i:,
~

1 .2
,......_
~
0

- 1.0 " --.....

o.8 o.6

~ "~\ i\

~ o.8

I I I
CDiJ: = 0.0034/deg 5e
Mach Number 0-0.2

~

~
Cl)

~

•rl
t)
•rl
t:
Cl)
0 0

o.6

. -100 -80
~

-6o

-40

bl)
m o.4 ~
0.2

"""~

~ ~

i=-,....__

V
/
I
o.j
;#~0.5 ~
~

o.o

-40 -32 -24 -16 -8

0

8

16

24

32

40

Angle of Attack (~H), Degrees

Figure 16. Horizontal Stabilizer Drag Coefficient Versus Angle of Attack

wing-pylon wake, rotor wake, and fuselage attitude. and angular velocities. Detailed input data requirements for the vertical stabilizers are described on Pages A-89 through A-94 in Appendix A.
2. Input Data Requirements
The constants required are generally geometric in nature and are a function of the empennage configuration of interest. The rudder effectiveness factors (Tr and Kr) can be measured both from a wind-tunnel model ( i ~ e., Ref. 23) or from sources such as Ref. 19. The roll and yaw rate correction coefficients which are a function of sideslip angle are determined from sources such as Ref. 19. The vertical fin dynamic pressure loss multiplier (KUNU) is included for the same general reason as was the horizontal stabilizer coefficient (KHNU).
The lift and drag coefficients of the fins should be determined from wind-tunnel data for angles of attack up to stall whenever. possible. Examples of the data requirements, as measured for the XV-15, are presented in Figs. 17 and 18. Otherwise, sources such as Ref. 19 can be used to compute these coefficients. Above stall, the coefficients are approximated using data from Ref. 21. The fuselage sidewash factor (1-acr/aa) at the fins is a function of flap setting, mast angle, fuselage angle of
attack, and sideslip angle. The rotor sidewash factor (KSR) is a. function
of the sideslip angle of the fin and the forward airspeed. Both of these groups of tables are best determined from powered model wind-tunnel data. If wind-tunnel data are not available, careful attention should be given to calculation of these parameters, or the XV-15 data values should be used.

•
•

•

TR-1195-2 (Rev. A)

38

•

•

•

..t..Ia....
\0 VI
I I\)
.........
!::1:1
.C<D:
>
'--'
\..N \0

1.2

o.8

--~-- o.4
C,) '--'

.p

Cl

(l)
•rl

0

C)

•rl ct-I ct-I

(l)

0
C,)

-o.4

(l)

C)

~

0

rr.i
~

-o.8

•rl

l1l

Figure From Ref. 1

AIRFOIL SECTION: NACA 0009 ASPECT RATIO-: AR::2.34 (per panel)
Cyf3 = 0 .053/deg

(Cy at f3 > 24 deg are approximate)

V

V

-

/ /0 ~-- V'".-.;:oo,,, ~ ~

~ -----~

-
~

//~V

½ ~ /2~

V½ ~ 5r (deg) /

~~ 10 I

~

:!:1-2
I ~t--...

V / "

~
For Data in this Range, See Appendix B

-10// -20
(..

~ I :~ --;;:
+o.8

~

V
/

I
I
-+0.4

~ ~

~.,

Insert for Cyv versus
f3v from
-+40 to +100 '
degrees

-1.2

-1.6

0 +4o

+6o

'-+So +100

-4o -32

-24 -16 -8

0

8

16

24

Sideslip Angle (f3 ), Degrees
V

32

4o

Figure 17. Vertical Stabilizer Side Force Coefficient Versus Sideslip Angle

Insert for Cnn versus f3u from +40 to +100 degrees
1-3

..!:..:I..c..l
\0

-

\Jl

\

I I\)
..---...
.!::d
CD
<l
..!_J_>,,

1 \

..---...
8

1

.0.__,

.-1

I\

'+100

f3v, deg +80 +6o

+4o

-DRA-G -PE-R PANEL
Sv = 25.25 ft2/panel

-

Cn = 0.003548/panel 0
Cnor = 0.0018/deg/panel

-

~ 0
ttl P-l

--- ...........

I

H

r---......

g

Cl)
P-l 0 ~
Cl)
•rl
t)
t•rl 0
Cl)
0

-

..........
"\ " Mach Number= 0-0.2

Mach Number
- 0-0.2
/
I

0

bO

ttl H

0

,::::i

0
-40

\ '\.

"' -32 -24

~ ~
-16 -8

0

o.4/
1/ o .5 ~
06 ~
~

8

16 24

32

40

Sideslip Angle (f3 ), Degrees
V

Figure 18. Vertical Stabilizer Drag Coefficient Versus Sideslip Angle

•

•

•

• •
•

G. SUBSYSTEM 7: LANDING GEAR

I. Model Structure

Two landing gear model structures are presented in Pages A-103 through A-122 of Appendix A; however, only the Subsystem 7A structure has been used for real-time simulation purposes due to computer cycle time limitations which have resulted in landing gear modeling instabilities ( the model is derived from Ref. 24). Use of the Subsystem 7A model structure requires careful tuning at NASA ARC; therefore, the input data provided for the XV-15 is for reference only, since it "works" for the XV-15. Any modeling of another tilt rotor would probably require modification to these coefficients. Therefore, a detailed discussion on most of the actual landing gear coefficients is not really useful.

2. Input Data Requirements

Most of the constants, as described on pages A-103 and A-104 of Appendix A, are geometric in nature and are primarily of value (especially in the batch version of the GTRS program) for computation of the location of landing gear drag. Both landing gear drag and landing gear pod drag are best determined from wind-tunnel data; however, numerous references exist (e.g., Ref. 25) which do provide guidance on landing gear drag for extended landing gear. Data for drag is input as a function of the percent of gear extension or retraction which, in turn, is a function of the "time" required for the landing gear to cycle up or down following the pilot's command to cycle the landing gear.

H. SUBSYSTEM 8: CONTROL SYSTEM

I. Model Structure

The control system mathematical model consists of a controls mixing model and a force gradient model. Details of the XV-15 control system are presented on Pages A-123 through A-164 of Appendix A. The flight control

TR-1195-2 (Rev. A)

41

system is illustrated schematically, and sign conventions are presented in Figs. ABa-1 and A8a-2, respectively. The mathematical model of the control system contains mixing for the pilot and automatic flight control system inputs, washout of the rotor controls as a function of mast angle and airspeed, and conversion, landing gear, and flap controls. The mathematical model does not include friction or free play, and the time constants of the control actuators are assumed zero, since, in practice, they are less than the computer frame time. This assumption was tested in a simulation of the XV-15, and results presented in Ref. 6 confirmed the assumption.
The pedal and cyclic stick longitudinal and lateral gradients are specified as a function of airspeed. The location of the gradient detent (zero force position) may be moved by the pilot in order to trim out steady stick forces.
2. Input Data Requireaents
Input data requirements, such as the control system gearing and control system limits, are generally self-explanatory as described and discussed in Appendix A. The force feel system, the control force trim system, and the pilot's control functions, as described in Subsystems 8b, Be, and 8d, respectively, are only applicable to the NASA ARC VMS simula~ tion version of the mathematical model. Therefore, further discussion on the control system is thought to be unwarranted, since most researchers will either use the XV-15 control system and the input values as described herein or will design their own control systems for replacement of the XV-15 control system.
I. SUBSYSTEM 9: CG AND INERTIA
The center of gravity and inertia subsystem, described on Pages A-165 through A-171 of Appendix A, provides modeling for the dynamic effects due to pylon acceleration. The changes in center-of-gravity location and inertia due to pylon tilt are also computed. Input data values fc;>r the

TR-1195-2 (Rev. A)

42

• • •

• •
•

subsystem are either geometric or are values of inertia which can be calculated or determined from several sources (i.e., Ref. 26).
J. SUBSYSTEMS 10 THROUGH 14: COORDINATE TRANSFORMATIONS AND EQUATIONS OF MOTION
The equations of motion used to solve for the six-degrees-of-freedom flight path are identical to the ones provided in Ref. 8. The pylon degrees of freedom are neglected, since the wing-pylon natural frequencies are well above the frequency capability of the simulation software and hardware.
Transformation of forces and moments from wind to body axes and from mast to body axes is required for a number of subsystems. These transformations are provided in Subsystems 10a through lOf. Tilt-rotor accelerations, velocities, force and moment calculations, and summations are provided by Subsystems 11, 12, 13, and 14, respectively. Except for Subsystem 14, only tilt-rotor geometric data is required for input. Input data values required for the empirical calculation of the unstable rolling moment and the pitching moment in ground effect were discussed previously in Section Bon the rotor-induced velocities.
K. SUBSYSTEM 15: FLIGHT ENVIRONMENT
The atmospheric model described on Pages A-235 through A-238 of Appendix A is the ICAO standard atmospheric model as described in Ref. 27.
L. SUBSYSTEM 16: PILOT'S INSTRUMENT PANEL
The pilot's instrument panel, as described in Pages A-239 through A-246 in Appendix A, is the instrument panel which is available at NASA ARC for use in the VMS cab. This instrument panel configuration provides important flight information and, in general, is a functional replica of the instruments of importance on the actual XV-15 instrument panel. Instruments such as radios, navigation aids, flight test instrumentation,

TR-1195-2 (Rev. A)

43

etc., which are not directly related to flying the XV-15, are either simulated by a cardboard replica or are omitted.
M. SUBSYSTEM 17: ROTOR COLLECTIVE GOVERNOR
I. Model Structure
The rotor rpm governor representation, described on Pages A-247 through A-255, consists of a single channel model of the actual flight rpm governor feedback network (Fig. Al 7-1). In the XV-15, the rotor blade collective pitch is changed so as to maintain constant rpm; the blade pitch is proportional to the integral of the error in rpm (e.g., the difference between the actual and the pilot-selected rpm) so that any steady error is completely washed out. The gain of the integral feedback is very low so that the governor will not destabilize structural modes.
A position gain is used in parallel with the integral gain in order to provide damping to the rotor rotational mode under conditions of low inflow, such as low power descents in the helicopter mode. The position gain is phased out as the pylons are converted to airplane mode in order to prevent destabilizing structural modes.
Control of the rpm governor consists of a thumb-operated, threeposition switch spring loaded to center, which is located on the power lever head. Pushing the switch forward increases the reference rpm by 20 rpm for each second that the switch is depressed; pulling aft decreases the reference rpm by 20 rpm/sec. A pointer on the rotor tachometer indicates the selected rpm. This system is modeled in the VMS cab.
2. Input Data Requirements
The input data required by the subsystem and provided in Appendix Bis for the XV-15, but it can be changed as desired by the researcher according to the block diagram in Fig. Al 7-1. At present, this model has been fully incorporated (with failure modes, etc.) and checked out only in the real-time simulation version of the GTRS program and not in the VAX

TR-1195-2 (Rev. A)

44

• •
•

• • •

version. The VAX version contains only a simplified governor for realistically maintaining control of rotor RPM.
N. SUBSYSTEMS 18 AND 19: ENGINES, FUEL CONTROLS, AND DRIVE SYSTEM DYNAMICS
1. Model Structure
The engine, fuel control, and drive system model is described on Pages A-256 through A-271 of Appendix A. The drive system is represented by the zero frequency symmetric mode, e.g., the rotors speed up or slow down in response to the imbalance between aerodynamic torque and engine torque. The frequencies of the flexible modes of the drive system (3.67 cps and 11.8 cps for the first antisymmetric and second symmetric modes, respectively) are too high to significantly influence the simulation .
The engine and power turbine (NII) governor models are composed of equations to calculate engine horsepower during transient and steady-state operation. The equations are based on the operating characteristics of the combined engine-fuel control system. This approach was taken rather than one involving time constants, inertias, and derivation of engine components to minimize the computational requirements.
The engine equations are derived in terms of the optimum power turbine speed and the horsepower developed at that speed. For a given throttle setting (or fuel flow rate), the engine will develop the maximum horsepower if the turbine is operating at the optimum speed. The commanded optimum power--referred to sea level, standard, static conditions--is given by equations presented in Fig. Al8-l where K8 through K14 are constants derived to fit the engine power versus throttle (XTH) setting characteristics given in the engine installation manual (Ref. 28).
The referenced optimum power, HPRO, at any time, t, after a power lever change is given by the equation

TR-1195-2 (Rev. A)

45

t dHP ROP

f

dt dt

t

0

where (HPRO)o is the power before the change in the power lever position and (dHPROP)/dt is the engine power acceleration schedule given as:

•

dHPROP dt

where f(HPR0 ,h) is the engine power acceleration schedule, derived to correlate with measured engine acceleration characteristics.
The actual horsepower, HP, is then computed by correcting the referred optimum horsepower, HPRO' for nonstandard conditions using the following equation

2 9.55 nRPT

9.ss n

HP

(---) + K (

RPT) + K]

fe RPMRO

z /e RPMRO

3

where Ki, K2 , and K3 are constants used to curve fit the power to the engine characteristics given in the installation manual, nR is the actual power turbine speed, RPMRO is the referred optimum power turbine speed,
and o and e are terms used to correct for nonstandard pressure and
temperature, respectively.
The equations used for the power turbine governor (NII) are similar to those for the engine except that the optimum power is referred to the NII speed commanded by the pilot rather than the throttle setting. It should be noted that in the XV-15, the NII governing speed is set at that corresponding to the rotor limit speed so that the NII governor is used only to prevent overspeeding.

TR-1195-2 (Rev. A)

46

• •

• • •

2. Input Data Requirements
Input data values provided for use of the engine, fuel control, and drive system are specifically for the T-53-L-ll engine and the XV-15. While some modifications to the input data for the model can be made in order to simulate a "larger" or "smaller" version of the T-53-1-11 engine, any need to simulate a significantly different engine should be accomplished by modifying the model to whatever extent necessary to accurately simulate the new engine instead of trying to change input data values for the model described herein.
O. SUBSYSTEM 20: STABILITY AND CONTROL AUGMENTATION SYSTEM (SCAS)
The SCAS mathematical model consists of a single channel representation of the electronic feedback network. The main feature of the SCAS mathematical model is the representation of the system gains. All gains are functions of pylon angle. The attitude-hold circuit is turned OFF or ON by a switch on the SCAS panel. SCAS actuator characteristics are not modeled; however, total system authorities are used. Simple failures can also be evaluated for the SCAS, even in the VAX version of the program. The decision not to model the actuator characteristics is discussed in more detail in Ref. 6. This evaluation verified that, when these characteristics are modeled, they are more than compensated for by the lag or reduction in bandwidth introduced into the system by the simulation computer cycle time delay.
Two different SCAS models are provided for use with the simulation version of the GTRS mathematical model. These models, the Bell developed S/N 702 model and the NASA ARC developed S/N 703 model (Ref. 29), are described in the block diagrams on Pages A-277 through A-282 of Appendix A. Gains and time constants shown on these block diagrams can be varied as desired by the researcher from those values used with the XV-15 (as tabulated in Appendix B). Presently, only the NASA ARC-developed SCAS is available for use in the VAX version of the GTRS model •

TR-1195-2 (Rev. A)

47

•

•

•

TR-1195-2 (Rev. A)

48

•

SECTION IV VALIDATION OF THE MATHEMATICAL MODEL

•

The accuracy of the GTRS mathematical model has been investigated with regard to rotor performance and force characteristics, airframe aerodynamics, rotor wake-airframe aerodynamic interaction, static and dynamic stability characteristics, and control power and damping. The majority of the data used in making this investigation has come from powered model wind-tunnel data, and Ref. 1 describes much of the early work conducted by BHT. Rotor test data has also been used for comparison, where available. Flight test data has been used more recently for correlation and validation efforts, and Refs. 6, 7, 30, and 31 provide correlation results between this version of GTRS and the XV-15. The most complete summary of correlation work accomplished in conjunction with this contract effort is presented in Ref. 7•

•

TR-1195-2 (Rev. A)

49

•

•

•

TR-1195-2 (Rev. A)

50

• • •

REFERENCES
1. Harendra, P. B., M. M. Joglekar, T. M. Gaffey, and R. L. Marr, V/STOL Tilt Rotor Study-Volume 5: A Mathematical Model for Real Time Flight Simulation of the Bell Model 301 Tilt Rotor Research Aircraft, NASA CR-114614, April 1973.
2. Schramm, M., and L. M. Landry, Jr., IFHC80 XV-15 Tilt Rotor Simulation Users' Manual, Bell Helicopter Company (Draft) Report prepared under Contract NAS2-10349.
3. Schramm, M., and L. M. Landry, Jr., IFHC80 XV-15 Tilt Rotor Simulation Programmer's Manual, Bell Helicopter Company (Draft) Report prepared under Contract NAS2-10349.
4. Hanson, Gregory D., and Samuel W. Ferguson, Generic Tilt-Rotor Simulation (GTRSIM) User's and Programmer's Guide. Volume 1: User's Guide, NASA CR-166535, October 1983.
5. Hanson, Gregory D., and Samuel W. Ferguson, Generic Tilt-Rotor Simulation (GTRSIM) User's and Programmer's Guide. Volume II: Programmer's Guide, NASA CR-166535, October 1983.
6. Ferguson, S. w., G. D. Hanson, and G. B. Churchill, "Simulation
Validation of the XV-15 Tilt-Rotor Research Aircraft," Preprint No. A-84-40-09-4000 presented at the 40th Annual Forum of the American Helicopter Society, Arlington, Virginia, May 16-18, 1984.
7. Ferguson, Samuel W., Development and Validation of the Generic TiltRotor Simulation (GTRSIM) Program, NASA CR-166537, March 1988.
8. Etkin, B., Dynamics of Flight, John Wiley and Sons, 1959.
9. Gess ow, A., and A. Crim, An Extension of Lifting Rotor Theory to Cover Operation at Large Angles of Attack and High Inflow Conditions, NACA TN 2665, April 1952.
10. Castles, w., Jr., and N. New, A Blade Element Analysis for Lifting
Rotors That is Applicable for Large Inflow and Blade Angles and Any Reasonable Blade Geometry, NACA TN 2656, 1952.
11. Drees, J. M., "A Theory of Airflow Through Rotors and Its Application to Some Helicopter Problems," The Journal of the He l i copter Association of Great Brita-in-, - -V-ol-. - ,3-, -N-o-. ~2-, September 1949.
12. Mil, M. L., et al., Helicopter Calculation and Design. Volume 1: Aerodynamics, NASA TTF-494, September 1967 .

TR-1195-2 (Rev. A)

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13. Hayden, James, "The Effect of the Ground on Helicopter Hovering Power Required," Preprint No. 1000 presented at the 32nd Annual Forum of the American Helicopter Society, Washington, D. C., May 1976.
14. Maisel, M., and D. Harris, "Hover Tests of the XV-15 Tilt Rotor Research Aircraft," AIAA Paper No. 81-2501, presented at the 1st AIAA Flight Testing Conference, November 11-13, 1981.
15. Landgrebe, A., An Analytical and Experimental Investigation of Helicopter Rotor Hover Performance and Wake Geometric Characteristics, USAAMRDL TR-71-024, (needs date).
16. Ball, LCDR J .c., and D. A. DuFresne, Shipboard Evaluation of the
XV-15 Tilt Rotor Research Aircraft, Naval Air Test Center Technical Report RW-54R-82, April 18, 1983.
17. Marr, R. L., K. W. Sambell, and G. T. Neal, V/STOL Tilt Rotor Study. Volume VI: Hover, Low Speed, and Conversion Tests of a Tilt Rotor Aeroelastic Model, NASA CR-114615, May 1973.
18. Wilson, J., R. Mineck, and C. Freeman, Aerodynamic Characteristics of a Powered Tilt Prop Rotor Wind Tunnel Model, NASA TM X-72818, March 1976.
19. Anon., USAF Stability and Control Datcom, Air Firce Flight Dynamics Laboratory, Wright-Patterson Air Force Base, Ohio, October 1960 (Revised September 1970).
20. Marr, R. L., Wind Tunnel Test Results of 25 ft Tilt Rotor Druing Autorotation, Bell Helicopter Report 301-099-005, NASA CR-137824, February 1, 1976.
21. Knight, Montgomery, and Wenzinger, Carl J., Wind Tunnel Tests on a Series of Wing Models Through a Large Angle of Attack Range. Part I-Force Tests, NACA Report No. 317,
22. Byrnes, A. L., et al., "Effect of Horizontal Stabilizer Vertical Location on the Design of Large Transport Aircraft," Journal of Aircraft, April 1966.
23. Anon., V/STOL Tilt Rotor Research Aircraft. Volume 2: Stability and Control and Handling Qualities Analyses, Bell Helicopter Report 301-199-002, 1973.
24. Stapleford, R. L., W. F. Clement, R. K. Heffley, et al., Flight Control/Flying Qualities Investigation for Lift/Cruise Fan V/STOL. Volume III: Simulation Model, NADC-77143-30, August 1979.

TR-1195-2 (Rev. A)

52

• •
•

• •

25. Hoerner, Dr.-Ing. S.F., Fluid-Dynamic Drag, published by author, 1958.
26. Lanham, Charles, Inertia Calculation Procedure for Preliminary Design, Aeronautical Systems Division Report, ASD-TR-79-5004, April 1979.
27. Anon., U. s. Standard Atmosphere, 1962, NASA, U. S. Government
Printing Office, 1962.
28. Anon., T-53-L-ll Gas Turbine Engine Installation Instructions, Lycoming Report No. 13611.1, August 1966.
29. Churchill, Gary B., and Ronald M. Gerdes, "Advanced AFCS Developments on the XV-15 Tilt Rotor Research Aircraft," Preprint No. A-84-40-10-4000 presented at the 40th Annual Forum of the American Helicopter Society, Arlington, Virginia, May 16-19, 1984.
30. Churchill, Gary B., and Daniel C. Dugan, "Simulation of the XV-15 Tilt Rotor Research Aircraft," NASA TM-84222, USAAVRADCOM TR82-A-4, March 1982.
31. Tischler, Mark B., Joseph G. M. Leung, and Daniel C. Dugan, "Frequency-Domain Identification of XV-15 Tilt-Rotor Aircraft Dynamics," Paper presented at the AIAA/AHS/IES/SETP/SFTE/ DGLR 2nd· Flight Testing Conference, Las Vegas, NE, November 16-18, 1983 .

•

TR-1195-2 (Rev. A)

53

•

•

•

TR-1195-2 (Rev. A)

54

•

•

APPENDIX A
GENERIC TILT-ROTOR SIMJLATION
MATHEMATI 00. MlDEL

•

TR-1195-2

A-1

TABLE OF CONTENTS

Subsystem Number

Subsystem Description

1

Rotor Aerodynamics

2

Rotor Induced Velocities

3

Fuselage Aerodynamics

4

Wing~Pylon Aerodynamics

5

Horizontal Stabilizer Aerodynamics

6

Vertical Stabilizer Aerodynamics

7a

Landing Gear (Present Sigma B Model)

7b

Landing Gear (Unused Bell Model)

Ba

Controls

Bb

Force Feel System

Be

Control Force Trim System

Bd

Pilot's Control Function

9

CG and Inertia

10a

Axes Transformation (Airframe Aerodynamic Forces and

Moments from Wind to Body Axis)

Axes Transformation (Rotor Forces and Moments from Wind to Body Axis)

10c

Axes Transformation (Euler Angles)

10d

Axes Transformation (Earth Based Velocity)

lOe

Axes Transformation (Ground Velocity Summation)

Axes Transformation (Ground Reference Distances)

11

Aircraft Angular Accelerations and Velocities

Page
--
A-5 A-34 A-41 A-45 A-7B A-B9 A-103 A-117 A-123 A-134 A-140 A-144 A-165
A-172
A-1B5 A-1B9 A-192 A-195 A-19B A-202

TR-1195-2 (Rev. A)

A-2

• • •

•
Subsystem Number

LIST OF SUBSYSTEMS (Concluded) Subsystem Description

•

12

Body Axis Linear Accelerations and Velocities

13

Force Summation

14

Moment Summation

15

Flight Environment

16

Pilot's Instrument Panel

17

Rotor Collective Governor

18

Engines and Fuel Controls

19

Drive System Dynamics

20

Stability and Control Augmentation System

LIST OF FIGURES

Page
--
A-210
A-217 A-222 A-235 A-239 A-247 A-256 A-265 A-272

Figure Number

Figure Description

Page

•

Al-1 Al-2 Al-3 A4-l

Rotor Axes Systems and Sign Conventions
Block Diagram of Ground Effect, Side-by-Side, and Tandem Rotor Calculations
Block Diagram of Induced Velocity and Thrust Calculations
Sign Convention and Notation for Mathematical Model of Rotor Wake-Wing Interaction

A-13 A-23 A-24 A-57

TR-1195-2 (Rev. A)

A-3

LIST OF FIGURES

Figure Number

Figure Description

Page

A4-2
A5-l
A6-l
A8a-l A8a-2 A8d-l A8d-2 A8d-3 A8d-4 A8d-5 Al6-l Al7-l Al8-l A20-l A20-2 A20-3 A20-4 A20-5 A20-6

Flow Chart of Tilt Rotor Wing Aerodynamics Affected

by the Rotor Wake

A-61

Sign Conventions and Notation for Horizontal Stabilizer

Aerodynamics

A-84

Sign Conventions and Notation for Vertical Stabilizer

Aerodynamics

A-96

Control System Block Diagram

A-129

Control Position/Force-Force and Moment Sign Convention A-130

XV-15 Collective Head Switches

A-147

XV-15 Cyclic Grip Switches

A-148

XV-15 Flap Switch Selector Control

A-149

XV-15 SCAS Control Panel

A-150

XV-15 Governor Control Panel

A-151

XV-15 Pilot's Control Panel

A-246

XV-15 Rotor RPM Governor Failure Logic Block Diagram A-254

Engine Model Block Diagram

A-262

Modified XV-15 Pitch SCAS Block Diagram (S/N 703)

A-277

Modified XV-15 Roll SCAS Block Diagram (S/N 703)

A-278

Modified XV-15 Yaw SCAS Block Diagram (S/N 703)

A-279

Bell XV-15 Pitch Axis SCAS Block Diagram (S/N 702)

A-280

Bell XV-15 Roll Axis SCAS Block Diagram (S/N 702)

A-281

Bell XV-15 Yaw Axis SCAS Block Diagram (S/N 702)

A-282

TR-1195-2 (Rev. A)

A-4

• • •

• •

1

ROTOR AERODYNAMICS

Inputs: Variables

Outputs:

From Subsystem 12

Symbol
u
V w VT

To Subsystem 2, 10b

11

p

q

1,2

r

2,4

15

p

MN

2

19

~

nL

9

hH

10b

SLCG

WLCG

8a

Sm

9oR

lOb,19

BlR

AIR

9oL BIL AIL
14

10b

XR

XL

Inputs: Constants, Coefficients, and Data Tables

•

Constants:

nb, m, Xm, 0m, R, 03, Cb' lb, lm, <l>m,
- BLCG' SLSP' BLSP' WLSP' KH, KHUB' ao
(continued on next page)

TR-1195-2 (Rev. A)

A-5

Symbol
TR HR YR TL HL YL
WiR wiL µR µL AR AL
nR'
nL'
M 1alR MblR 1alL blL
QR QL
TR TL

1

ROTOR AERODYNAMICS (CONCLUDED)

Inputs: Constants, Coefficients, and Data Tables (Concluded)

Coefficients: aa, al, a2 , o0 , o1 , o2, B, a01 , CDMACH, CDMAX,
CDALPH, CDLIM, CDFACT, CTMAXM, GECONl, GECON2, GEWASH, SFWASH, MULO, MUHl, KMUl, KMU2, KMUSF

Data Tables:

C"j:fa = f(µ,8m)
-
XsF = f( Iv I> Xss = f(i:i>

Table 1-I Table 1-II Table 1-III

•

•

•

TR-1195-2 (Rev. A)

A-6

• • •

SUBSYSTEM NO. 1: ROTOR AERODYNAMICS

Inputs: Variables

Symbol
u
V
w
p q r
p
~
nr,

Description
x-velocity (longitudinal) of the aircraft e.g. in body axis with respect to the air
y-velocity (lateral) of the aircraft e.g. in body axis with respect to the air
z-velocity (vertical) of the aircraft e.g. in body axis with respect to the air
Total linear velocity of the aircraft e.g. with respect to the air
Body axis roll rate
Body axis pitch rate
Body axis yaw rate
Air density
Mach number
Instantaneous right rotor speed
Instantaneous left rotor speed
Rotor hub height above ground
Station line of e.g.
Water line of e.g.
Mast conversion angle(+ fwd, 0 deg= vertical or helicopter, 90 deg= horizontal or airplane)
Right rotor root collective pitch
Right rotor forward cyclic input
Right rotor lateral cyclic,input
Left rotor root collective pitch

TR-1195-2 (Rev. A)

A-7

Units
ft/sec
ft/sec
ft/sec
ft/sec
rad/sec rad/sec rad/sec slug/ft3
ND
rad/sec rad/sec ft in in rad
rad rad rad rad

SUBSYSTEM NO. I-ROTOR DYNAMICS (Continued)

Inputs: Constants, Coefficients, and Data Tables

Symbol

Description

m
Xm
R
fj>m
BLcG SLgp BLgp WLgp

Left rotor forward cyclic input Left rotor lateral cyclic input Right rotor x-force (body axis) Left rotor x-force (body axis) Number of rotor blades Number of rotor segments Blade station/R Blade twist Rotor radius Pitch flap coupling Blade chord Blade flapping inertia Mast length Lateral mast tilt Butt line of e.g. Station line of engine nacelle shaft pivot point Butt line of engine nacelle shaft pivot point Water line of engine nacelle shaft pivot point Flapping spring rate Coning hubspring

TR-1195-2

A-8

Units
rad rad lb lb
ND ND ND
deg ft deg in slug-ft ft deg in in
in
in
ft-lb/deg ft-lb/deg

• • •

• • •

SUBSYSTEM NO. I: ROTOR AERODYNAMICS (Continued)

Inputs: Constants, Coefficients, and Data Tables (Continued)

Symbol
a
0
B
CDMACH CDMAX CDALPH CDLIM CDFACT CTMAXM GECONl GECON2 GEWASH SFWASH

Description Precone angle
Blade lift coefficient Blade lift coefficient
Blade lift coefficient Blade drag coefficient
Blade drag coefficient
Blade drag coefficient Blade tip loss factor
Blade zero lift coefficient
Coefficient for lower limit of rotor mach effects Maximum rotor drag coefficient
Rotor drag equation coefficient (slope with alpha)
Onset of profile drag rise
Rotor drag equation coefficient Rotor CT max multiplier coefficient
Constant in the rotor ground effect equation
Constant in the rotor ground effect equation
Airspeed washout for rotor ground effects Airspeed washout for side-by-side rotor effects

Units deg 1/rad 1/µ
11i
ND 1/rad 1/rad2
ND
deg
ND
ND ND
ND ND ND
ft/sec
ft/sec
ft/sec
ft/sec

TR-1195-2 (Rev. A)

A-9

SUBSYSTEM NO. 1: ROTOR AERODYNAMICS (Continued)

Inputs: Constants, Coefficients, and Data Tables (Concluded)

Symbol MURO MUHl KMUl KMU2 KMUSF
Outputs:

Description
Induced velocity distribution equation coefficient
Induced velocity distribution equation coefficient
Induced velocity distribution equation coefficient
Induced velocity distribution equation coefficient
Induced velocity distribution equation coefficient for sideward flight
Maximum available rotor thrust coefficient,= f(µ,Bm)
Sideward fligh.t rotor correction
factor, = f( !vi)
Side-by-side rotor effect correction factor,= f(u)
Mast axis right rotor thrust (+ up for helicopter)
Mast axis H-force right rotor thrust(+ aft for helicopter)
Mast axis Y-force right rotor thrust(+ right for helicopter)
Mast axis left rotor thrust (+ up for helicopter)
Mast axis H-force left rotor - thrust (+ aft for helicopter)
Mast axis Y-force left rotor thrust(+ right for helicopter)

Units
ND ND ND ND ND
ND ND
ND
lb lb lb lb lb lb

TR-1195-2 (Rev. A)

A-10

• • •

•
• •

SUBSYSTEM NO. 1-ROTOR AERODYNAMICS (Continued)

Outputs: Continued

Symbol

Description

Mast axis_ uniform component of induced velocity at right rotor (+ downward for helicopter)

Mast axis uniform component of induced velocity at left rotor (+ downward for helicopter)

Tip speed (advance) ratio, right rotor

Tip speed (advance) ratio, left rotor

Inflow ratio, right rotor

Inflow ratio, left rotor

QR'

Total right rotor speed (corrected for aircraft angular

rate)

QL'

Total left rotor speed

(corrected for aircraft angular

rate)

Mast axis longitudinal flapping restraint exerted by right rotor on airframe(+ nose up for helicopter)

Mast axis lateral flapping restraint exerted by right rotor on airframe (+ outboard for helicopter)

Mast axis longitudinal flapping restraint exerted by left rotor on airframe(+ nose up for helicopter)

Mast axis lateral flapping restraint exerted by left rotor on airframe (+ outboard for helicopter)

Units ft/sec ft/sec
ND ND ND ND
rad/sec rad/sec ft-lb
ft-lb ft-lb ft-lb

TR-1195-2

A-11

SUBSYSTEM NO. I-ROTOR AERODYNAMICS (Concluded)

Outputs: Concluded

Symbol

Description

Mast axis right rotor torque (+ trying to slow rotor down)
Mast axis left rotor torque (+ trying to slow rotor down)

•
Units ft-lb ft-:-lb

•

TR-1195-2

A-12

•

MAST RIGHT ROTOR SHOWN
SIGN CO'NVENTIONS ON RIGHT TPP AND LEFT SIDE ARE COMPLETELY _ - SYMMETRICAL ABOUT AIRCRAFT
PLANE-OF-SYMMETRY

• •

...
~WMR = SWASHPLATE
301-099-001

A. MAST AXIS SYSTEM .

(1) U, V, WHMR ARE TRANS-

MAST

FORMED FROM BODY (FUSELAGE) AXIS SYSTEM -

WIND-MAST

(2) FLAPPING AND PERFORMANCE CALCULATIONS ARE MADE IN WIND-MAST SYSTEM OBTAINED BY RESOLVING UHMR AND
VHMR INTO A SINGLE WIND
VECTOR AS SHOWN IN (B) BELOW

(3) T, H, Y, \ AND bl ARE
CALCULATED IN WIND-MAST AXIS SYSTEM - THEN TRANSFORMED TO MAST AXIS SYSTEM TO OBTAIN T, H, Y, a 1 AND b1.
B. "WIND-MAST" AXIS SYSTEM.
Figure A1-1. Rotor Axes Systems and Sign Conventions
A-13

EQUATIONS SUBSYSTEM NO. 1--ROTOR AERODYNAMICS
A. ~ Twist Constants (One Time Per Rotor)
Kco,m = 01i [sin( er Xm-t )- sin ( er Xm)]

•

K c2, m= ; ; (K s1,m) + 01j [ ( X m- 1)2 Sin ( 0 ~ X m- 1) ~ ( Xm)2 Sin ( 0 ~ X m) ]

Kc3,m = ;:(Ks2,m)+ 01m[(xm-1)3sin(erxm-1)-(xm)3 sin(erxm)]

I

I

.

er= where

twist rate of m th segment= (em --((em-1)))

Xm Xm-1

Xm = Radial station of m th segment

em= Blade pitch angle at m th segment

TR-1195-2 (Rev. A)

A-14

• •

EQUATIONS (CONTINUED)

•

SUBSYSTEM NO. 1--ROTOR AERODYNAMICS

A. Blade Twist Constants (Concluded)
.Ken m "" Blade twist constants

(n = 0, 1, 2, 3)

m = number of geometric segments, starting from tip (r/R - 1.0) to root (r/R = 0.0)

(;}~=(;JR+ aOL

Define blade pitch constant components as:
L l
TW 1 n = Kcn,m COS Ll0 0 m
m•l

•

L l
TW2n = Kcn,m sin Ll0 0 m
m•l
L l
TW3n = Ksn,m sin Ll0 0 m
m•l
L l
TW4n = Ksn,mcosLl0 0m
m•l

8R = Blade pitch at the rotor center
1 = Number of m aerodynamic segments to account for
blade root cutout.
B. Initial Transformation Equations
(One Time Per Rotor)
A=nR 2

DN'=AR 2 =nR 4

•

TR-1195-2 (Rev. A)

A-15

EQUATIONS (CONTINUED) SUBSYSTEM NO. 1--ROTOR AERODYNAMICS
B. Initial Transformation Equations (Concluded) (One Time Per Rotor)

•

Yrn = PY
C. Long Term Transformations 1. Rotor Angular Velocity in Space
.Q ~ = .QR + p Sin /3 m COS tp m + q COS /3 m Sin tp m - f COS /3 m COS tp m

2. "Wind-Mast" Axis Angular Rates

f f p WMR = p HMR cos WMR + q HMR sin WMR f f q WMR = - p HMR sin WMR + q HMR cos WMR
where
/3 /3 /3 p HMR = p cos m - q sin m sin 'Pm+ r sin m cos 'Pm

qHMR =

q COS 'Pm

+rsint/>m

TR-1195-2 (Rev. A)

A-16

• •

EQUATIONS (CONTINUED)

•

SUBSYSTEM NO. 1--ROTOR AERODYNAMICS

2. "Wind-Mast" Axis Angular Rates: Right Rotor (Concluded)

f wMR = wind

azimuth

angle

defined

to

be

equal

to

t

an

-

IV HMR --

UHMR

_

PwMR

PwMR = .Q~

•

q WML = - p HML sin f WML + q HML cos f WML
where
p HML = - p cos /3 m- q sin /3 msin¢ m- r sin /3 mcos <pm

qHML=

-rsin¢m

f WML = wind

azimuth

angle

defined

to

be

equal

to

t

an

-

IV HML --

U HML

_

PwML

PwML = .Q~

_

qWML

qWML = .Q~

•

TR-1195-2 (Rev. A)

A-17

EQUATIONS (CONTINUED) SUBSYSTEM NO. 1--ROTOR AERODYNAMICS

3. Rotor Hub Velocity--Mast Axes

= UHMR U HBR cos Pm - V HBR sin /3 m sin 'Pm+ WHBR sin /3 m cos <pm

= V HMR

V HBR COS 'Pm

+ \,J HBR sin 'Pm

w HMR = - u HBR sin pm - V HBR cos pm sin"' m + w HBR cos pm cos"' m

Where,

• •

•

TR-1195-2 (Rev. A)

A-18

EQUATIONS (CONTINUED)

•

SUBSYSTEM NO. 1--ROTOR AERODYNAMICS

•

3. Rotor Hub Velocity--Mast Axes: Right Rotor (Concluded) Left Rotor
u u w HML = HBL cos /3 m + V HBL sin /3 m sin <Pm+ HBL sin /3 m cos <Pm

VHML=

- V HBL COS </>rn

w u w HML = - HBL sin /3 m + V HBL cos /3 m sin <Pm+ HBL cos /3 m cos <Pm

Where,
UHBL = U-q(LZH) + r(LYH)

V Hat = V + P ( L ZH) + r ( L xH)

\.J HBL = \.J - p ( LYH )- q ( L XH)

•

TR-1195-2 (Rev. A)

A-19

EQUATIONS (CONTINUED) SUBSYSTEM NO. 1--ROTOR AERODYNAMICS

4. Aerodynamic Coefficients

Right Rotor

DNR

=

pf

2

,
R

2

DN

,

V (

2 UHMR

+

2 ) 112 HMR

µR=

n~ A,

WHMR =---

OR

R

(V f -I

HMR)

WMR = tan

UHMR

Where a0 , a1 , a2 = blade lift coefficients

C = (_2_/_3_)_K_H

KFAR

I b n~ 2

= YMR Ym aR
TR-1195-2 (Rev. A)

A-20

• • •

•

EQUATIONS (CONTINUED) SUBSYSTEM NO. 1--ROTOR AERODYNAMICS
4. Aerodynamic Coefficients: Right Rotor (Concluded) Define,

•

(For left rotor, replace subscript R with L) D. Short Term Transformations
(Every Update Cycle) 1. "Wind-Mast" Axis Cyclic Inputs
Right Rotor
r r A IR= A IR cos WMR - BIR sin WMR
r f BIR= A IR sin WMR +BIR cos WMR
(For left rotor, replace subscript R with L) 2. Blade Pitch Constants
Right Rotor

(For left rotor, replace subscript R with L) 3. Performance Parameters
Right Rotor

a

?CTR
=--

rR aaR

•

TR-1195-2 (Rev. A)

A-21

EQUATIONS (CONTINUED) SUBSYSTEM NO. 1--ROTOR AERODYNAMICS
3. Performance Parameters: Right Rotor (Concluded)
+ CDFACT[ CDLIM +max(MTIP, CDMACH)]}}
(For left rotor, replace subscript R with L) 4. Ground Effect, Side-by-Side and Tandem Rotor Factors
(See Fig. Al-2) 5. Thrust and Induced Velocity
(See Fig. Al-3) 6. Rotor Flapping (Wind-Mast Axis System)
Right Rotor
TW34 = twist at 3/4 radius (starting at root) (For the XV-15, TW34 = 34.525 degrees) The first-order flapping equations used are described in matrix form as follows:

• •

•

TR-1195-2 (Rev. A)

A-22

•

Figure Al-2. Block Diagram of Ground Effect, Side-by-Side, and Tandem Rotor Calculations
/GEWASH * ✓u 2 + v21 ) WASHOUT = exp \
< 0.001

GECON3 = hH /2R

•

G = 1 - (GECONl) ( expGECON2*GECON3 )
>

G= 1

u = (l{fBR + UHBL )/(2 0if)
V = (VHBR+ VHBL)/(20if)

Xss = t(lul)

XsFL = O

<

XsFR = r(v)

'>-=>=----.i XsFL = f(V) XsFR= O

•

TR-1195-2 (Rev. A)

>
A-23

Figure Al-3. Block Diagram of Induced Velocity and Thrust Calculations

Replace R with L for Left Rotor

IT2R = 0

1

0.5

/lir = (SIGN %R )(%R)

T2T = TR I= 0 2
%R = ITR l/(2B DNR) C1R = Ti/DNR ).iR = (Q3R)~
Q5R = 0-6 * IQ3Ri1-5/(IQ3RI + aµ~)
). R = ).OR+ \R

•

Q3R !1.0-[(1.0-G)(WASHOUT)]l( i+:Xss + XsF)

Qrn= ).OR + - - - - - - - - - - - - - - -

2

2 o.5 Q1rn(IQ3R l+(B/3)AR 111R I)

WR+QOO~R) + - - - - - - - -

(IQ3R I + a11!)

"rn = ~R -"oR
).R = o.7Qrn+ o.3AR
1= I + 1

•

NO

TR-1195-2 (Rev. A)

A-24

•

• •

Figure Al-3 (Concluded)
T2R = Q6R !Cs2R+ C'soR µ:1 2 - CsrnµR qWMR - CC1R[/\ R- (!1R/2)PWMR]
- Brn[Ccrn µR+ Cs2R (arn/2 - PwMR/2)] + Arn [(1/2) Cs2R(b1R- (4/3) KR), rn+ qWMR)]
+ tan 63 [Ccrn µRbrn]l
IT2R = IT2R + 1
NO

TR= T2R CT = 'Jk/DNR
R

•

NOTE: Cycle through rotor four times to insure flapping convergence (update on A with
better AiR)

Where
TR = (Cr /a )(DNR )(OR) Cr, /a = f(µ .~m)

TR-1195-2 (Rev. A)

A-25

EQUATIONS (CONTINUED) SUBSYSTEM NO. 1--ROTOR AERODYNAMICS
6. Rotor Flapping (Wind-Mast Axis System) Right Rotor (Continued) A simplified zero-order (algebraic) flapping equation is used at the user's option (switch incorporated) by solving the following:

•

The above coefficients are as follows:

•

•

TR-1195-2 (Rev. A)

A-26

•

EQUATIONS (CONTINUED) SUBSYSTEM NO. 1--ROTOR AERODYNAMICS
6. Rotor Flapping (Wind-Mast Axis System) Right Rotor (Continued)

C21 = - C 12
•

•

TR-1195-2 (Rev. A)

A-27

EQUATIONS (CONTINUED) SUBSYSTEM NO. 1--ROTOR AERODYNAMICS
6. Rotor Flapping (Wind-Mast Axis System) Right Rotor (Concluded)

•

6 c c c c a: c c = 11 B 2 - 21 B 1 + ( 21 A 11 - 11 A 21 ) 1R + ( 21 A 12 - 11 A 22) 5 IR

IR

C11C22-C12C21

•

•

TR-1195-2 (Rev. A)

A-28

•

EQUATIONS (CONTINUED)

•

SUBSYSTEM NO. 1--ROTOR AERODYNAMICS

•

7. Inflow Distribution Factor KR

(For left rotor, replace subscript R with L) At low airspeeds KR - f(µ, PF) where:
KRl - IOOJSF + (IOOJl - IOOJSF)( Icos 3 /3 FI )
At higher airspeeds, KR= f(µ) where the following table summarizes the options for the calculation of KR

O<~<MULO MULO < ~ < MUHl ~ > MUHl

KRl (~) KRl(MULO) + IOOJ2(~ - MULO) KRl(MULO) + IOOJ2(MUH1 - MULO)

8. Rotor Flapping in Mast Axis System Right Rotor

(For left rotor, replace subscript R with L)
a a IR= IR cos f WMR + b IR sin f WMR
a b IR = - IR sin f WMR + b IR cos f WMR

•

TR-1195-2 (Rev. A)

A-29

EQUATIONS (CONTINUED) SUBSYSTEM NO. 1--ROTOR AERODYNAMICS
9. Rotor Inplane Forces in Wind-Mast Axis System Right Rotor (For left rotor, replace subscript R with L)

•

•

TR-1195-2 (Rev. A)

A-30

•

•

EQUATIONS (CONTINUED) SUBSYSTEM NO. 1--ROTOR AERODYNAMICS
9 . Rotor Inplane Forces in Wind-Mast Axis System Right Rotor (Concluded)

•

9 8

2-

+ Cs2R .l R a 1R ] }

10. Rotor Inplane Forces in Mast Axis System Right Rotor (For left rotor, replace subscript R with L)

•

r r y R = - HR sin WMR + y R cos WMR

TR-1195-2 (Rev. A)

A-31

EQUATIONS (CONTINUED) SUBSYSTEM NO. 1--ROTOR AERODYNAMICS 11. Rotor Power and Torque Required (For left rotor, replace subscript R with L)
2 3 - Cc 1R ( 'A. R2 - µ R 'A. R -a 1R ) - 1 C c3R { -a 21R + (b- 1R - 4 K R 'A. iR ) 2

• •

TR-1195-2 (Rev, A)

A-32

•