NASA/TP–2009-215402

NDARC

NASA Design and Analysis of Rotorcraft

Wayne Johnson

Ames Research Center

Moffett Field, California

December 2009
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NASA/TP–2009-215402

NDARC

NASA Design and Analysis of Rotorcraft

Wayne Johnson

Ames Research Center

Moffett Field, California

National Aeronautics and

Space Administration

Ames Research Center

Moffett Field, California 94035-1000

December 2009
Available from:

NASA Center for AeroSpace Information National Technical Information Service

7115 Standard Drive 5285 Port Royal Road

Hanover, MD 21076-1320 Springfield, VA 22161

(301) 621-0390 (703) 487-4650
TABLE OF CONTENTS

CHAPTER 1 – INTRODUCTION ...................................................................................................................... 1

1 –1 Background ....................................................................................................................................... 1

1 –2 Requirements ..................................................................................................................................... 3

1 –3 Overview ........................................................................................................................................... 4

1 –4 Terminology ...................................................................................................................................... 4

1 –5 Analysis Units ................................................................................................................................... 6

1 –6 Outline of Report .............................................................................................................................. 6

1 –7 References ........................................................................................................................................ 6

CHAPTER 2 – NOMENCLATURE .................................................................................................................. 9

CHAPTER 3 – TASKS .................................................................................................................................... 15

3–1 Size Aircraft for Design Conditions and Missions ........................................................................ 15

3–2 Mission Analysis ............................................................................................................................ 18

3–3 Flight Performance Analysis .......................................................................................................... 18

3–4 Maps ............................................................................................................................................... 19

CHAPTER 4 – OPERATION .......................................................................................................................... 21

4–1 Flight Condition ............................................................................................................................. 21

4–2 Mission ........................................................................................................................................... 22

4–3 Takeoff Distance ............................................................................................................................ 25

4–4 Flight State ..................................................................................................................................... 29

4–5 Environment and Atmosphere ........................................................................................................ 31

4–6 References ...................................................................................................................................... 34

CHAPTER 5 – SOLUTION PROCEDURES .................................................................................................. 35

5–1 Iterative Solution Tasks ................................................................................................................... 38

5–2 Theory ............................................................................................................................................ 41

CHAPTER 6 – COST ....................................................................................................................................... 47

6–1 CTM Rotorcraft Cost Model .......................................................................................................... 47

6–2 References ...................................................................................................................................... 50

CHAPTER 7 –AIRCRAFT .............................................................................................................................. 53

7–1 Disk Loading and Wing Loading ................................................................................................... 53

7–2 Controls .......................................................................................................................................... 54

7–3 Trim ................................................................................................................................................ 56

7–4 Geometry ........................................................................................................................................ 56

7–5 Aircraft Motion .............................................................................................................................. 58

7–6 Loads and Performance .................................................................................................................. 59

7–7 Aerodynamics ................................................................................................................................. 60

7–8 Trailing-Edge Flaps ........................................................................................................................ 61

7–9 Drag ................................................................................................................................................ 62

7–10 Performance Indices ....................................................................................................................... 65

7–11 Weights ........................................................................................................................................... 65

7–12 Weight Statement ........................................................................................................................... 66

7–13 References ...................................................................................................................................... 66

CHAPTER 8 – SYSTEMS ............................................................................................................................... 69

8–1 Weights ........................................................................................................................................... 69

iii
TABLE OF CONTENTS (cont.)

CHAPTER9 – FUSELAGE ............................................................................................................................ 71

9–1 Geometry ........................................................................................................................................ 71

9–2 Control and Loads .......................................................................................................................... 71

9–3 Aerodynamics ................................................................................................................................. 72

9–4 Weights ........................................................................................................................................... 73

CHAPTER 10 – LANDING GEAR ................................................................................................................. 75

10–1 Geometry ........................................................................................................................................ 75

10–2 Weights ........................................................................................................................................... 75

CHAPTER11 – ROTOR ................................................................................................................................. 77

11 –1 Drive System .................................................................................................................................. 78

11 –2 Geometry ........................................................................................................................................ 79

11 –3 Control and Loads .......................................................................................................................... 80

11 –4 Aerodynamics ................................................................................................................................. 84

11 –5 Power .............................................................................................................................................. 95

11 –6 Performance Indices ..................................................................................................................... 102

11 –7 Interference ................................................................................................................................... 102

11 –8 Drag .............................................................................................................................................. 104

11 –9 Weights ......................................................................................................................................... 105

11 –10 References .................................................................................................................................... 106

CHAPTER12 – FORCE ................................................................................................................................ 107

12–1 Control and Loads ........................................................................................................................ 107

12–2 Performance and Weights ............................................................................................................. 108

CHAPTER13 – WING ................................................................................................................................... 109

13–1 Geometry ....................................................................................................................................... 109

13–2 Control and Loads ......................................................................................................................... 112

13–3 Aerodynamics ................................................................................................................................ 112

13–4 Wind Extensions ........................................................................................................................... 115

13–5 Weights .......................................................................................................................................... 115

CHAPTER 14 – EMPENNAGE ..................................................................................................................... 117

14–1 Geometry ....................................................................................................................................... 117

14–2 Control and Loads ......................................................................................................................... 117

14–3 Aerodynamics ................................................................................................................................ 117

14–4 Weights .......................................................................................................................................... 119

CHAPTER15 – FUEL TANK ........................................................................................................................ 121

15–1 Fuel Capacity ................................................................................................................................. 121

15–2 Geometry ....................................................................................................................................... 121

15–3 Fuel Reserves ................................................................................................................................ 121

15–4 Auxiliary Fuel Tank ...................................................................................................................... 122

15–5 Weights .......................................................................................................................................... 122

iv
TABLE OF CONTENTS (cont.)

CHAPTER16 – PROPULSION ..................................................................................................................... 125

16–1 Drive System ................................................................................................................................. 125

16–2 Power Required ............................................................................................................................. 126

16–3 Geometry ....................................................................................................................................... 127

16–4 Drive-System Rating ..................................................................................................................... 127

16–5 Weights .......................................................................................................................................... 127

CHAPTER 17 – ENGINE GROUP ................................................................................................................. 129

17–1 Engine Performance ...................................................................................................................... 129

17–2 Control and Loads ......................................................................................................................... 130

17–3 Nacelle Drag .................................................................................................................................. 131

17–4 Engine Scaling ............................................................................................................................... 132

17–5 Weights .......................................................................................................................................... 132

CHAPTER 18 – REFERRED PARAMETER TURBOSHAFT ENGINE MODEL ...................................... 135

18–1 Operating Environment ................................................................................................................. 135

18–2 Engine Ratings .............................................................................................................................. 136

18–3 Performance Characteristics .......................................................................................................... 136

18–4 Installation ..................................................................................................................................... 137

18–5 Power-Turbine Speed .................................................................................................................... 137

18–6 Power Available ............................................................................................................................ 138

18–7 Performance at Power Required .................................................................................................... 139

18–8 Scaling ........................................................................................................................................... 139

18–9 Engine Speed ................................................................................................................................. 140

18–10 Weight ........................................................................................................................................... 140

18–11 Units .............................................................................................................................................. 141

18–12 Typical Parameters ........................................................................................................................ 141

CHAPTER 19 – AFDD WEIGHT MODELS ................................................................................................. 147

19–1 Wing Group ................................................................................................................................... 147

19–2 Rotor Group ................................................................................................................................... 152

19–3 Empennage Group ......................................................................................................................... 153

19–4 Fuselage Group .............................................................................................................................. 154

19–5 Alighting Gear Group .................................................................................................................... 156

19–6 Engine Section or Nacelle Group and Air Induction Group ......................................................... 156

19–7 Propulsion Group .......................................................................................................................... 157

19–8 Flight Controls Group ................................................................................................................... 159

19–9 Hydraulic Group ............................................................................................................................ 161

19–10 Anti-Icing Group ........................................................................................................................... 161

19–11 Other Systems and Equipment ...................................................................................................... 162

19–12 Folding Weight .............................................................................................................................. 163

19–13 Parametric Weight Correlation ...................................................................................................... 163

19–14 References ..................................................................................................................................... 163

v
vi
LIST OF FIGURES

Figure 1-1. Outline of NDARC tasks ............................................................................................................ 5

Figure 4-1. Takeoff distance and accelerate-stop distance elements .......................................................... 27

Figure 5-1. Outline of NDARC tasks .......................................................................................................... 36

Figure 5-2. Design and analysis tasks, with nested loops and solution methods ........................................ 37

Figure 6-1. Statistical estimation of rotorcraft flyaway cost ($/lb) ............................................................. 51

Figure 6-2. Statistical estimation of rotorcraft flyaway cost ($M) .............................................................. 52

Figure 7-1. Aircraft geometry ...................................................................................................................... 57

Figure 7-2a. Weight statement (*indicates extension of RP8A) ................................................................... 67

Figure 7-2b. Weight statement (*indicates extension of RP8A) ................................................................... 68

Figure 11-1. Tail-rotor radius scaling ............................................................................................................ 78

Figure 11-2. Ground-effect models (hover) .................................................................................................. 89

Figure 11-3a. Tip-path plane tilt with cyclic pitch. ......................................................................................... 91

Figure 11-3b. Thrust-vector tilt with cyclic pitch ........................................................................................... 91

Figure 11-4. Induced power factor for rotor in hover ................................................................................... 97

Figure 11-5. Induced power factor for rotor in axial flight ........................................................................... 98

Figure 11-6. Induced power factor for rotor in edgewise flight .................................................................... 98

Figure 11-7. Stall function ............................................................................................................................. 99

Figure 11-8. Mean drag coefficient for rotor in hover .................................................................................. 99

Figure 11-9a. Mean drag coefficient for rotor in forward flight, high stall .................................................. 100

Figure 11-9b. Mean drag coefficient for rotor in forward flight, low stall ................................................... 100

Figure 13-1. Wing geometry (symmetric, only right half-wing shown)...................................................... 110

Figure 15-1. Outline of Nauxtank calculation ................................................................................................ 123

Figure 17-1. Historical data for turboshaft-engine specific fuel consumption............................................ 133

Figure 17-2. Historical data for turboshaft-engine weight .......................................................................... 133

Figure 17-3. Historical data for turboshaft-engine specific power ............................................................. 134

Figure 18-1. Fuel flow, mass flow, and net jet thrust variation with power ............................................... 143

Figure 18-2. Power variation with turbine speed ........................................................................................ 143

Figure 18-3. Specific power variation with temperature ratio, static .......................................................... 144

Figure 18-4. Specific power variation with temperature ratio, 200 knots ................................................... 144

Figure 18-5. Mass-flow variation with temperature ratio, static ................................................................. 145

Figure 18-6. Mass-flow variation with temperature ratio, 200 knots .......................................................... 145

Figure 18-7. Power variation with temperature ratio, static ........................................................................ 146

Figure 18-8. Power variation with temperature ratio, 200 knots ................................................................. 146

vii
LIST OF FIGURES (cont.)

Figure 19-1. Wing group (AFDD93) ........................................................................................................... 164

Figure 19-2. Rotor group, blade weight (AFDD82) .................................................................................... 164

Figure 19-3. Rotor group, hub weight (AFDD82) ....................................................................................... 165

Figure 19-4. Rotor group, blade weight (AFDD00) .................................................................................... 165

Figure 19-5. Rotor group, hub weight (AFDD00) ....................................................................................... 166

Figure 19-6. Empennage group, horizontal tail weight (AFDD82) ............................................................. 166

Figure 19-7. Empennage group, vertical tail weight (AFDD82) ................................................................. 167

Figure 19-8. Empennage group, tail rotor weight (AFDD82) ..................................................................... 167

Figure 19-9. Fuselage group, fuselage weight (AFDD84) .......................................................................... 168

Figure 19-10. Fuselage group, fuselage weight (AFDD82) .......................................................................... 168

Figure 19-11. Alighting gear group, landing gear weight (AFDD82) ........................................................... 169

Figure 19-12. Engine section or nacelle group, engine support weight (AFDD82) ...................................... 169

Figure 19-13. Engine section or nacelle group, cowling weight (AFDD82) ................................................ 170

Figure 19-14. Air induction group, air induction weight (AFDD82) ............................................................ 170

Figure 19-15. Propulsion group, accessories weight (AFDD82) .................................................................. 171

Figure 19-16. Propulsion group, fuel tank weight (AFDD82) ...................................................................... 171

Figure 19-17. Propulsion group, gear box and rotor shaft weight (AFDD83) .............................................. 172

Figure 19-18. Propulsion group, gear box and rotor shaft weight (AFDD00) .............................................. 172

Figure 19-19. Propulsion group, drive shaft weight (AFDD82) ................................................................... 173

Figure 19-20. Propulsion group, rotor brake weight ..................................................................................... 173

Figure 19-21. Flight controls group, rotor non-boosted control weight (AFDD82) ..................................... 174

Figure 19-22. Flight controls group, rotor boost mechanisms weight (AFDD82) ........................................ 174

Figure 19-23. Flight controls group, rotor boosted control weight (AFDD82) ............................................. 175

Figure 19-24 Sum of all parametric weight .................................................................................................. 175

viii
LIST OF TABLES

Table 4-1. Mission segment calculations ..................................................................................................... 25

Table 4-2. Mission segments for takeoff calculation ................................................................................... 26

Table 4-3. Typical friction coefficient μ ..................................................................................................... 28

Table 4-4. Temperatures and vertical temperature gradients ...................................................................... 33

Table 4-5. Constants adopted for calculation of the ISA ............................................................................. 33

Table 5-1. Maximum-effort solution ........................................................................................................... 40

Table 5-2. Maximum-effort solution ........................................................................................................... 40

Table 5-3. Trim solution .............................................................................................................................. 40

Table 5-4. Trim solution .............................................................................................................................. 41

Table 6-1. DoD and CP1 inflation factors ................................................................................................... 49

Table 6-2. Cost model parameters ............................................................................................................... 50

Table 7-1. Geometry conventions ................................................................................................................ 57

Table 7-2. Component contributions to drag.. ............................................................................................. 64

Table 7-3. Component contributions to nominal drag area ......................................................................... 64

Table 11-1. Principal configuration designation ........................................................................................... 77

Table 11-2. Rotor-shaft axes .......................................................................................................................... 83

Table 11-3. Principal configuration designation ......................................................................................... 105

Table 12-1. Force orientation ....................................................................................................................... 107

Table 17-1. Engine orientation .................................................................................................................... 131

Table 18-1. Typical engine ratings .............................................................................................................. 136

Table 18-2. Conventional units .................................................................................................................... 141

Table 18-3. Typical engine-performance parameters .................................................................................. 142

Table 18-4 Typical parameter ratios for various ratings (percent) ............................................................. 142

Table 19-1. Parameters for tiltrotor-wing weight ........................................................................................ 150

Table 19-2. Parameters for aircraft-wing weight ......................................................................................... 151

Table 19-3. Parameters for rotor weight ...................................................................................................... 152

Table 19-4. Parameters for lift-offset rotor weight ...................................................................................... 153

Table 19-5. Parameters for tail weight ......................................................................................................... 154

Table 19-6. Parameters for fuselage weight (AFDD84 model) ................................................................... 155

Table 19-7. Parameters for fuselage weight (AFDD82 model) ................................................................... 155

Table 19-8. Parameters for landing-gear weight .......................................................................................... 156

Table 19-9. Parameters for engine section, nacelle, and air induction weight ............................................ 157

Table 19-10. Parameters for engine-system weight ....................................................................................... 157

Table 19-11. Parameters for fuel-system weight ........................................................................................... 158

Table 19-12. Paremeters for drive-system weight ......................................................................................... 159

Table 19-13. Parameters for drive shaft and rotor brake weight ................................................................... 159

Table 19-14. Parameters for fixed-wing flight-control weight ...................................................................... 160

Table 19-15. Parameters for rotary-wing flight-control weight ..................................................................... 160

Table 19-16. Parameters for conversion-control weight ................................................................................ 161

Table 19-17. Parameters for hydraulic group weight .................................................................................... 161

Table 19-18. Parameters for anti-icing group weight .................................................................................... 162

Table 19-19. Other systems and equipment weight ....................................................................................... 162

ix

Chapter 1

Introduction

The NASA Design and Analysis of Rotorcraft (NDARC) software is an aircraft system analysis tool

intended to support both conceptual design efforts and technology impact assessments. The principal

tasks are to design (or size) a rotorcraft to meet specified requirements, including vertical takeoff and

landing (VTOL) operation, and then analyze the performance of the aircraft for a set of conditions. For

broad and lasting utility, it is important that the code have the capability to model general rotorcraft

configurations, and estimate the performance and weights of advanced rotor concepts. The architecture

of the NDARC code accommodates configuration flexibility; a hierarchy of models; and ultimately

multidisciplinary design, analysis, and optimization. Initially the software is implemented with lowfidelity models, typically appropriate for the conceptual design environment.

An NDARC job consists of one or more cases, each case optionally performing design and analysis

tasks. The design task involves sizing the rotorcraft to satisfy specified design conditions and missions.

The analysis tasks can include off-design mission performance calculation, flight performance calculation for point operating conditions, and generation of subsystem or component performance maps. For

analysis tasks, the aircraft description can come from the sizing task, from a previous case or a previous

NDARC job, or be independently generated (typically the description of an existing aircraft).

The aircraft consists of a set of components, including fuselage, rotors, wings, tails, and propulsion.

For each component, attributes such as performance, drag, and weight can be calculated; and the

aircraft attributes are obtained from the sum of the component attributes. Description and analysis

of conventional rotorcraft configurations is facilitated, while retaining the capability to model novel

and advanced concepts. Specific rotorcraft configurations considered are single main-rotor and tailrotor helicopter; tandem helicopter; coaxial helicopter; and tiltrotors. The architecture of the code

accommodates addition of new or higher-fidelity attribute models for a component, as well as addition

of new components.

1–1 Background

The definition and development of NDARC requirements benefited substantially from the experiences and computer codes of the preliminary design team of the U.S. Army Aeroflightdynamics

Directorate (AFDD) at NASA Ames Research Center.

In the early 1970s, the codes SSP-1 and SSP-2 were developed by the Systems Research Integration

Office (SRIO, in St. Louis) of the U.S. Army Air Mobility Research and Development Laboratory.

SSP-1 performed preliminary design to meet specified mission requirements, and SSP-2 estimated the

performance for known geometry and engine characteristics, both for single main-rotor helicopters

(ref. 1). Although similar tools were in use in the rotorcraft community, these computer programs were

independently developed, to meet the requirements of government analysis. The Advanced Systems
2 Introduction

Research Office (ASRO, at Ames Research Center) of USAAMRDL produced in 1974 two Preliminary

Systems Design Engineering (PSDE) studies (refs. 2 and 3) using SSP-1 and SSP-2. These two codes

were combined into one code called PSDE by Ronald Shinn.

The MIT Flight Transportation Laboratory created design programs for helicopters (ref. 4) and

tiltrotors (ref. 5). Michael Scully, who wrote the helicopter design program and was significantly

involved in the development of the tiltrotor design program, joined ASRO in 1975; then ideas from the

MIT programs began to be reflected in the continuing development of PSDE. An assessment of design

trade-offs for the Advanced Scout Helicopter (ASH) used a highly modified version of PSDE (ref. 6).

A U.S. Department of Defense Joint Study Group was formed in April 1975 to perform an Interservice Helicopter Commonality Study (HELCOM) for the Director of Defense Research and Engineering.

The final HELCOM study report was published in March 1976 (ref. 7). A result of this study was an

assessment by ASRO that PSDE needed substantial development, including better mathematical models

and better technical substantiation; more flexible mission analysis; and improved productivity for both

design and analysis tasks. Thus began an evolutionary improvement of the code, eventually named

RASH (after the developer Ronald A. Shinn, as a consequence of the computer system identification

of output by the first four characters of the user name). RASH included improvements in flight performance modeling, output depth, mission analysis, parametric weight estimation, design sensitivity

studies, off-design cases, and coding style. The code was still only for single main-rotor helicopters.

In the early 1980s, tool development evolved in two separate directions at the Preliminary Design

Team of ASRO. RASH was developed into the HELO (or PDPAC) code, for conventional and compound

single main-rotor helicopters. With the addition of conversion models and wing weight-estimation

methods (refs. 8 and 9), RASH became the TR code, for tiltrotor aircraft. The JVX Joint Technology

Assessment of 1982 utilized the HELO and TR codes. A special version called PDABC, including

a weight-estimation model for lift-offset rotors (ref. 10), was used to analyze the Advancing Blade

Concept. The JVX JTA report (ref. 11) documented the methodology implemented in these codes.

Work in support of the LHX (Light Helicopter Experimental) program from 1983 on led to a

requirement for maneuver analysis of helicopters and tiltrotors, implemented in the MPP code (Maneuver

Performance Program) by John Davis. The core aircraft model in MPP was similar to that in TR and

HELO, but the trim strategy in particular was new. A design code does not require extensive maneuver

analysis capability, but MPP had an impact on the design-code development, with the MPP performance

and trim methods incorporated into TR87. The sizing analysis of TR88 and the aircraft flight model from

MPP were combined into the VAMP code (VSTOL Design and Maneuver Program). VAMP combined

the capability to analyze helicopters and tiltrotors in a single tool, although the capability of HELO to

analyze compound helicopters was not replicated.

In the early 1990s, the RC code (for RotorCraft) emerged from the evolution of VAMP, with John

Preston as the lead developer (refs. 12 and 13). Some maneuver analysis capabilities from MPP were

added, and the analysis capability extended to helicopters. The models were confirmed by comparison

with results from TR and HELO. RC was operational by 1994, although HELO and RC continued to

be used into the mid 1990s. RC97 was a major version, unifying the tiltrotor and helicopter analyses.

The RC code introduced new features and capabilities, productivity enhancements, as well as coding

standards and software configuration control. Special versions of RC were routinely produced to meet

the unique requirements of individual projects (such as ref. 14).

NASA, with support from the U.S. Army, in 2005 conducted the design and in-depth analysis
Introduction

of rotorcraft configurations that could satisfy the Vehicle Systems Program technology goals (ref. 15).

These technology goals and accompanying mission were intended to identify enabling technology for

civil application of heavy-lift rotorcraft. The emphasis was on efficient cruise and hover, efficient

structures, and low noise. The mission specified was to carry 120 passengers for 1200 nautical miles, at

a speed of 350 knots and 30000-foot altitude. The configurations investigated were a Large Civil Tiltrotor

(LCTR), a Large Civil Tandem Compound (LCTC), and a Large Advancing Blade Concept (LABC). The

results of the NASA Heavy Lift Rotorcraft Systems Investigation subsequently helped define the content

and direction of the Subsonic Rotary Wing project in the NASA Fundamental Aeronautics program. The

design tool used was the AFDD RC code. This investigation is an example of the role of a rotorcraft sizing

code within NASA. The investigation also illustrated the difficulties involved in adapting or modifying

RC for configurations other than conventional helicopters and tiltrotors, supporting the requirement for

a new tool.

1–2 Requirements

Out of this history, the development of NDARC was begun in early 2007. NDARC is entirely new

software, built on a new architecture for the design and analysis of rotorcraft. From the RC theoretical

basis, the equations of the parametric weight equations and the Referred Parameter Turboshaft Engine

Model were used with only minor changes. Use was also made of the RC component aerodynamic

models and rotor performance model. The current users of RC, informed by past and recent applications,

contributed significantly to the requirements definition.

The principal tasks are to design (size) rotorcraft to meet specified requirements, and then analyze

the performance of the aircraft for a set of flight conditions and missions. Multiple design requirements,

from specific flight conditions and various missions, must be used in the sizing task. The aircraft

performance analysis must cover the entire spectrum of the aircraft capabilities, and allow general and

flexible definition of conditions and missions.

For government applications and to support research, it is important to have the capability to model

general rotorcraft configurations, including estimates of the performance and weights of advanced rotor

concepts. In such an environment, software extensions and modifications will be routinely required

to meet the unique requirements of individual projects, including introduction of special weight and

performance models for particular concepts.

Thus the code architecture must accommodate configuration flexibility and alternate models, including a hierarchy of model fidelity. Although initially implemented with low-fidelity models, typical

of the conceptual design environment, ultimately the architecture must allow multidisciplinary design,

analysis, and optimization. The component performance and engine models must cover all operating conditions. The software design and architecture must facilitate extension and modification of the

software.

Complete and thorough documentation of the theory and its software implementation is essential, to

support development and maintenance and to enable effective use and modification. Most of the history

described previously supports this requirement by the difficulties encountered in the absence of good

documentation. Documentation of the methodology was often prompted only by the need to substantiate

conclusions of major technology assessments, and occasionally by the introduction of new users and

developers. For a new software implementation of a new architectures, documentation is required from

the beginning of the development.
4 Introduction

1–3 Overview

The NDARC code performs design and analysis tasks. The design task involves sizing the rotorcraft

to satisfy specified design conditions and missions. The analysis tasks can include off-design mission

performance analysis, flight performance calculation for point operating conditions, and generation of

subsystem or component performance maps. Figure 1-1 illustrates the tasks. The principal tasks (sizing,

mission analysis, and flight performance analysis) are shown in the figure as boxes with heavy borders.

Heavy arrows show control of subordinate tasks.

The aircraft description (fig. 1-1) consists of all the information, input and derived, that defines

the aircraft. The aircraft consists of a set of components, including fuselage, rotors, wings, tails, and

propulsion. This information can be the result of the sizing task; can come entirely from input, for a

fixed model; or can come from the sizing task in a previous case or previous job. The aircraft description

information is available to all tasks and all solutions (indicated by light arrows).

The sizing task determines the dimensions, power, and weight of a rotorcraft that can perform a

specified set of design conditions and missions. The aircraft size is characterized by parameters such as

design gross weight, weight empty, rotor radius, and engine power available. The relationships between

dimensions, power, and weight generally require an iterative solution. From the design flight conditions

and missions, the task can determine the total engine power or the rotor radius (or both power and radius

can be fixed), as well as the design gross weight, maximum takeoff weight, drive system torque limit,

and fuel-tank capacity. For each propulsion group, the engine power or the rotor radius can be sized.

Missions are defined for the sizing task and for the mission performance analysis. A mission

consists of a number of mission segments, for which time, distance, and fuel burn are evaluated. For

the sizing task, certain missions are designated to be used for design gross-weight calculations; for

transmission sizing; and for fuel-tank sizing. The mission parameters include mission takeoff gross

weight and useful load. For specified takeoff fuel weight with adjustable segments, the mission time

or distance is adjusted so the fuel required for the mission (burned plus reserve) equals the takeoff fuel

weight. The mission iteration is on fuel weight.

Flight conditions are specified for the sizing task and for the flight performance analysis. For the

sizing task, certain flight conditions are designated to be used for design gross-weight calculations; for

transmission sizing; for maximum takeoff-weight calculations; and for antitorque or auxiliary-thrust

rotor sizing. The flight-condition parameters include gross weight and useful load.

For flight conditions and mission takeoff, the gross weight can be maximized, such that the power

required equals the power available.

A flight state is defined for each mission segment and each flight condition. The aircraft performance

can be analyzed for the specified state, or a maximum effort performance can be identified. The maximum

effort is specified in terms of a quantity such as best endurance or best range, and a variable such as

speed, rate of climb, or altitude. The aircraft must be trimmed, by solving for the controls and motion

that produce equilibrium in the specified flight state. Different trim-solution definitions are required for

various flight states. Evaluating the rotor-hub forces may require solution of the blade-flap equations of

motion.

1–4 Terminology

The following terminology is introduced as part of the development of the NDARC theory and
Flight

Performance

Analysis V

design

conditions design

missions

Mission

adjust & fuel wt iteration

max takeoff GW

each segment

Flight Condition

max GW

Introduction 5

fixed model or

previous job or

DESIGN previous case 4

Sizing Task

size iteration Aircraft

Description

ANALYZE

Airframe

r

0014: 

Aerodynamics Map

Engine

Performance Map

Mission Analysis

Flight State

max effort / trim aircraft / flap equations

Figure 1-1. Outline of NDARC tasks.

software. Relationships among these terms are reflected in figure 1-1.

a) Job: An NDARC job consists of one or more cases.

b) Case: Each case performs design and/or analysis tasks. The analysis tasks can include off-design

mission performance calculation, flight performance calculation for point operating conditions, and

generation of airframe aerodynamics or engine performance maps.

c) Design Task: Size rotorcraft to satisfy specified set of design flight conditions and/or design missions.

Key aircraft design variables are adjusted until all criteria are met. The resulting aircraft description can

be the basis for mission analysis and flight performance analysis tasks.

d) Mission Analysis Task: Calculate aircraft performance for one off-design mission.

e) Flight Performance Analysis Task: Calculate aircraft performance for point operating condition.

f) Mission: Ordered set of mission segments, for which time, distance, and fuel burn are evaluated.
6 Introduction

Gross weight and useful load are specified for the beginning of the mission, and adjusted for fuel burn

and useful load changes at each segment. Missions are defined for the sizing task and for the mission

performance analysis.

g) Flight Condition: Point operating condition, with specified gross weight and useful load. Flight

conditions are specified for the sizing task and for the flight performance analysis.

h) Flight State: Aircraft flight condition, part of definition of each flight condition and each mission

segment. Flight state solution involves rotor-blade motion, aircraft trim, and perhaps a maximum-effort

calculation.

i) Component: The aircraft consists of a set of components, including fuselage, rotors, wings, tails, and

propulsion. For each component, attributes such as performance, drag, and weight are calculated.

j) Propulsion Group: A propulsion group is a set of components and engine groups, connected by a drive

system. An engine group consists of one or more engines of a specific type. The components define the

power required. The engine groups define the power available.

1–5 Analysis Units

The code can use either English or SI units for input, output, and internal calculations. A consistent

mass-length-time-temperature system is used, except for weight and power:

length mass time temperature weight power

English: foot slug second OF pound horsepower

SI: meter kilogram second °C kilogram kiloWatt

In addition, the default units for flight conditions and missions are: speed in knots, time in minutes,

distance in nautical miles, and rate of climb in feet-per-minute. The user can specify alternate units for

these and other quantities.

1–6 Outline of Report

This document provides a complete description of the NDARC theoretical basis and architecture.

Chapters 3–5 describe the tasks and solution procedures, while chapters 7–17 present the models for the

aircraft and its components. The cost model is described in chapter 6; the engine model in chapter 18;

and the weight model in chapter 19. The accompanying NDARC Input Manual describes the use of the

code.

1–7 References

1) Schwartzberg, M.A.; Smith, R.L.; Means, J.L.; Law, H.Y.H.; and Chappell, D.P.: Single-Rotor

Helicopter Design and Performance Estimation Programs. USAAMRDL Report SRIO 77-1, June

1977.

2) Wheatley, J.B.; and Shinn, R.A.: Preliminary Systems Design Engineering for a Small Tactical Aerial

Reconnaissance System-Visual. USAAMRDL, June 1974.

3) Shinn, R.A.: Preliminary Systems Design Engineering for an Advanced Scout Helicopter. USAAMRDL, August 1974.
Introduction 7

4) Scully, M.; and Faulkner, H.B.: Helicopter Design Program Description. MIT FTL Technical Memo

71-3, March 1972.

5) Faulkner, H.B.: A Computer Program for the Design and Evaluation of Tilt Rotor Aircraft. MIT FTL

Technical Memo 74-3, September 1974.

6) Scully, M.P.; and Shinn, R.A.: Rotor Preliminary Design Trade-Offs for the Advanced Scout Helicopter. American Helicopter Society National Specialists’ Meeting on Rotor System Design, Philadelphia, Pennsylvania, October 1980.

7) Interservice Helicopter Commonality Study, Final Study Report. Director of Defense Research and

Engineering, Office of the Secretary of Defense, March 1976.

8) Chappell, D.P.: Tilt-rotor Aircraft Wing Design. ASRO-PDT-83-1, 1983.

9) Chappell, D.; and Peyran, R.: Methodology for Estimating Wing Weights for Conceptual Tilt-Rotor

and Tilt-Wing Aircraft. SAWE Paper No. 2107, Category No. 23, May 1992.

10)Weight Trend Estimation for the Rotor Blade Group, Rotor Hub Group, and Upper Rotor Shaft of

the ABC Aircraft. ASRO-PDT-83-2, 1983.

11) Technology Assessment of Capability for Advanced Joint Vertical Lift Aircraft (JVX), Summary

Report. U.S. Army Aviation Research and Development Command, AVRADCOM Report, May 1983.

12) Preston, J.: and Peyran, R.: Linking a Solid-Modeling Capability with a Conceptual Rotorcraft

Sizing Code. American Helicopter Society Vertical Lift Aircraft Design Conference, San Francisco,

California, January 2000.

13) Preston, J.: Aircraft Conceptual Design Trim Matrix Selection. American Helicopter Society

Vertical Lift Aircraft Design Conference, San Francisco, California, January 2006.

14) Sinsay, J.D.: The Path to Turboprop Competitive Rotorcraft: Aerodynamic Challenges. American

Helicopter Society Specialists’ Conference on Aeromechanics, San Francisco, Californi, January 2008.

15)Johnson, W.; Yamauchi, G.K.; and Watts, M.E.: NASA Heavy Lift Rotorcraft Systems Investigation.

NASA TP 2005-213467, December 2005.
Introduction
Chapter 2

Nomenclature

The nomenclature for geometry and rotations employs the following conventions. A vector x is a

column matrix of three elements, measuring the vector relative to a particular basis (or axes, or frame).

The basis is indicated as follows:

a) xA is a vector measured in axes A;

b) xEF/A is a vector from point F to point E, measured in axes A.

A rotation matrix C is a three-by-three matrix that transforms vectors from one basis to another:

c) CBA transforms vectors from basis A to basis B, so x

B = CBAxA.

The matrix CBA defines the orientation of basis B relative to basis A, so it also may be viewed as rotating

the axes from A to B. For a vector u, a cross-product matrix u^ is defined as follows:

⎡ ⎤

0 

-u3 u2

u^ = ⎣

u3 0 -u1

⎦

-u2u1 0

such that iv is equivalent to the vector cross-product u x v. The cross-product matrix enters the relation

between angular velocity and the time derivative of a rotation matrix:

C ̇ AB = -

WAB/AC

AB = CABWBA/B

(the Poisson equations). For rotation by an angle α about the x, y, or z axis (1, 2, or 3 axis), the following

notation is used: ⎡ ⎤

1 0 0

Xα = ⎣0 cos α sin α ⎦

0 - sin α cos α

⎡ ⎤

cos α 0 - sin α

Yα = ⎣ 0 1 0 ⎦

sin α 0 cos α

⎡ ⎤

cos α sin α 0

Zα = ⎣- sin α cos α 0 ⎦

0 0 1

Thus for example, CBA = XφYθZ ψ means that the axes B are located relative to the axes A by first

rotating by angle ψ about the z-axis, then by angle θ about the y-axis, and finally by angle φ about the

x-axis.
10 Nomenclature

Acronyms

AFDD U.S. Army Aeroflightdynamics Directorate

ASM available seat mile

CAS calibrated airspeed

CPI consumer price index

CRP contingency rated power

DoD Department of Defense

ECU environment control unit

EG engine group

ERP emergency rated power

FCE flight control electronics

ICAO International Civil Aviation Organization

IGE in-ground-effect

IRP intermediate rated power

IRS infrared suppressor

ISA International Standard Atmosphere

ISO International Organization for Standardization

JVX Joint Vertical Experimental

MCP maximum continuous power

MEP mission equipment package

MRP maximum rated power

OEI one-engine inoperative

OGE out-of-ground-effect

PG propulsion group

RPTEM referred parameter turboshaft engine model

SDGW structural design gross weight

SLS sea-level, standard day

TAS true airspeed

WMTO maximum takeoff weight

Weights

WD design gross weight

WE empty weight

WMTO maximum takeoff weight

WSD structural design gross weight

WG gross weight, WG = WE + WUL = WO + Wpay + Wfuel

WO operating weight, WO = WE + WFUL

WUL useful load, WUL = WFUL + Wpay + Wfuel

Wpay payload

Wfuel fuel weight

WFUL fixed useful load

Wburn mission fuel burn

Wvib vibration control weight

Wcont contingency weight

χ technology factor
Nomenclature 11

Fuel Tanks

Nauxtank number of auxiliary fuel tanks

Vfuel—cap fuel capacity, volume

Waux—cap auxiliary-fuel-tank capacity

Wfuel—cap fuel capacity, maximum usable fuel weight

Power

Ninop number of inoperative engines, engine group

PavPG power available, propulsion group; min(E fPPavEG, (Ωprim/Ωref)PDSlimit)

PavEG power available, engine group; (Neng — Ninop ) Pav

PreqPG power required, propulsion group; Pcomp + Pxmsn + Pacc

PreqEG power required, engine group

Pcomp component power required

Pxmsn transmission losses

Pacc accessory power

PDSlimit drive-system torque limit (specified as power limit at reference rotor speed)

PESlimit engine-shaft rating

PRSlimit rotor-shaft rating

Engine

Daux momentum drag

FN net jet thrust

m ̇ mass flow (conventional units)

N specification turbine speed

Neng number of engines in engine group (EG)

Peng sea-level static power available per engine at specified takeoff rating

Pav power available, installed; min(Pa — Ploss, Pmech)

Pa power available, uninstalled

Preq power required, installed; Pq - Ploss

Pq power required, uninstalled

Ploss installation losses

Pmech mechanical power limit

sfc specific fuel consumption, w/P  ̇ (conventional units)

SP specific power, P/m ̇ (conventional units)

SW specific weight, P/W

w ̇ fuel flow (conventional units)

Tip Speed and Rotation

Nspec specification engine-turbine speed (rpm)

r gear ratio; Ωdep/Ωprim for rotor, Ωspec /Ωprim for engine

Vtip—ref reference tip speed, propulsion group primary rotor; each drive state

Ωprim primary-rotor rotational speed, Ω = Vtip—ref /R

Ωdep dependent-rotor rotational speed, Ω = Vtip—ref /R

Ωspec specification engine-turbine speed
12 Nomenclature

Mission

D mission segment distance

dR mission segment range contribution

E endurance

R range

T mission segment time

w ̇ fuel flow

Environment

cs speed of sound

g gravitational acceleration

h altitude

T temperature, O R or °K

Vw wind speed

μ viscosity

ν kinematic viscosity

ρ density

τ temperature, O F or °C

Axis Systems

A component aerodynamic

B component

F aircraft

I inertial

V velocity

Geometry

B length

L reference length (fuselage length, rotor radius, or wing span)

Swet wetted area

SL, BL, WL fixed input position (station line, buttline, waterline)

positive aft, right, up; arbitrary origin

x/L, y/L, z/L scaled input position; positive aft, right, up; origin at reference point

x, y, z calculated position, aircraft axes; positive forward, right, down;

origin at reference point for geometry,

origin at center of gravity for motion and loads

z

F

component position vector, in aircraft axes, relative reference point
Nomenclature 13

Motion

F

aAC

n

V

F

vAC

Vc

Vcal

Vf

Vh

Vs

' φF, θF, /)F

,

θV, ψV

WF

FωAC

aircraft linear acceleration

load factor

aircraft velocity magnitude

aircraft velocity

climb velocity

calibrated airspeed, V ρ/ρ0

forward velocity

horizontal velocity

sideward velocity

roll, pitch, yaw angles; orientation airframe axes F relative inertial axes

climb, sideslip angles; orientation velocity axes V relative inertial axes

turn rate

aircraft angular velocity

Aerodynamics and Loads

cd, c^ section drag, lift coefficients

CD, CY, CL component drag, side, lift force coefficients

C^, CM, CN component roll, pitch, yaw moment coefficients

D, Y, L aerodynamic drag, side, lift forces (component aerodynamic axes A)

D/q drag area, SCD (S = Reference area of component)

F force

if ratio flap chord to airfoil chord, c f

/c

M moment

Mx, My, Mz aerodynamic roll, pitch, yaw moments (component aerodynamic axes A)

q dynamic pressure, 1

/2ρ|v|2

v component velocity relative air (including interference)

α angle of attack, component axes B relative aerodynamic axes A

β sideslip angle, component axes B relative aerodynamic axes A

δfflap deflection

Aircraft

Aref reference rotor area, E fAA; typically projected area of lifting rotors

c component control, c = STcAC + c0

cAC aircraft control

De aircraft effective drag, P/V

DL disk loading, WD /Aref

L/De aircraft effective lift-to-drag ratio, WV/P

M aircraft hover figure of merit, W W/2ρAref/P

Sref reference wing area, E S; sum area all wings

T control matrix

WL wing loading, WD / Sref

αtilt tilt control variable
14 Nomenclature

Rotor

A disk area

cdmean profile power mean drag coefficient, CPo = (σ/8)cdmeanFP

CT/σ thrust coefficient divided by solidity, T/ρA(ΩR)

2σ

CW/σ design blade loading, W/ ρAV2tipσ (Vtip = hover tip speed)

H, Y, T drag, side, thrust force on hub (shaft axes)

L/De rotor effective lift-to-drag ratio, VL/ (Pi + Po )

M rotor hover figure of merit, T fD v/P

Mat advancing tip Mach number

Mx

, My roll, pitch moment on hub

Pi , Po , Pp

induced, profile, parasite power

Q shaft torque

R blade radius

r direction of rotation ( 1 for counter-clockwise, —1 for clockwise)

r blade span coordinate

Tdesign design thrust of antitorque or auxiliary thrust rotor

W/A disk loading, W = fW WD

βc, βs

longitudinal, lateral flapping (tip-path plane tilt relative shaft)

γ blade Lock number

η propulsive efficiency, TV/P

κ induced power factor, Pi = κPideal

λ inflow ratio

μ advance ratio

ν blade flap frequency (per-rev)

θ0.75 blade collective pitch angle (at 75% radius)

θc, θs lateral, longitudinal blade pitch angle)

σ solidity (ratio blade area to disk area)

ψ blade azimuth coordinate

Wing

AR aspect ratio, b2 /S

b span

c chord, S/b

S area

W/S wing loading, W = fW WD
Chapter 3

Tasks

The NDARC code performs design and analysis tasks. The design task involves sizing the rotorcraft

to satisfy specified design conditions and missions. The analysis tasks can include mission performance

analysis, flight performance calculation for point operating conditions, and generation of subsystem or

component performance maps.

3–1 Size Aircraft for Design Conditions and Missions

3-1.1 Sizing Method

The sizing task determines the dimensions, power, and weight of a rotorcraft that can perform a

specified set of design conditions and missions. The aircraft size is characterized by parameters such as

design gross weight (WD ) or weight empty (WE), rotor radius (R), and engine power available (Peng).

The relationships between dimensions, power, and weight generally require an iterative solution. From

the design flight conditions and missions, the task can determine the total engine power or the rotor

radius (or both power and radius can be fixed), as well as the design gross weight, maximum takeoff

weight, drive-system torque limit, and fuel-tank capacity. For each propulsion group, the engine power

or the rotor radius can be sized.

a) Engine power: Determine Peng, for fixed R. The engine power is the maximum

of the power required for all sizing flight conditions and sizing missions (typically

including vertical flight, forward flight, and one-engine inoperative). Hence the

engine power is changed by the ratio max(PreqPG/PavPG ) (excluding flight states for

which zero power margin is calculated, such as maximum gross weight or maximum

effort). This approach is the one most commonly used for the sizing task.

b) Rotor radius: Determine R for input Peng. The maximum power required for

all sizing flight conditions and sizing missions is calculated, and then the rotor

radius determined such that the power required equals the input power available.

The change in radius is estimated as R = Rold PreqPG/PavP G (excluding flight

states for which zero power margin is calculated, such as maximum gross weight or

maximum effort). For multi-rotor aircraft, the radius can be fixed rather than sized

for some rotors.

Alternatively, Peng and R can be input rather than sized. Aircraft parameters can be determined by a

subset of the design conditions and missions.

a) Design gross weight WD : maximum gross weight from designated conditions

and missions (for which gross weight is not fixed).
16 Tasks

b) Maximum takeoff gross weight WMTO : maximum gross weight from designated

conditions (for which gross weight is not fixed).

c) Drive-system torque limit PDSlimit: maximum torque from designated conditions

and missions (for each propulsion group; specified as power limit at reference rotor

speed).

d) Fuel-tank capacity Wfuel_cap: maximum fuel weight from designated missions

(without auxiliary tanks).

e) Antitorque or auxiliary-thrust rotor design thrust Tdesign: maximum rotor thrust

from designated conditions.

Alternatively, these parameters can be fixed at input values. The design gross weight (WD ) can be fixed.

The weight empty can be fixed (achieved by adjusting the contingency weight).

A successive substitution method is used for the sizing iteration, with an input tolerance E. Relaxation is applied to Peng or R, WD , WMTO, PDSlimit, Wfuel-cap, and Tdesign. Convergence is tested in terms

of these parameters, and the aircraft weight empty WE . Two successive substitution loops are used. The

outer loop is an iteration on performance: engine power or rotor radius, for each propulsion group. The

inner loop is an iteration on parameters: WD , WMTO, PDSlimit, Wfuel-cap, and Tdesign. Either loop can

be absent, depending on the definition of the size task.

For each flight condition and each mission, the gross weight and useful load are specified. The

gross weight can be input, maximized, or fallout. For flight conditions, the payload or fuel weight can be

specified, and the other calculated; or both payload and fuel weight specified, with gross weight fallout.

For missions, the payload or fuel weight can be specified, the other fallout, and then time or distance of

mission segments adjusted; or fuel weight calculated from mission, and payload fallout; or both payload

and fuel weight specified (or payload specified and fuel weight calculated from mission), with gross

weight fallout. For each flight condition and mission segment, the following checks are performed.

a) The power required does not exceed the power available: PreqPG <_ ( 1 + E) PavPG

(for each propulsion group).

b) The torque required does not exceed the drive-system limit: For each propulsion

group PreqPG/Ω < (1+E)PDSlimit/Ωprim. Rotor-shaft torque and engine-shaft torque

are also checked.

c) The fuel weight does not exceed the fuel capacity: Wfuel <_ (1 + E) (Wfuel-cap +

E NauxtankWaux-cap) (including auxiliary tanks).

These checks are performed using an input tolerance E.

Sizing flight conditions typically include takeoff (hover or specified vertical rate-of-climb), oneengine-inoperative, cruise or dash, perhaps transmission, and perhaps mission midpoint hover. Sizing

missions typically include a design mission and a mission to determine fuel-tank capacity.

3-1.2 Component Sizing

3-1.2.1 Engine Power

The engine size is described by the power Peng, which is the sea-level static power available per

engine at a specified takeoff rating. The number of engines Neng is specified for each engine group.

If the sizing task determines the engine power for a propulsion group, the power Peng of at least one
Tasks 17

engine group is found (including the first engine group). The total power required is PPG = r E NengPeng,

where r = max(PreqPG/PavPG). The sized power is Psized = PPG — Efixed NengPeng, where the sum is

over the engine groups for which the power is fixed. Then the sized engine power is Peng = fnPsized /Neng

for the n-th engine group (with fn an input ratio and f1 = 

^

no1,sized fn for the first group).

3-1.2.2 Main Rotor

The main-rotor size is defined by the radius R or disk loading W/A, thrust-weighted solidity σ,

hover tip speed Vtip, and blade loading CW/σ = W/ ρAV2tip σ. With more than one main rotor, the disk

loading and blade loading are obtained from an input fraction of design gross weight, W = fW WD . The

air density ρ for CW / σ is obtained from a specified takeoff condition.

If the rotor radius is fixed for the sizing task, three of (R or W/A), CW/σ, Vtip, σ are input and the

other parameters are derived. Optionally the radius can be calculated from a specified ratio to the radius

of another rotor.

If the sizing task determines the rotor radius (R and W/A), then two of CW/σ, Vtip, σ are input and

the other parameter is derived. The radius can be sized for just a subset of the rotors, with fixed radius

for the others. The radii of all sized rotors are adjusted by the same factor.

3-1.2.3 Antitorque or Auxiliary-Thrust Rotor

For antitorque and auxiliary-thrust rotors, three of (R or W/A), CW/σ, Vtip, σ are input and the

other parameters are derived. Optionally the radius can be calculated from a specified ratio to the radius

of another rotor. Optionally the radius can be scaled with the main-rotor radius. The disk loading and

blade loading are based on fTTdesign, where fT is an input factor and Tdesign is the maximum thrust from

designated design conditions.

3-1.2.4 Wing

The wing size is defined by the wing area S or wing loading W/S, span (perhaps calculated from

other geometry), chord, and aspect ratio. With more than one wing, the wing loading is obtained from

an input fraction of design gross weight, W = fW WD .

Two of the following parameters are input: area (or wing loading), span, chord, and aspect ratio;

the other parameters are derived. Optionally the span can be calculated from the rotor radius, fuselage

width, and clearance (typically used for tiltrotors). Optionally the span can be calculated from a specified

ratio to the span of another wing.

3-1.2.5 Fuel Tank

The fuel-tank capacity Wfuel-cap (maximum usable fuel weight) is determined from designated

sizing missions. The maximum mission fuel required, Wfuel-miss (excluding reserves and any fuel in

auxiliary tanks), gives

Wfuel-cap = max(ffuel-capWfuel-miss, Wfuel-miss + Wreserve)

where ffuel-cap > 1 is an input factor. Alternative, the fuel-tank capacity Wfuel-cap can be input.

3-1.2.6 Weights

The structural design gross weight WSD and maximum takeoff weight WMTO can be input, or
18 Tasks

specified as an increment d plus a fraction f of a weight W:

J dSDGW + fSDGWWD

WSD = dSDGW + fSDGWW = 11 dSDGW + fSDGW (WD — Wfuel + ffuelWfuel— cap)

^dWMTO + fWMTOWD

WMTO = dWMTO + fWMTOW = dWMTO + fWMTO (WD — Wfuel + Wfuel— cap)

This convention allows the weights to be input directly ( f = 0), or scaled with WD. For WSD, W is the

design gross weight WD, or WD adjusted for a specifed fuel state (input fraction of fuel capacity). For

WMTO, W is the design gross weight WD, or WD adjusted for maximum fuel capacity. Alternatively,

WMTO can be calculated as the maximum gross weight possible at a designated sizing flight condition.

3-1.2.7 Drive-System Rating

The drive-system rating is defined as a power limit, PDSlimit. The rating is properly a torque limit,

QDSlimit = PDSlimit/Ω, but is expressed as a power limit for clarity. The drive-system rating can be

specified as follows:

a) Input PDSlimit.

b) From the engine takeoff power rating, PDSlimit = flimit E NengPeng (summed over

all engine groups).

c) From the power available at the transmission sizing flight conditions, PDSlimit =

flimit (Ωref/Ωprim) E NengPav (largest of all conditions).

d) From the power required at the transmission sizing flight conditions, PDSlimit =

flimit (Ωref/Ωprim) E NengPreq (largest of all conditions).

with flimit an input factor. The drive-system rating is a limit on the entire propulsion system. To account

for differences in the distribution of power through the drive system, limits are also used for the torque

of each rotor shaft (PRSlimit) and of each engine group (PESlimit). The engine-shaft rating is calculated

as for the drive-system rating, without the sum over engine groups. The rotor-shaft rating is either input

or calculated from the rotor power required at the transmission sizing flight conditions. The power limit

is associated with a reference rotational speed, and when applied it is scaled with the rotational speed of

the flight state. The rotation speed for the drive-system rating PDSlimit is the hover speed of the primary

rotor of the propulsion group (for the first drive state). The rotation speed for the engine-shaft rating

PESlimit is the corresponding engine turbine speed. The rotation speed for the rotor-shaft rating PRSlimit

is the corresponding speed of that rotor.

3–2 Mission Analysis

For the mission analysis, the fuel weight or payload weight is calculated. Power required, torque

(drive system, engine shaft, and rotor shaft), and fuel weight are then verified to be within limits.

Missions can be fixed or adjustable.

3–3 Flight Performance Analysis

For each performance flight condition, the power required is calculated or maximum gross weight

is calculated. Power required, torque (drive system, engine shaft, and rotor shaft), and fuel weight are

then verified to be within limits.
Tasks 19

3–4 Maps

3-4.1 Engine Performance

The engine performance can be calculated for a specified range of power, altitude, and speed.

3-4.2 Airframe Aerodynamics

The airframe aerodynamic loads can be calculated for a specified range of angle of attack, sideslip

angle, and control angles. The aerodynamic analysis evaluates the component lift, drag, and moments

given the velocity. The aircraft velocity is here v

F

AC == CFA (v 0 0)T; interference velocity from the rotors

is not considered. From the angle of attack α and sideslip angle β, the transformation from wind axes to

airframe axes is CFA = YαZ−β (optionally CFA = Z−βYα can be used, for better behavior in sideward

flight). The loads are summed in the airframe axes (with and without tail loads), and then the wind axis

loads are:

( −D Mx

F

A

= Y = C

AF FF MA

=

⎛

⎝My⎠ = CAFMF

—L Mz

The center of action for the total loads is the fuselage location zfuse. The ratio of the loads to dynamic

pressure is required, so a nominal velocity v = 100 (ft/sec or m/sec) and sea-level standard density are

used.
20 Tasks
Chapter 4

Operation

4–1 Flight Condition

Flight conditions are specified for the sizing task and for the flight performance analysis. For each

condition, a flight state is also defined. For the sizing task, certain flight conditions are designated

to be used for design gross-weight calculations, for transmission sizing, for maximum takeoff-weight

calculations, or for rotor thrust sizing. The flight-condition parameters include gross weight and useful

load. The gross weight can be specified as follows, consistent with the sizing method:

a) Design gross weight, WD (calculated or input).

b) Structural design gross weight, WSD, or maximum takeoff weight, WMTO (which

may depend on WD) .

c) Function of WD : W = d + fWD (with d an input weight and f an input factor).

d) Function of WSD (W = d + fWSD); or function of WMTO (W = d + fWMTO).

e) Input W.

f) Gross weight from specified mission segment (such as midpoint hover).

g) Gross weight maximized, such that power required equals power available: zero

power margin, min(PavPG - PreqPG) = 0 (minimum over all propulsion groups).

h) Gross weight maximized, such that power required equals input power:

min((d + fPavPG) - PreqPG) = 0 (minimum over all propulsion groups, with d an

input power and f an input factor; this convention allows the power to be input

directly, f = 0, or scaled with power available).

i) Gross weight fallout from input payload and fuel weights: WG = WO + Wpay +

Wfuel.

Only the last three options are available for WD design conditions in the sizing task. The gross weight

can be obtained from a mission segment only for the sizing task. Optionally the altitude can be obtained

from the specified midpoint mission segment. The secant method or the method of false position is used

to solve for the maximum gross weight. A tolerance a and a perturbation Δ are specified.

The useful load can be specified as follows, consistent with the sizing method and the gross-weight

specification.

a) Input payload weight Wpay, fuel-weight fallout: Wfuel = WG - WO - Wpay.

b) Input fuel weight Wfuel, payload-weight fallout: Wpay = WG - WO - Wfuel.

c) Input payload and fuel weights, gross-weight fallout (must match gross-weight

option): WG = WO + Wpay + Wfuel.

The input fuel weight is Wfuel = min(dfuel + ffuel Wfuel−cap , Wfuel−cap) + ^ Nauxtank Waux−cap . For fallout
22 Operation

fuel weight, Nauxtank is adjusted (optionally only increased). If the auxiliary tank weight is greater than

the increment in fuel weight needed, then the fallout fuel weight Wfuel = WG — WO — Wpay can not

be achieved; in such a case, the fuel weight is capped at the maximum fuel capacity and the payload

weight adjusted instead. The fixed useful load can have increments, including crew-weight increment;

furnishings-weight increment; and installed folding, wing extension, and other kits. These increments

are reflected in the fallout weight.

4–2 Mission

Missions are defined for the sizing task and for the mission performance analysis. A mission

consists of a specified number of mission segments. A flight state is defined for each mission segment.

For the sizing task, certain missions are designated to be used for design gross-weight calculations,

for transmission sizing, or for fuel-tank sizing. The mission parameters include mission takeoff gross

weight and useful load. The gross weight can be specified as follows, consistent with the sizing method:

a) Design gross weight, WD (calculated or input).

b) Structural design gross weight, WSD, or maximum takeoff weight, WMTO (which

may depend on WD) .

c) Function of WD : W = d + fWD (with d an input weight and f an input factor).

d) Function of WSD, W = d + fWSD; or function of WMTO , W = d + f WMTO.

e) Input W.

f) Gross weight maximized at specified mission segments, such that power required

equals power available: zero power margin, min(PavPG — PreqPG) = 0 (minimum

over all propulsion groups).

g) Gross weight maximized at specified mission segments, such that power required

equals input power: min((d + fPavPG) — PreqPG) = 0 (minimum over all propulsion

groups, with d an input power and f an input factor; this convention allows the

power to be input directly, f = 0, or scaled with power available).

h) Gross-weight fallout from input initial payload and fuel weights: WG = WO +

Wpay + Wfuel.

i) Gross-weight fallout from input initial payload weight and calculated mission fuel

weight: WG = WO + Wpay + Wfuel.

If maximum gross weight is specified for more than one mission segment, then the minimum takeoff

gross-weight increment is used, so the power margin is zero for the critical segment and positive for

other designated segments. Only the last four options are available for WD design conditions in the

sizing task. The secant method or the method of false position is used to solve for the maximum gross

weight. A tolerance a and a perturbation Δ are specified.

The useful load can be specified as follows, consistent with the sizing method and the gross-weight

specification:

a)Input initial payload weight Wpay, fuel-weight fallout: Wfuel = WG — WO — Wpay.

b) Input fuel weight Wfuel, initial payload-weight fallout: Wpay = WG — WO — Wfuel .

c) Calculated mission-fuel weight, initial payload-weight fallout: Wpay = WG —

WO — Wfuel.

d) Input payload and fuel weights, takeoff gross-weight fallout (must match grossweight option): WG = WO + Wpay + Wfuel.
Operation 23

e) Input payload weight and calculated mission fuel weight, takeoff gross-weight

fallout (must match gross-weight option): WG = WO + Wpay + Wfuel.

The input fuel weight is Wfuel = min(dfuel + ffuelWfuel−cap, Wfuel—cap) + E NauxtankWaux−cap. If the

fuel weight is not calculated from the mission, then the mission is adjusted. The fixed useful load can

have increments, including installed folding kits; other increments are specified for individual mission

segments.

The takeoff gross weight is evaluated at the start of the mission, perhaps maximized for zero power

margin at a specified mission segment (either takeoff conditions or midpoint). Then the aircraft is flown

for all segments. For calculated mission fuel weight, the fuel weight at takeoff is adjusted to equal the fuel

required for the mission (burned plus reserve). For specified takeoff fuel weight with adjustable segments,

the mission time or distance is adjusted so the fuel required for the mission (burned plus reserve) equals

the takeoff fuel weight. The mission iteration is thus on mission fuel weight. Range credit segments

(described at the end of this section) can also require an iteration. A successive substitution method

is used if an iteration is required, with a tolerance a specified. The iteration to maximize takeoff gross

weight could be an outer loop around the mission iteration, but instead it is executed as part of the mission

iteration. At the specified mission segment, the gross weight is maximized for zero power margin, and

the resulting gross-weight increment added to the takeoff gross weight for the next mission iteration.

Thus takeoff gross weight is also a variable of the mission iteration.

Each mission consists of a specified number of mission segments. The following segment types

can be specified:

a) Taxi or warm-up (fuel burned but no distance added to range).

b) Distance: fly segment for specified distance (calculate time).

c) Time: fly segment for specified time (calculate distance).

d) Hold: fly segment for specified time (loiter, so fuel burned but no distance added

to range).

e) Climb: climb or descend from present altitude to next segment altitude (calculate

time and distance).

f) Spiral: climb or descend from present altitude to next segment altitude; fuel

burned but no distance added to range.

For each mission segment a payload weight can be specified; or a payload weight change can be specified,

as an increment from the initial payload or as a fraction of the initial payload.

The number of auxiliary fuel tanks can change with each mission segment: Nauxtank is adjusted

based on the fuel weight (optionally only increased relative to the input number at takeoff, optionally fixed

during mission). For input fuel weight, Nauxtank is specified at takeoff. For fallout fuel weight, the takeoff

fuel weight is adjusted for the auxiliary-fuel-tank weight given Nauxtank (fixed WG − Wpay = WO +Wfuel) .

If the auxiliary-tank weight is greater than the increment in fuel weight needed, then the fallout fuel

weight Wfuel = WG — WO — Wpay can not be achieved; in such a case, the fuel weight is capped at the

maximum fuel capacity and the takeoff payload weight adjusted instead. For fuel-tank design missions,

Nauxtank and fuel-tank capacity is determined from Wfuel. Optionally the aircraft can refuel (either on

the ground or in the air) at the start of a mission segment, by either filling all tanks to capacity or adding

a specified fuel weight. Optionally fuel can be dropped at the start of a mission segment. The fixed

useful load can have adjustments, including crew-weight increment; furnishings-weight increment; and

installed wing extension and other kits.
24 Operation

For calculation of the time or distance in a mission segment, a headwind or tailwind can be specified.

The wind velocity is a linear function of altitude h: Vw = f(max(0, dwind + fwind h)), with the plus sign

for a headwind and the minus sign for a tailwind. For example, California-to-Hawaii 85 th percentile

winter quartile headwind profile is Vw = 9. 59 + 0. 00149h (with altitude h in ft).

Mission fuel reserves can be specified in several ways for each mission. Fuel reserves can be

defined in terms of specific mission segments, for example 200 miles plus 20 minutes at speed for best

endurance. Fuel reserves can be an input fraction of the fuel burned by all (except reserve) mission

segments, so Wfuel = (1 + fres)Wburn. Fuel reserves can be an input fraction of the fuel capacity, so

Wfuel = Wburn + fres Wfuel−cap . If more than one criterion for reserve fuel is specified, the maximum

reserve is used. Time and distance in reserve segments are not included in endurance and range.

To facilitate specification of range, range calculated for a group of segments (typically climb and

descent segments) can be credited to a designated distance segment. For mission analysis, missions can

be fixed or adjustable. In an adjustable mission, the fuel is input, so the time or distance in specified

segments is adjusted based on the calculated fuel burned. If more than one segment is adjusted, all must

be distance or all must be time or hold. Each segment can have only one special designation: reserve,

adjustable, or range credit.

A segment with a large distance, time, or altitude change can be split into several segments, for

more accurate calculation of the performance and fuel burned. The number of segments n can be input

or calculated from an input increment Δ: n = [x/Δ] + 1, where the brackets indicate integer truncation

and x is the total distance, time, or altitude change. Then the change for each split segment is Δ = x/ n.

Table 4-1 summarizes the time T, distance D, and range dR calculations for each segment. The

segment fuel burned is dWburn = Tw,  ̇ where w ̇ is the fuel flow. The horizontal velocity is Vh , and the

vertical velocity (climb or descent) is Vc . The altitude at the start of the segment is h, and at the end

of the segment (start of next segment) hend. The wind speed is Vw , and the ground speed is Vh - Vw .

In an adjusted mission, the distances or times are changed at the end of the mission such that the sum

of the fuel-burned increments will equal the difference between takeoff fuel weight (plus any added

fuel) and the calculated mission fuel: E dWburn = E w ̇ dT = E w ̇ dD/ (Vh - Vw ) = ΔWfuel. The

increments are apportioned among the adjusted segments by the factor f, determined from the ratio of

the input distances or times: dD = fΔD or dT = fΔT. Hence ΔD = ΔWfuel/

(E

fw/  ̇ (Vh - W)) or

ΔT = Δ Wfuel/

(E 

f w ̇

)

. For a segment that is a source of range credit, the range increment is set to zero

and the distance D is added to Dother of the destination segment. For the destination segment, the range

contribution remains fixed at the input value, but the time and hence fuel burned are calculated from

(dist - Dother). It is necessary to separately accumulate Dother from earlier segments and Dother from

later segments; Dother from later segments are estimated initially from the last iteration. At the end of

the mission, the times and fuel burned are recalculated for all range credit destination segments.

The segment time, distance, and fuel burned are evaluated by integrating over the segment duration.

This integration can be performed by using the horizontal velocity, climb velocity, and fuel flow obtained

for the flight state with the gross weight and altitude at the start of the segment, or at the middle of the

segment, or the average of the segment start and segment end values (trapezoidal integration). The gross

weight at the segment middle equals the gross weight at the segment start, less half the segment fuel

burned (obtained from the previous mission iteration). The gross weight at the segment end equals the

gross weight at the segment start, less the segment fuel burned. With trapezoidal integration, for the

output the flight state is finally evaluated at the segment middle.
Operation 25

Table 4-1. Mission segment calculations.

Segment kind Time T Distance D Range dR

taxi time 0 D

distance D/(Vh - Vw) dist D

time time T (Vh - Vw ) D

hold time 0 D

climb (h - hend )/Vc T (Vh - Vw ) D

spiral (h - hend )/Vc 0 D

range credit

source T D 0

destination D/(Vh - Vw) dist - Dother dist

adjusted

distance T + dD/ (Vh - Vw ) D + dD = D + fΔD Dnew

time T + dT = T + fΔT D + dT (Vh - Vw ) Dnew

hold T + dT = T + fΔT 0 Dnew

The mission endurance (block time), range, and fuel burned are E = E T, R = E dR, Wburn =

E

dWburn (sum over all non-reserve segments). The reserve fuel from mission segments is Wres =

E

dWburn (sum over all reserve segments). Optionally the reserve fuel is the maximum of that from

mission segments and the fraction fresWburn, or the fraction fresWfuel−cap. The calculated mission fuel

is then Wfuel = Wburn + Wres.

4–3 Takeoff Distance

The takeoff distance can be calculated, either as ground run plus climb to clear an obstacle or

accelerate-stop distance in case of engine failure. The obstacle height ho is typically 35 ft for commercial

transport aircraft, or 50 ft for military aircraft and general aviation. This calculation allows determination

of the balanced field length: engine failure at critical speed, such that the distance to clear the obstacle

equals the distance to stop. Landing and VTOL takeoff calculations are not implemented, as these are

best solved as an optimal control problem.

The takeoff distance consists of a ground run, from zero ground speed to liftoff speed VLO, perhaps

including engine failure at speed VEF; then rotation, transition, and climb; or decelerate to stop. Figure

4-1 describes the elements of the takeoff distance and the accelerate-stop distance, with the associated

speeds. The ground is at angle γG relative to the horizontal (inertial axes), with γG positive for takeoff up

hill. The takeoff profile is defined in terms of ground speed or climb speed, input as calibrated airspeed

(CAS). The aircraft speed relative to the air is obtained from the ground speed, wind, and ground slope.

The aircraft acceleration as a function of ground speed is integrated to obtain the ground distance, as

well as the time, height, and fuel burned. Usually the speed increases from the start to liftoff (or engine

failure), but the calculated acceleration depends on the flight-state specification. The analysis checks

for consistency of the input velocity and the calculated acceleration (on the ground) and for consistency

of the input height and input or calculated climb angle (during climb).

The takeoff profile consists of a set of mission segments. The first segment is at the start of the

takeoff, V = 0. Subsequent segments correspond to the ends of the integration intervals. The last

segment has the aircraft at the required obstacle height, or stopped on the ground. The mission can
26 Operation

consist of just one takeoff; more than one takeoff; or both takeoff and non-takeoff segments. Takeoff

segments contribute to the mission fuel burned, but do not contribute to the mission time, distance, or

range. The takeoff-distance calculation is performed for a set of adjacent segments, the first segment

specified as the takeoff start, and the last segment identified as before a non-takeoff segment or before

another takeoff start. The takeoff distance is calculated if a liftoff segment (with VLO) is specified;

otherwise the accelerate-stop distance is calculated. Table 4-2 summarizes the mission segments for

takeoff calculations. There can be only one liftoff, engine failure, rotation, and transition segment (or

none). The engine-failure segment must occur before the liftoff segment, and rotation and transition

segments must be after liftoff. All ground-run segments must be before liftoff, and all climb segments

must be after liftoff. Takeoff segments (except start, rotation, and transition) can be split, in terms

of height for climb and velocity for other segments. Splitting the takeoff or engine-failure segment

produces additional ground-run segments. Separately defining multiple ground run, climb, or brake

segments allows configuration variation during the takeoff.

Table 4-2. Mission segments for takeoff calculation.

Takeoff distance Accelerate-stop distance

start V = 0 start V = 0

ground run V ground run V

engine failure VEF engine failure VEF

ground run V brake V

liftoff VLO brake V = 0

rotation VR

transition VTR

climb, to h VCL

climb, to ho VCL

Each takeoff segment requires that the flight state specify the appropriate configuration, trim option,

and maximum effort. In particular, the number of inoperative engines for a segment is part of the flightstate specification, regardless of whether or not an engine-failure segment is defined. The engine-failure

segment (if present) serves to implement a delay in decision after failure: for a time tl after engine

failure, the engine rating, power fraction, and friction of the engine-failure segment are used (so the

engine-failure segment corresponds to conditions before failure). The number of inoperative engines

specified must be consistent with the presence of the engine-failure segment. The takeoff is assumed

to occur at fixed altitude (so the maximum-effort variable can not be altitude). The flight-state velocity

specification is superseded by the ground or climb speed input for the takeoff segment. The flight-state

height above ground-level specifcation is superseded by the height input for the takeoff segment.

The ground distance, time, height, and fuel burned are calculated for each takeoff segment. The

takeoff distance or accelerate-stop distance is the sum of the ground distance of all segments. Takeoff

segments do not contribute to mission time, distance, or range.

4-3.1 Ground Run

The takeoff starts at zero ground speed and accelerates to liftoff ground speed VLO (input as CAS).

Possibly an engine failure speed VEF < VLO is specified. Start, liftoff, and engine-failure segments

designate events, but otherwise are analyzed as ground-run segments. The decision speed Vl is tl

seconds after engine failure (typically ti = 1 to 2 sec). Up to ti after engine failure, conditions of
Operation 27

horizpntai

ground yG ground slope

Yin ertial

zinertial

obstacle

h o

y relative ground

ground run rotation

hTR

distance sGsRsTR s CL

ground V=0 VEF V1 VR

VLO VTR VCL

or climb o^ Q ^. ^^ O o^ y . 6 speed ^} ^ ^+^s^

wo

o*

^`^^

^6

G

accelerate stop

distance sAs S

ground V=0

speed

VEF V1=0

Figure 4-1. Takeoff distance and accelerate-stop distance elements.

the engine-failure segment are used (so the engine-failure segment corresponds to conditions before

failure). The aircraft acceleration is obtained from the thrust minus drag ( T — D in airplane notation),

plus a friction force proportional to the weight on the wheels (W — L in airplane notation):

Ma = T—D— μ(W—L) =

E

Fx — μ

E

Fz

from the force components in ground axes (rotated by the ground-slope angle γG from inertial axes).

Table 4-3 gives typical values of the friction coefficient μ. The velocity of the aircraft relative to the

air is obtained from the ground velocity V, wind velocity Vw (assumed parallel to the ground here),

and the ground slope: Vh = (V + Vw) cos γG and Vc = (V + Vw ) sin γG . The takeoff configuration is

specified, including atmosphere, in-ground-effect, gear down, power rating, nacelle tilt, flap setting,

and number of inoperative engines. An appropriate trim option is specified, typically fixed attitude

with longitudinal force trimmed using collective, for a given longitudinal acceleration. Perhaps the net

aircraft yaw moment is trimmed with the pedal. The maximum-effort condition is specified: maximum

longitudinal acceleration (ground axes) for zero power margin. The aircraft acceleration as a function

of ground speed is integrated to obtain the segment time, ground distance, height, and fuel burned:

tG =

J

dt =

J

d

vd v =

J

 av =

E 1

(

2+ai

)

(v2 — v1 )=

E

Δt

seg seg

f f dt f v dv 

= 

f d(v

2

) — 1( 1 1 2 2

v2 + v1

sG = J v dt = J v dv dv 

J a J 2a — E 2 \ 2a2 + 2a1

) 

(v2 — v1) =

E

g ( 2 ) Δt

seg seg
28 Operation

hG =0

wG =

J

wfdt  ̇

(

wf2

2

wf1

)

Δt

seg

Trapezoidal integration is used; each segment corresponds to the end of an integration integral.

Table 4-3. Typical friction coefficient μ.

surface rolling braking

dry and hard 0.03–0.05 0.30–0.50

grass 0.08 0.20

ice 0.02 0.06–0.10

4-3.2 Brake

After engine failure, the aircraft can decelerate to a stop. The operating engines are at idle. Reverse

thrust is not permitted for the accelerate-stop distance calculation. The braking configuration is specified.

Typically no trim option is executed; rather the aircraft has fixed attitude with controls for zero rotor

thrust (such as zero collective and pedal). The aircraft acceleration as a function of ground speed is

integrated, as for ground run.

4-3.3 Rotation

Rotation occurs at speed VR; usually VR = VLO is used. The duration tR is specified; then sR = VRtR,

hR = 0, and wR = w  ̇ ftR are the ground distance, height, and fuel burned. Typically tR = 1 to 3 sec.

4-3.4 Transition

Transition from liftoff to climb is modeled as a constant load factor pull-up to the specified climb

angle γ, at speed VTR. Usually VTR = VLO is used, and typically nTR ∼= 1 . 2. From the load factor nTR =

1 + V

2

TR/gRTR, the flight-path radius is RTR = V

2

TR/(g(nTR — 1)) and the pitch rate is θ

 ̇ = VTR/ RT R.

Then

tTR = γ/θ ̇= γ 

RTR

= γ

VTR

/

VTR g )

(nTR − 1

sTR = RTR sin γ

hTR = RTR(1 — cos γ)

wT R = w ftT R  ̇

are the time, ground distance, height, and fuel burned.

4-3.5 Climb

Climb occurs at angle γ relative to the ground and air speed VCL, from the transition height hTR

to the obstacle height ho (perhaps in several climb segments). The climb configuration is specified,

including atmosphere, in-ground-effect, gear down or retracted, power rating, nacelle tilt, flap setting,

and number of inoperative engines. An appropriate trim option is specifed, typically aircraft force and

moment trimmed with attitude and controls. The climb angle and air speed can be fixed or a maximumeffort condition can be specified. The maximum-effort options are fixed air speed and maximum rate of
Operation 29

climb for zero power margin; or airspeed for best climb rate or best climb angle with maximum rate of

climb for zero power margin. Not implemented is a maximum-effort calculation of maximum flight-path

acceleration for zero power margin and specified climb angle; this calculation would require integration

of the acceleration as a function of flight speed. For the climb segment, the input VCL is the magnitude

of the aircraft velocity relative to the air, and the climb angle relative to the horizontal is θV = γ + γG .

Hence from the maximum-effort calculation, the climb angle relative to the ground is γ = θV − γG and

the ground speed is Vground = VCL cos γ − Vw (from the wind speed Vw). Then

tCL = sCL/Vground

sCL = (h − hlast)/ tan γ

hCL = h

wCL = w ̇ f

tCL

are the time, ground distance, height, and fuel burned.

4–4 Flight State

A flight state is defined for each flight condition (sizing-task design conditions and flight performance analysis) and for each mission segment. The following parameters are required:

a) Speed: flight speed and vertical rate of climb, with the following options:

1)Specify horizontal speed (or forward speed or velocity magnitude), rate

of climb (or climb angle), and sideslip angle.

2) Hover or vertical flight (input vertical rate of climb; climb angle 0 or

±90 degrees).

3) Left or right sideward flight (input velocity and rate of climb; sideslip

angle ±90 degrees).

4) Rearward flight (input velocity and rate of climb; sideslip angle 180

degrees).

b) Aircraft motion.

1)Pitch and roll angles (Aircraft values or flight-state input; initial values

for trim variables, fixed otherwise).

2) Turn, pull-up, or linear acceleration.

c) Altitude: For mission segment, optionally input, or from last mission segment;

climb segment end altitude from next segment.

d) Atmosphere:

1) Standard day at specifed altitude.

2) Standard day plus temperature increment.

3) Standard day and specifed temperature.

4) Input density and temperature.

5) Input density, speed of sound, and viscosity.

e) Height of landing gear above ground level. Landing gear state (extended or

retracted).
30 Operation

f) Aircraft control state: input; or conversion schedule.

g) Aircraft control values (Aircraft values or flight-state input; initial values for trim

variables, fixed otherwise).

h) Aircraft center-of-gravity position (increment or input value).

For each propulsion group, the following parameters are required:

i) Drive-system state.

j) Rotor tip speed for primary rotor:

1)Input.

2) Reference.

3) Conversion schedule or function speed.

4) Default for hover, cruise, maneuver, OEI, or transmission sizing condition.

5) From input CT/σ = t0 — μt1 , or μ, or Mat (where μ is the rotor advance

ratio and Mat is the rotor advancing tip Mach number).

And for each engine group of each propulsion group:

k) Number of inoperative engines.

l) Infrared suppressor state: off (hot exhaust) or on (suppressed exhaust).

m) Engine rating and fraction of rated engine power available.

Aircraft and rotor performance parameters for each flight state:

o) Aircraft drag: forward-flight drag increment, accounting for payload aerodynamics.

p) Rotor performance: induced power factor κ and profile power mean cd .

The aircraft trim state and trim targets are also specified.

The aircraft performance can be analyzed for the specified state, or a maximum-effort performance

can be identified. For the maximum effort, a quantity and variable are specified. The available maximumeffort quantities include:

a) Best endurance: maximum 1/w ̇ .

b) Best range: 99% maximum V/w ̇ (high side); or low side; or 100%.

c) Best climb or descent rate: maximum Vz or 1/P.

d) Best climb or descent angle: maximum Vz/Vh or V/P.

e) Ceiling: maximum altitude.

f) Power limit: zero power margin, min(PavPG — PreqPG) = 0 (minimum over all

propulsion groups).

g) Torque limit: zero torque margin, min(Qlimit — Qreq) = 0 (minimum over all

propulsion groups, engine groups, and rotors; Qlimit expressed as power at reference

rotation speed).

h) Wing stall: zero wing-lift margin, CLmax — CL = 0 (for designated wing).

i) Rotor stall: zero rotor-thrust margin, (CT/σ)max − CT/σ = 0 (for designated rotor,

steady or transient limit).

Here w ̇ is the aircraft fuel flow and P the aircraft power. The available maximum-effort variables include:
Operation 31

a) Horizontal velocity Vh or vertical rate of climb Vz (times an input factor).

b) Aircraft altitude.

c) Aircraft angular rate, θ ̇ (pull-up) or ψ ̇ (turn).

d) Aircraft linear acceleration (airframe, inertial, or ground axes).

If the variable is velocity, first the velocity is found for the specifed maximum effort; then the performance

is evaluated at that velocity times an input factor. For endurance, range, or climb, the slope of the quantity

to be maximized must be zero; hence in all cases the target is zero. The slope of the quantity is evaluated

by first-order backward difference. For the range, first the variable is found such that V/ w ̇ is maximized

(slope zero); then the variable is found such that V/w ̇ equals 99% of that maximum. Two maximum

effort quantity/variable pairs can be specified, and solved in nested iterations. The secant method or

the method of false position is used to solve for the maximum effort. The task of finding maximum

endurance, range, or climb is usually solved using the golden-section method. A tolerance c and a

perturbation Δ are specified.

Given the gross weight and useful load (from the flight condition or mission specification), the

performance is calculated for this flight state. The calculated state information includes weight, speed

and velocity orientation (climb and sideslip), aircraft Euler angles, rotor tip speeds, and aircraft controls.

The aircraft weight statement defines the fixed useful load and operating weight for the design

configuration. For each flight state, the fixed useful load may be different from the design configuration,

because of changes in auxiliary-fuel-tank weight or kit weights or increments in crew or furnishings

weights. Thus the fixed useful-load weight is calculated for the flight state; and from it the useful-load

weight and operating weight. The gross weight, payload weight, and usable-fuel weight (in standard

and auxiliary tanks) completes the weight information for the flight state.

4–5 Environment and Atmosphere

The aerodynamic environment is defined by the speed of sound cs, density ρ, and kinematic viscosity

ν = μ/ ρ of the air (or other fluid). These quantities can be obtained from the standard day (International

Standard Atmosphere), or input directly. The following options are implemented:

a) Input the altitude hgeom and a temperature increment ΔT. Calculate the temperature and pressure for the standard day, add ΔT, and then calculate the density from

the equation of state for a perfect gas. Calculate the speed of sound and viscosity

from the temperature.

b) Input the pressure altitude hgeom and the temperature τ (°F or °C). Calculate the

pressure for the standard day, and then the density from the equation of state for a

perfect gas. Calculate the speed of sound and viscosity from the temperature.

c) Input the density ρ and the temperature τ (°F or °C). Calculate the speed of sound

and viscosity from the temperature.

d) Input the density ρ, sound speed cs , and viscosity μ. Calculate the temperature

from the sound speed.

Here hgeom is the geometric altitude above mean sea level.

The International Standard Atmosphere (ISA) is a model for the variation with altitude of pressure,
32 Operation

temperature, density, and viscosity, published as International Standard ISO 2533 by the International

Organization for Standardization (ISO) (ref. 1). The ISA is intended for use in calculations and design

of flying vehicles, to present the test results of flying vehicles and their components under identical

conditions, and to allow unifcation in the field of development and calibration of instruments. The ISA

is defined up to 80 kilometer geopotential altitude and is identical to the ICAO Standard Atmosphere up

to 32 kilometers.

Dry air is modeled in the ISA as a perfect gas with a mean molecular weight, and hence a gas

constant R, defined by adopted values for sea-level pressure, temperature, and density ( p0 , T0

, ρ0). The

speed of sound at sea level cs0is defined by an adopted value for the ratio of specific heats γ. The variation

of temperature with geopotential altitude is defined by adopted values for vertical temperature gradients

(lapse rates, Lb ) and altitudes ( hb). The variation of pressure with geopotential altitude is further defined

by an adopted value for the standard acceleration of free-fall (g). The variation of dynamic viscosity μ

with temperature is defined by adopted values for Sutherland’s empirical coefficients β and S.

The ISA consists of a series of altitude ranges with constant lapse rate Lb (linear temperature change

with altitude). Thus at altitude h, the standard day temperature is

Tstd = Tb + Lb (h — hb )

for h > hb . The altitude ranges and lapse rates are given in table 4-4. Note that h0 is sea level, and h1 is

the boundary between the troposphere and the stratosphere. This altitude h is the geopotential height,

calculated assuming constant acceleration due to gravity. The geometric height hgeom is calculated using

an inverse square law for gravity. Hence h = rhgeom/(r + hgeom), where r is the nominal radius of

the Earth. The standard-day pressure is obtained from hydrostatic equilibrium ( dp = — ρg dh) and the

equation of state for a perfect gas (p = ρRT, so dp/p = —(g/RT)dh). So in isothermal regions ( Lb = 0)

the standard-day pressure is

pstd = 

e

−(g/RT)(h−hb

)

pb

and in gradient regions (Lb =^ 0)

pstd 

= (

T

/ 

g/RLb

pb \ Tb

where pb is the pressure at hb , obtained from these equations by working up from sea level. Let T0

,

p0, ρ0 , cs0,μ0 be the temperature, pressure, density, sound speed, and viscosity at sea-level standard

conditions. Then the density, sound speed, and viscosity are obtained from

p

1

ρ = ρ0

(

p0

)

T0

( )

1/2

cs = cs0 1

^,0

( T

/

μ= μ0

(

α(T/T0)+1 — α)

where μ0 = βTo

/2

/(T0 + S) and α = T0 /(T0 + S). For the cases using input temperature, T = Tzero + τ.

The density altitude and pressure altitude are calculated for reference. From the density and the standard

day (troposphere only), the density altitude is:

T0 ρ 

1/(g/R|L0 |−1)

hd =

jL0 j 1 — 

(

ρ0 )
Operation 33

From the pressure p = ρRT and the standard day, the pressure altitude is:

^11/(g/RIL0I)

hp =1 −

|L0 |

\

ρ0 T0

T0

The required parameters are given in table 4-5, including the acceleration produced by gravity, g. The

gas constant is R = p0 / ρ0T0, and μ0 is actually obtained from S and β. The ISA is defined in SI units.

Although table 4-5 gives values in both SI and English units, all the calculations for the aerodynamic

environment are performed in SI units. As required, the results are converted to English units using the

exact conversion factors for length and force.

The gravitational acceleration g can have the standard value or an input value.

Table 4-4. Temperatures and vertical temperature gradients.

level base altitude hb lapse rate Lb temperature Tb

km °K/km °K

troposphere -2 -6.5 301.15

0 troposphere 0 -6.5 288.15

1 tropopause 11 0 216.65

2 stratosphere 20 +1.0 216.65

3 stratosphere 32 +2.8 228.65

4 stratopause 47 0 270.65

5 mesosphere 51 -2.8 270.65

6 mesosphere 71 -2.0 214.65

7 mesopause 80 0 196.65

Table 4-5. Constants adopted for calculation of the ISA.

parameter SI units English units

units h

units τ

mper ft

kg per lbm

T0

Tzero

p0

ρ0

cs0

μ0

S

β

γ

m

°C

288.15 °K

273.15 °K

101325.0 N/m2

1.225 kg/m3

340.294 m/sec

1.7894E-5 kg/m-sec

110.4 °K

1.458E-6

1.4

ft

°F

0.3048

0.45359237

518.67 °R

459.67 °R

2116.22 lb/ft2

0.002377 slug/ft3

1116.45 ft/sec

3.7372E-7 slug/ft-sec

g 9.80665 m/sec2

32.17405 ft/sec 2

r 6356766 m 20855531 ft
34 Operation

4–6 References

1) International Organization for Standardization: Standard Atmosphere. ISO 2533-1975(E), May

1975.
Chapter 5

Solution Procedures

The NDARC code performs design and analysis tasks. The design task involves sizing the rotorcraft

to satisfy specified design conditions and missions. The analysis tasks can include off-design mission

performance analysis, flight performance calculation for point operating conditions, and generation of

subsystem or component performance maps. Figure 5-1 illustrates the tasks. The principal tasks (sizing,

mission analysis, and flight performance analysis) are shown in the figure as boxes with dark borders.

Dark arrows show control of subordinate tasks.

The aircraft description (fig. 5-1) consists of all the information, input and derived, that defines

the aircraft. The aircraft consists of a set of components, including fuselage, rotors, wings, tails, and

propulsion. This information can be the result of the sizing task; can come entirely from input, for a

fixed model; or can come from the sizing task in a previous case or previous job. The aircraft description

information is available to all tasks and all solutions (indicated by light arrows).

Missions are defined for the sizing task and for the mission performance analysis. A mission consists

of a specified number of mission segments, for which time, distance, and fuel burn are evaluated. For

specified takeoff fuel weight with adjustable segments, the mission time or distance is adjusted so the

fuel required for the mission (burned plus reserve) equals the takeoff fuel weight. The mission iteration

is on fuel weight.

Flight conditions are specified for the sizing task and for the flight performance analysis.

For flight conditions and mission takeoff, the gross weight can be maximized such that the power

required equals the power available.

A flight state is defined for each mission segment and each flight condition. The aircraft performance

can be analyzed for the specified state, or a maximum-effort performance can be identified. The

maximum effort is specified in terms of a quantity such as best endurance or best range, and a variable

such as speed, rate of climb, or altitude. The aircraft must be trimmed, by solving for the controls

and motion that produce equilibrium in the specified flight state. Different trim-solution definitions are

required for various flight states. Evaluating the rotor hub forces may require solution of the blade-flap

equations of motion.

The sizing task is described in more detail in chapter 3; the flight condition, mission, and flightstate calculations are described in chapter 4; and the solution of the blade-flap equations of motion is

described in chapter 11. The present chapter provides details of the solution procedures implemented

for each iteration of the analysis.

The nested iteration loops involved in the solution process are indicated by the subtitles in the

boxes of Figure 5-1 and illustrated in more detail in Figure 5-2. The flight-state solution involves up
Sizing Task

size iteration

design

conditions design

missions

36 Solution Procedures

DESIGN

Flight Condition

max GW

fixed model or

previous job or

previous case ANALYZE

Airframe

T r n Aerodynamics Map

Aircraft Engine

Description Performance Map

Mission Analysis

Flight

Performance

Analysis

Jr

Mission

adjust & fuel wt iteration

II max takeoff GW

each segment

Flight State

max effort / trim aircraft / flap equations

Figure 5-1. Outline of NDARC tasks.

to three loops. The innermost loop is the solution of the blade-flap equations of motion, needed for an

accurate evaluation of the rotor hub forces. The next loop is the trim solution, which is required for

most flight states. The flight state optionally has one or two maximum-effort iterations. The flight-state

solution is executed for each flight condition and for each mission segment. A flight-condition solution

or any mission-segment solution can optionally maximize the aircraft gross weight. The mission usually

requires an iterative solution, for fuel weight or for adjustable segment time or distance. Thus each flightcondition solution involves up to four nested iterations: maximum gross weight (outer), maximum effort,

trim, and blade motion (inner). Each mission solution involves up to five nested iterations: mission

(outer), and then for each segment maximum gross weight, maximum effort, trim, and blade motion

(inner). Finally, the design task introduces a sizing iteration, which is the outermost loop of the process.
Solution Procedures

Sizing Task

Size Iteration

method: successive

substitution

Flight Conditions

Missions

Mission Analysis

Missions

Flight Performance Analysis

Flight Conditions

37

Flight Condition

Maximum GW

method: secant or false position

Flight State

Mission

Mission Iteration

fuel weight, adjust time/distance

method: successive substitution

Segments

Maximum GW

method: secant or false

position

Flight State

Flight State

Maximum Effort

method: golden section search for maximum

endurance, range, or climb;

otherwise secant or false position

Trim

method: Newton-Raphson

Component Performance Evaluation

Blade Flapping

method: Newton-Raphson

Figure 5-2. Design and analysis tasks, with nested loops and solution methods.
38 Solution Procedures

5–1 Iterative Solution Tasks

5-1.1 Tolerance and Perturbation

For each solution procedure, a tolerance e and a perturbation Δ may be required. Single values are

specified for the task, and then scaled for each element tested or perturbed.

The scaling is based on a reference weight W (design gross weight, or derived from aircraft

CT/σ = 0. 07), a reference length L (fuselage length, rotor radius, or wing span), and a reference power

P (aircraft installed power, or derived from P = W W/2ρA). Then the force reference is F = W,

the moment reference is M = WL/10, and the angle reference is A = 1 deg. The velocity reference is

V = 400 knots. The angular velocity reference is Ω = V/L (in deg/sec). The coefficient reference is

C = 0. 6 for wings and C = 0. 1 for rotors. Altitude scale is H = 10000 ft. Acceleration scale is G = g

(acceleration due to gravity). The range scale is X = 100 nm. These scaling variables are referred to in

the subsections that follow, and in tables 5-1, 5-3, and 5-4.

5-1.2 Size Aircraft

The sizing task determines the dimensions, power, and weight of a rotorcraft that can perform a

specified set of design conditions and missions. The aircraft size is characterized by parameters such as

design gross weight, weight empty, rotor radius, and engine power available. The relationships between

dimensions, power, and weight generally require an iterative solution. From the design flight conditions

and missions, the task can determine the total engine power or the rotor radius (or both power and radius

can be fixed), as well as the design gross weight, maximum takeoff weight, drive-system torque limit,

and fuel-tank capacity. For each propulsion group, the engine power or the rotor radius can be sized.

A successive substitution method is used for the sizing iteration, with an input tolerance e. Relaxation is applied to Peng or R, WD , WMTO, PDSlimit, Wfuel—cap, and Tdesign. Two successive substitution

loops are used. The outer loop is an iteration on performance: engine power or rotor radius, for each

propulsion group. The inner loop is an iteration on parameters: WD , WMTO, PDSlimit, Wfuel—cap, and

Tdesign . Either loop can be absent, depending on the definition of the sizing task. Convergence is tested

in terms of these parameters and the aircraft weight empty WE . The tolerance is 0. 1Pe for engine power

and drive-system limit; 0.01We for gross weight, maximum takeoff weight, fuel weight, and design rotor

thrust; and 0. 1Le for rotor radius.

5-1.3 Mission

Missions consist of a specified number of segments, for which time, distance, and fuel burn are

evaluated. For calculated mission fuel weight, the fuel weight at takeoff is adjusted to equal the fuel

required for the mission (burned plus reserve). For specified takeoff fuel weight with adjustable segments,

the mission time or distance is adjusted so the fuel required for the mission (burned plus reserve) equals

the takeoff fuel weight. The mission iteration is thus on fuel weight. Range credit segments can also

require an iteration.

A successive substitution method is used if an iteration is required, with a tolerance e specified.

The principal iteration variable is takeoff fuel weight, for which the tolerance is 0.01 We. For calculated

mission fuel weight, the relaxation is applied to the mission fuel value used to update the takeoff fuel

weight. For specified takeoff fuel weight, the relation is applied to the fuel weight increment used to

adjust the mission segments. The tolerance for the distance flown in range credit segments is Xe. The
Solution Procedures 39

relaxation is applied to the distance flown in the destination segments for range credit.

5-1.4 Maximum Gross Weight

Flight conditions are specified for the sizing task and for the flight performance analysis. Mission

takeoff conditions are specified for the sizing task and for the mission analysis. Optionally for flight

conditions and mission takeoff, the gross weight can be maximized, such that the power required equals

the power available, min(PavPG −PreqPG) = 0 (zero power margin, minimum overall propulsion groups);

or such that the power required equals an input power, min((d + fPavPG) — PreqPG) = 0 (minimum over

all propulsion groups, with d an input power and f an input factor; this convention allows the power to

be input directly, f = 0, or scaled with power available).

The secant method or the method of false position is used to solve for the maximum gross weight.

A tolerance a and a perturbation Δ are specified. The variable is gross weight, with initial increment

of WΔ, and tolerance of 0.01Wa. Note that the convergence test is applied to the magnitude of the

gross-weight increment.

5-1.5 Maximum Effort

The aircraft performance can be analyzed for the specified state or a maximum-effort performance

can be identified. The secant method or the method of false position is used to solve for the maximum

effort. The task of finding maximum endurance, range, or climb is usually solved using the goldensection method. A tolerance a and a perturbation Δ are specified.

A quantity and variable are specified for the maximum-effort calculation. Tables 5-1 and 5-2

summarize the available choices, with the tolerance and initial increment used for the variables. Note

that the convergence test is applied to the magnitude of the variable increment. Optionally two quantity/

variable pairs can be specified, solved in nested iterations. The two variables must be unique. The two

variables can maximize the same quantity (endurance, range, or climb). If the variable is velocity, first

the velocity is found for the specified maximum effort; the performance is then evaluated at that velocity

times an input factor. For endurance, range, or climb, the slope of the quantity to be maximized must be

zero; hence in all cases the target is zero. The slope of the quantity is evaluated by first-order backward

difference. For the range, first the variable is found such that V/w ̇ is maximized (slope zero), and then

the variable is found such that V/ w ̇ equals 99% of that maximum; for the latter the variable perturbation

is increased by a factor of 4 to ensure that the solution is found on the correct side of the maximum.

5-1.6 Trim

The aircraft trim operation solves for the controls and motion that produce equilibrium in the

specified flight state. A Newton—Raphson method is used for trim. The derivative matrix is obtained

by numerical perturbation. A tolerance a and a perturbation Δ are specified.

Different trim-solution definitions are required for various flight states. Therefore one or more

trim states are defined for the analysis, and the appropriate trim state selected for each flight state of

a performance condition or mission segment. For each trim state, the trim quantities, trim variables,

and targets are specified. Tables 5-3 and 5-4 summarize the available choices, with the tolerances and

perturbations used.
40 Solution Procedures

Table 5-1. Maximum-effort solution.

maximum effort variable initial increment tolerance

horizontal velocity Vh 0.1VΔ 0. 1Ve

vertical rate of climb Vz 0.1VΔ 0. 1Ve

aircraft altitude HΔ He

aircraft angular rate θ ̇ (pullup), ψ ̇ (turn) ΩΔ Ωe

aircraft linear acceleration ax , ay , az GΔ Ge

Table 5-2. Maximum-effort solution.

maximum effort quantity

best endurance maximum 1/w ̇

best range 99% maximum V/w ̇ high or low side, or 100%

best climb or descent rate maximum Vz or 1/P

best climb or descent angle maximum Vz /V or V/P

ceiling maximum altitude

power limit power margin, min(PavPG — PreqPG) = 0 over all propulsion groups

torque limit torque margin, min(Qlimit — Qreq) = 0 over all limits

wing stall lift margin, CLmax — CL = 0 for designated wing

rotor stall thrust margin, (CT/σ)max — CT/σ = 0 for designated rotor

Table 5-3. Trim solution.

trim quantity target tolerance

aircraft total force x, y, z components 0 Fe

aircraft total moment x, y, z components 0 Me

aircraft load factor x, y, z components Flight State e

propulsion group power Flight State Pe

power margin PavPG — PreqPG Flight State Pe

rotor force lift, vertical, propulsive Flight State, component schedule Fe

rotor thrust CT/σ Flight State Ce

rotor thrust margin (CT/σ)max — CT/σ Flight State Ce

rotor flapping βc

, βs Flight State Ae

rotor hub moment x (roll), y (pitch) Flight State Me

rotor torque Flight State Me

wing force lift Flight State, component schedule Fe

wing lift coefficient CL Flight State Ce

wing lift margin CLmax — CL Flight State Ce

tail force lift Flight State Fe
Solution Procedures 41

Table 5-4. Trim solution.

trim variable perturbation

aircraft orientation θ (pitch), φ (roll) 100AΔ

aircraft velocity Vh (horizontal velocity) VΔ

aircraft velocity Vz (vertical velocity) VΔ

aircraft velocity β (sideslip) 100AΔ

aircraft angular rate θ ̇ (pullup), ψ ̇ (turn) ΩΔ

aircraft control angle 100AΔ

5-1.7 Rotor-Flap Equations

Evaluating the rotor hub forces may require solution of the flap equations E(v) = 0. For tippath plane command, the thrust and flapping are known, so v = (θ0.75 θc θs)T . For no-feathering plane

command, the thrust and cyclic pitch are known, so v = (θ0.75 βc βs)T . A Newton–Raphson solution

method is used: from E(vn+1) = E(vn ) + (dE/dv)(vn+1 - vn ) = 0, the iterative solution is

vn+1 = vn - C E(vn )

where C = f (dE/dv)

-1 , including the relaxation factor f. The derivative matrix for axial flow can be

used. Alternatively, the derivative matrix dE/ dv can be obtained by numerical perturbation. Convergence

of the Newton–Raphson iteration is tested in terms of I EI < E for each equation, where E is an input

tolerance.

5–2 Theory

The analysis uses several methods to solve nonlinear algebraic equations. Such equations may be

written in two forms:

a) fixed point, x = G(x)

b) zero point, f (x) = 0

where x, G, and f are vectors. The analysis provides operations that implement the function G or f.

Solution procedures appropriate for the zero-point form can be applied to equations in fixed-point form

by defining f (x) = x - G(x). In this context, f can be considered the iteration error.

Efficient and convergent methods are required to find the solution x = α of these equations. Note

that f

'

(α) = 0 or G'(α) = 1 means that α is a higher-order root. For nonlinear problems, the method will

be iterative: xn+1 = F(xn ). The operation F depends on the solution method. The solution error is:

^n+1 = α - xn+1 = F(α) - F(xn ) = (α - xn )F'(ξn) - ^nF'(α)

Thus the iteration will converge if F is not too sensitive to errors in x: I F'(α) I < 1 for scalar x. For

x a vector, the criterion is that all the eigenvalues of the derivative matrix ∂F/∂x have magnitude less

than one. The equations in this section are generally written for scalar x; the extension to vector x

is straightforward. Convergence is linear for F' nonzero, quadratic for F' = 0. Iterative methods

have a relaxation factor (and other parameters) to improve convergence, and a tolerance to measure

convergence.
42 Solution Procedures

The following subsections describe the solution methods used for the various iterations, as shown

in Figure 5-2.

5-2.1 Successive-Substitution Method

The successive-substitution method (with relaxation) is an example of a fixed point solution. A

direct iteration is simply xn+1 = G(xn ), but jG'j > 1 for many practical problems. A relaxed iteration

uses F = (1 - λ)x + λG:

xn+1 = (1- λ)xn + λG(xn) = xn - λf (xn)

with relaxation factor λ. The convergence criterion is then

jF'(α) j = j1 - λ + λG' j < 1

so a value of λ can be found to ensure convergence for any finite G'. Specifically, the iteration converges

if the magnitude of λ is less than the magnitude of 2/(1 - G') = 2/f' (and λ has the same sign as

1- G' = f'). Quadratic convergence (F' = 0) is obtained with λ = 1/(1 - G') = 1/f'. Over-relaxation

(λ > 1) can be used if jG'j < 1. Since the correct solution x = α is not known, convergence must be

tested by comparing the values of two successive iterations:

error = 11xn+1 - xn 11 < tolerance

where the error is some norm of the difference between iterations (typically absolute value for scalar x).

Note that the effect of the relaxation factor is to reduce the difference between iterations:

xn+1 - xn = λ(G(xn ) - xn

)

Hence the convergence test is applied to (xn+1 - xn )/λ in order to maintain the definition of tolerance

independent of relaxation. The process for the successive-substitution method is shown in Figure 5-3.

5-2.2 Newton–Raphson Method

The Newton–Raphson method (with relaxation and identification) is an example of a zero-point

solution. The Taylor series expansion of f (x) = 0 leads to the iteration operator F = x - f /f':

xn+1 = xn - [f'(xn )]-1

f(xn )

which gives quadratic convergence. The behavior of this iteration depends on the accuracy of the

derivative f'. Here it is assumed that the analysis can evaluate directly f, but not f'. It is necessary to

evaluate f' by numerical perturbation of f, and for efficiency the derivatives may not be evaluated for

each xn . These approximations compromise the convergence of the method, so a relaxation factor λ is

introduced to compensate. Hence a modified Newton–Raphson iteration is used, F = x - Cf :

xn+1 = xn - Cf(xn) = xn- λD-1f(xn)

where the derivative matrix D is an estimate of f'. The convergence criterion is then

jF'(α) j = j1 - Cf' j = j1 - λD

-1 f' j < 1

since f (α) = 0. The iteration converges if the magnitude of λ is less than the magnitude of 2D/ f

'

(and λ

has the same sign as D/ f'). Quadratic convergence is obtained with λ = D/ f

'

(which would require λ
D =

f

···

J

=

f

···

∂xi

f (xi + δxi) − f (xi

)

δxi

Solution Procedures 43

successive substitution iteration

save: xold = x

evaluate x

relax: x = λx + (1 − λ)xold

test convergence: error = 11 x − xold 11 ≤ λtolerance × weight

Figure 5-3. Outline of successive-substitution method.

to change during the iteration however). The Newton–Raphson method ideally uses the local derivative

in the gain factor, C = 1/ f', so has quadratic convergence:

F'(α) = 

ff

22

' = 0

since f (α) = 0 (if f' =^ 0 and f

'' is finite; if f' = 0, then there is a multiple root, F' = 1/2 , and the

convergence is only linear). A relaxation factor is still useful, since the convergence is only quadratic

sufficiently close to the solution. A Newton–Raphson method has good convergence when x is sufficiently close to the solution, but frequently has difficulty converging elsewhere. Hence the initial

estimate x0 that starts the iteration is an important parameter affecting convergence. Convergence of the

solution for x may be tested in terms of the required value (zero) for f:

error = 1f 11 ≤ tolerance

where the error is some norm of f (typically absolute value for scalar f).

The derivative matrix D is obtained by an identification process. The perturbation identification

can be performed at the beginning of the iteration, and optionally every MPID iterations thereafter. The

derivative matrix is calculated from a one-step finite-difference expression (first order). Each element

xi of the vector x is perturbed, one at a time, giving the i-th column of D:

Alternatively, a two-step finite-difference expression (second order) can be used:

f J f J

· · ·

∂f

· · ·

f (xi + δxi) − f (xi

 − δxi

)

D= · · · = · · ·

∂xi 2δxi

With this procedure, the accuracy of D (hence convergence) can be affected by both the magnitude and

sign of the perturbation (only the magnitude for a two-step difference).

The process for the Newton–Raphson method is shown in Figure 5-4. A problem specified as

h(x) = htarget becomes a zero-point problem with f = h − htarget. A successive-substitution problem,

x = G(x), becomes a zero-point problem with f = x − G. At the beginning of the solution, x has an

initial value. The perturbation identification can optionally never be performed (so an input matrix is

required), performed at the beginning of the iteration, or performed at the beginning and every MPID

iterations thereafter.
44 Solution Procedures

initialize

evaluate h

test convergence: error = 1 hj - htargetj 1 < tolerance x weightj

initialize derivative matrix D to input matrix

calculate gain matrix: C = λD-1

iteration

identify derivative matrix

optional perturbation identification

perturb each element of x: δxi = Δ x weighti

evaluate h

calculate D

calculate gain matrix: C = λD-1

increment solution: δx = -C(h - htarget)

evaluate h

test convergence: error = 1 hj

 - htargetj 1 < tolerance x weightj

Figure 5-4. Outline of Newton-Raphson method.

5-2.3 Secant Method

The secant method (with relaxation) is developed from the Newton–Raphson method. The modified

Newton–Raphson iteration is:

xn+1 = xn - C f (xn ) = xn - λD

-1

 f (xn)

where the derivative matrix D is an estimate of f'. In the secant method, the derivative of f is evaluated

numerically at each step:

f(xn

) - f(xn-1)

f'(xn ) =

xn - xn- 1

It can be shown that then the error reduces during the iteration according to:

En

+1 1 - 

1 f''/2f' 1 1 En 1 1En-1 1 - 1f''/2 f'1

.62 1En 11.62

which is slower than the quadratic convergence of the Newton–Raphson method ( E

2

n), but still better than

linear convergence. In practical problems, whether the iteration converges at all is often more important

than the rate of convergence. Limiting the maximum amplitude of the derivative estimate may also be

appropriate. Note that with f = x - G(x), the derivative f

'

is dimensionless, so a universal limit (say

maximum 1

f' 1 = 0.3) can be specified. A limit on the maximum increment of x (as a fraction of the x

value) can also be imposed. The process for the secant method is shown in Figure 5-5.

5-2.4 Method of False Position

The method of false position is a derivative of the secant method, based on calculating the derivative

with values that bracket the solution. The iteration starts with values of x0 and x1 such that f (x0 ) and

f (x1 ) have opposite signs. Then the derivative f' and new estimate xn+1 are

f'(xn ) - f 

(xn) - f (xk)

xn - xk

xn+1 = xn - λD

-1

 f (xn)
Solution Procedures 45

initialize

evaluate f0 at v0 , f1 at v1 = v0 +Δv , f2 at v2 = v1 +Δv

iteration

calculate derivative f'

secant: from f0 and f1

false position: from f0 ,

calculate gain: C = λ/f'

increment solution: δv = — Cf

shift: f2 = f1 , f1 = f0

evaluate f

test convergence

and f1 or f2 (opposite sign from f1 )

Figure 5-5. Outline of secant method or method of false position.

initialize

evaluate f0 at x0 , f1 at x1 = x0 +Δx , f2 at x2 = x1 +Δx

bracket maximum: while not f1 > f0 , f2

if f2 > f0 , then x3 = x2 +(x2 — x1 ) ; 1,2,3 —> 0,1,2

if f0 > f2 , then x3 = x0 — (x1 — x0 ) ; 3, 0,1 —> 0,1, 2

iteration (search)

if x2 — x1 >x1 — x0 , then x3 = x1 + W(x2 — x1 )

if f3 < f1 , then 0,1,3 —> 0,1,2

if f3 > f1 , then 1,3,2 —> 0,1,2

if x1 — x0 > x2 — x1 , then x3 = x1 — W (x1 — x0 )

if f3 < f1 , then 3,1,2 —> 0,1,2

if f3 > f1 , then 0, 3,1 —> 0,1, 2

test convergence

Figure 5-6. Outline of golden-section search.

using k = n — 1 or k = n — 2 such that f (xn) and f (xk) have opposite signs. The convergence is

slower (roughly linear) than for the secant method, but by keeping the solution bracketed convergence

is guaranteed. The process for the method of false position is shown in Figure 5-5.

5-2.5 Golden-Section Search

The golden-section search method can be used to find the solution x that maximizes f (x). The

problem of maximizing f (x) can be attacked by applying the secant method or method of false position to

the derivative f

'

(x) = 0, but that approach is often not satisfactory as it depends on numerical evaluation

of the second derivative. The golden-section search method begins with a set of three values x0 < x 1 < x2

and the corresponding functions f0 , f1 , f2. The x value is incremented until the maximum is bracketed,

f1 > f0 , f2. Then a new value x3 is selected in the interval x0 to x2 ; f3 evaluated; and the new set of

x0 < x1 < x2 determined such that the maximum is still bracketed. The new value x3 is a fraction

W = (3 — v

f

5) /2 = 0. 38197 from x1 into the largest segment. The process for the golden-section search

is shown in Figure 5-6.
46 Solution Procedures
Chapter 6

Cost

Costs are estimated using statistical models based on historical aircraft price and maintenance cost

data, with appropriate factors to account for technology impact and inflation. The aircraft flyaway

cost (CAC, in $) consists of airframe, mission equipment package (MEP), and flight-control-electronics

(FCE) costs. The direct operating cost plus interest (DOC + I, in cents per available seat mile (ASM))

is the sum of maintenance cost (Cmaint , in $ per flight hour), flight crew salary and expenses, fuel and

oil cost, depreciation, insurance cost, and finance cost. Inflation factors can be input, or internal factors

used. Table 6-1 gives the internal inflation factors for DoD (ref. 1) and consumer price index (CPI,

ref. 2). For years beyond the data in the table, optionally the inflation factor is extrapolated based on the

last yearly increase.

6–1 CTM Rotorcraft Cost Model

The CTM rotorcraft cost model (refs. 3–5) gives an estimate of aircraft flyaway cost and direct

operating cost plus interest. The basic statistical relations for airframe purchase price and maintenance

cost per hour are:

cAF = 739. 66 KETKENKLGKR WAF

619 /

P WAF 0.5887N

0 .1465

l I blade + ccomp

cmaint = 0. 49885 WE3746P

0 .4635

with WAF = WE − WMEP − WFCE. The term ccomp = LWcomp accounts for additional costs for composite

construction; Wcomp is the composite structure weight, obtained as an input fraction of the component

weight, with separate fractions for body, tail, pylon, and wing weight. The configuration factors are:

KET = 1.0 for turbine aircraft

KEN = 1.0 for multi-engine aircraft

KLG = 1.0 for retractable landing gear

KR = 1.0 for single main rotor

0.557 for piston aircraft

0.736 for single-engine aircraft

0.884 for fixed landing gear

1.057 for twin main rotors, 1.117 for four main rotors

The MEP and FCE costs are obtained from input cost-per-weight factors: CMEP = rMEP WMEP and

CFCE = rFCEWFCE. These cost equations are based on 1994 dollars and current technology levels.

Including an inflation factor Fi and technology factors χ gives the unit flyaway cost CAC and maintenance

cost per flight hour Cmaint:

CAC = Fi (χAFcAF + CMEP + CFCE)

Cmaint = Fi χmaint cmaint

The statistical equation for cAF predicts the price of 123 out of 128 rotorcraft within 20% (figs. 6-1 and 6-

2). These equations also serve to estimate turboprop airliner flyaway costs by setting Nrotor = Nblade = 1
48 Cost

and using additional factors 0 .8754 (pressurized) or 0 . 7646 (unpressurized). The other terms in the

operating cost are:

Cfuel = G(Wfuel/ρfuel)Ndep

Ccrew = 3. 19 KcrewW

0 .4

MTOB

Cins = 0. 0056 CAC

Cdep = CAC 1DS

(1− V)

1+ S 2L +1 i

Cfin = CAC D 4 100

The fuel burn Wfuel, block time Tmiss, and block range Rmiss are obtained for a designated mission. The

number of departures per year is Ndep = B/Tmiss. The flight time per trip is Ttrip = Tmiss — TNF. The

flight hours per year are TF = Ttrip Ndep. The yearly operating cost COP and DOC + I are then:

COP = TFCmaint + Cfuel + Ccrew + Cdep + Cins + Cfin

DOC + I = 100 COP/ASM

where the available seat miles per year are ASM = 1 . 1508NpassRmissNdep. Parameters are defined in

table 6-2, including units as used in these equations.
Cost 49

Table 6-1. DoD and CPI inflation factors.

inflation factors inflation factors

year DoD CPI year DoD CPI

1911

1912

1913 6.68

1914 6.75

1915 6.82

1916 7.35

1917 8.64

1918 10.19

1919 11.67

1920 13.50

1921 12.08

1922 11.34

1923 11.54

1924 11.54

1925 11.81

1926 11.94

1927 11.74

1928 11.54

1929 11.54

1930 11.27

1931 10.26

1932 9.24

1933 8.77

1934 9.04

1935 9.24

1936 9.38

1937 9.72

1938 9.51

1939 9.38

1940 9.45

1941 9.92

1942 11.00

1943 11.67

1944 11.88

1945 12.15

1946 13.16

1947 15.05

1948 16.26

1949 16.06

1950 15.37 16.26

1951 17.02 17.54

1952 16.26 17.88

1953 16.31 18.02

1954 16.02 18.15

1955 17.29 18.08

1956 17.58 18.35

1957 18.17 18.96

1958 18.02 19.50

1959 18.11 19.64

1960 18.34 19.97

1961 18.31 20.18

1962 18.84 20.38

1963 19.09 20.65

1964 19.76 20.92

1965 20.36 21.26

1966 21.97 21.86

1967 22.77 22.54

1968 24.03 23.48

1969 25.10 24.76

1970 25.76 26.18

1971 27.24 27.33

1972 28.98 28.21

1973 31.34 29.96

1974 34.13 33.27

1975 38.10 36.30

1976 42.14 38.39

1977 43.75 40.89

1978 47.82 43.99

1979 53.02 48.99

1980 58.52 55.60

1981 63.80 61.34

1982 68.35 65.11

1983 71.92 67.21

1984 74.57 70.11

1985 76.86 72.60

1986 79.19 73.95

1987 81.87 76.65

1988 85.01 79.82

1989 88.19 83.67

1990 91.30 88.19

inflation factors

year DoD CPI

1991 93.99 91.90

1992 96.21 94.67

1993 98.16 97.50

1994 100.00 100.00

1995 101.73 102.83

1996 103.25 105.87

1997 104.41 108.30

1998 105.47 109.99

1999 106.88 112.42

2000 108.46 116.19

2001 109.99 119.50

2002 111.63 121.39

2003 113.88 124.16

2004 116.81 127.46

2005 120.04 131.78

2006 123.05 136.03

2007 125.67 139.91

2008 127.67 145.28

2009* 129.37

2010* 131.22

2011* 133.41

2012* 135.77

2013* 138.26

2014* 140.72

2015* 143.25

* projected
50 Cost

Table 6-2. Cost model parameters.

parameter definition units

WE weight empty lb

WMTO maximum takeoff weight lb

Nblade number of blades per rotor

P rated takeoff power (all engines) hp

WMEP fixed useful load weight, mission equipment package lb or kg

WFCE fixed useful load weight, flight control electronics lb or kg

rMEP cost factor, mission equipment package $/lb or $/kg

rFCE cost factor, flight control electronics $/lb or $/kg

L additional labor rate for composite construction $/lb or $/kg

Wfuel mission fuel burned lb or kg

Tmiss mission time hr

Rmiss mission range nm

G fuel cost $/gallon or $/liter

B available block hours hr

S spares per aircraft (fraction purchase price)

D depreciation period yr

V residual value (fraction)

L loan period yr

i interest rate %

TNF non-flight time per trip hr

Npass number of passengers

ρfuel fuel density (weight per volume) lb/gal or kg/liter

6–2 References

1) National Defense Budget Estimates for FY 1998/2010. Office of the Under Secretary of Defense

(Comptroller), March 1997/2008. Department of Defense deflators, for Total Obligational Authority

(TOA), Procurement.

2) Consumer Price Index. U.S. Department Of Labor, Bureau of Labor Statistics, 2009. All Urban

Consumers (CPI-U), U.S. city average, all items.

3) Harris, F.D.; and Scully, M.P.: Rotorcraft Cost Too Much. Journal of the American Helicopter

Society, vol. 43, no. 1, January 1998.

4) Harris, F.D.: An Economic Model of U.S. Airline Operating Expenses. NASA CR 2005-213476,

December 2005.

5) Coy, J.J.: Cost Analysis for Large Civil Transport Rotorcraft. American Helicopter Society Vertical

Lift Aircraft Design Conference, San Francisco, California, January 2006.
Cost 51

1000.

900.

800.

0.

0 100. 200. 300. 400. 500. 600. 700. 800. 900. 1000.

actual base price (1994$/lb)

700.

o, 600.

500.

400.

U

U

300.

200.

100.

Figure 6-1. Statistical estimation of rotorcraft flyaway cost ($/lb).
52 Cost

100.0

10.0

o,

o,

c^

w

UM

U

1.0

a

0.1

0.1 1.0 10.0 100.0

actual base price (1994 $M)

Figure 6-2. Statistical estimation of rotorcraft flyaway cost ($M).
Chapter 7

Aircraft

The aircraft consists of a set of components, including rotors, wings, tails, fuselage, and propulsion. For each component, attributes such as performance, drag, and weight can be calculated. The

aircraft attributes are obtained from the sum of the component attributes. Description and analysis of

conventional rotorcraft configurations is facilitated, while retaining the capability to model novel and advanced concepts. Specific rotorcraft configurations considered include: single-main-rotor and tail-rotor

helicopter; tandem helicopter; and coaxial helicopter.

The following components form the aircraft:

a) Systems: The systems component contains weight information (fixed useful load, vibration, contingency, and systems and equipment) for the aircraft.

b) Fuselage: There is one fuselage for the aircraft.

c) Landing Gear: There is one landing gear for the aircraft.

d) Rotors: The aircraft can have one or more rotors, or no rotors. In addition to main rotors, the

component can model tail rotors, propellers, proprotors, and ducted fans.

e) Forces: The force component is a simple model for a lift, propulsion, or control subsystem.

f) Wings: The aircraft can have one or more wings, or no wings.

g) Tails: The aircraft can have one or more horizontal or vertical tail surfaces, or no tails.

h) Fuel Tank: There is one fuel tank component for the aircraft. There can be one or more sizes of

auxiliary fuel tanks.

i) Propulsion Groups: There are one or more propulsion groups. Each propulsion group is a set of

components (rotors) and engine groups, connected by a drive system. The engine model describes a

particular engine, used in one or more engine groups. The components define the power required. The

engine groups define the power available.

j) Engine Groups: An engine group consists of one or more engines of a specific type. For each engine

type an engine model is defined.

7–1 Disk Loading and Wing Loading

The aircraft disk loading is the ratio of the design gross weight and a reference rotor area: DL =

WD/Arse . The reference area is a sum of specified fractions of the rotor areas, Arse = E fAA (typically

the projected area of the lifting rotors). The disk loading of a rotor is the ratio of a specified fraction of
54 Aircraft

the design gross weight and the rotor area:

TfWWD fW WD

(DL) rotor =

A

=

A =A/Aref Aref

where probably Erotor fW = 1, and the lifting rotors are all rotors not designated antitorque or auxiliarythrust. If there are N lifting rotors of the same area, with no overlap, then fA = 1, Aref = NA,

fW = A/Aref = 1/N, and (DL) rotor = DL. For rotors designated antitorque or auxiliary-thrust, the disk

loading is calculated from the design rotor thrust: (DL) rotor = Tdesign/A.

For coaxial rotors, the default reference area is the area of one rotor: fA = 1/2 , Aref = A , fW = 1/2,

and (DL) rotor = 1/2DL. For tandem rotors, the default reference area is the projected area: Aref =

(2 — m)A, where mA is the overlap area (m = 0 for no overlap, m = 1 for coaxial). Then fA = 

2−m

2 ,

fW = 1/2 , and (DL) rotor = 2 2

m DL. Optionally, the reference area for tandem rotors can be total rotor

area instead: Aref = 2A.

The aircraft wing loading is the ratio of the design gross weight and a reference wing area: WL =

WD/Sref . The reference area is a sum of the wing areas, Sref = E S. The wing loading of an individual

wing is the ratio of a specified fraction of the design gross weight and the wing area:

W fWWD fW WD

(WL)wing = S=

S =

S/Sref Sref

where probably Ewing fW =1. If there are N wings of the same area, then fW = S/Sref = 1/N, and

(WL)wing = WL.

7–2 Controls

A set of aircraft controls cAC are defined; they are connected to the component controls. The

connection to the component control c is typically of the form c = STcAC + c0 , where T is an input

matrix and c0 the component control for zero aircraft control. The connection (matrix T) is defined for

a specified number of control-system states (allowing change of control configuration with flight state).

The factor S is available for internal scaling of the matrix. The control state and initial control values

are specified for each flight state.

Typical (default) aircraft controls are the pilot’s controls: collective stick, lateral and longitudinal

cyclic sticks, pedal, and tilt. Units and sign convention of the pilot’s controls are contained in the matrix

T. For the single-main-rotor and tail-rotor configuration, it is often convenient for the collective and

cyclic stick motion to equal the collective and cyclic pitch input of the main rotor, and the pedal motion to

equal the collective pitch input of the tail rotor. The aircraft controls should be scaled to approximately

the same amplitude, by appropriate definition of the T matrix and scale factor S.

These aircraft controls are available for trim of the aircraft. Any aircraft controls not selected for

trim will remain fixed at the values specified for the flight state. Thus by defining additional aircraft

controls, component controls can be specified as required for a flight state.

Each aircraft control variable cAC can be zero, constant, or a function of flight speed (piecewise

linear input). The flight-state input can override this value of the aircraft control. The input value is an

initial value if the control is a trim variable.

Each component control variable c0 (value for zero aircraft control) can be zero, constant, or a

function of flight speed (piecewise linear input). The component control from aircraft control ( TcAC) is

a fixed value, or a function of speed, or a linear function of another control (perhaps a trim variable).
Aircraft 55

The tilt control variable αtilt is intended for nacelle tilt angle or conversion control, particularly for

tiltrotors. The convention is αtilt = 0 for cruise, and αtilt = 90 degree for helicopter mode. If αtilt exists

as a control, it can be zero, constant, or a function of flight speed (piecewise linear input).

An optional conversion schedule of control is defined in terms of conversion speeds: hover and

helicopter mode for speeds below VChover, cruise mode for speeds above VCcruise, and conversion mode

between. The nacelle angle is αtilt = 90 in helicopter mode and αtilt = 0 in airplane mode, and it varies

linearly with speed in conversion mode. The tip speed is Vtip−hover in helicopter and conversion mode,

and Vtip−cruise in airplane mode. Control states and drive-system states are defined for helicopter, cruise,

and conversion-mode flight. The flight state specifies the nacelle tilt angle, tip speeds, control state,

and drive-system state, including the option to obtain any or all of these quantities from the conversion

schedule.

The flight speed used for control scheduling is usually the calibrated airspeed (CAS): Vcal = V 

V

ρ--/ρ0

(hence variation with dynamic pressure). Velocity schedules are used for conversion, controls and

motion, rotor tip speed, landing gear retraction, and trim targets. Optionally these velocity schedules

use either calibrated airspeed Vcal or the true airspeed V.

The control matrices T can be defined based on the configuration. Let cAC0 , 

cACc, cACs, cACp be

the pilot’s controls (collective, lateral cyclic, longitudinal cyclic, and pedal). For the helicopter, the first

rotor is the main rotor and the second rotor is the tail rotor; then

⎛ ⎞ ⎡ ⎤ ⎛ ⎞

TMcoll 1 0 0 0 cAC0

⎜ TMlat ⎟ = 0 ⎢ —r 0 0 cACc

TMlng

⎣ ⎦ ⎝

0 0 —1 0 cACs

TTcoll 0 0 0 —r cACp

where r is the main-rotor direction of rotation (r = 1 for counter-clockwise rotation, r = —1 for clockwise

rotation). For the tandem configuration, the first rotor is the front rotor and the second rotor is the rear

rotor; then ⎛ ⎞ ⎡ ⎤ ⎛

TFcoll 1 0 —1 0 cAC0

⎞

⎜ TFlat ⎟ ⎢ 0 —rF 0 — rF ⎥ ⎜ cACc ⎝ ⎠ = ⎣ ⎦ ⎝ ⎠

TRcoll 1 0 1 0 cACs

TRlat 0 —rR 0 rR

cACp

For the coaxial configuration:

T1coll 1 0 0 r1

⎛ ⎞

⎜ T1lat ⎟

⎡

⎢ 0 —r1 0 0 

⎤

⎥ cAC0

⎛ ⎞

⎜

T1lng

⎟

=

⎢

0

1

0

0

—1 0

0

⎥

cACc

T2coll r2

⎥

⎥ cACs 

⎠

T2lat

⎝ ⎠

0

⎣ —r2 0 0⎦

cACp

T2lng 0 0 —1 0

For the tiltrotor, the first rotor is the right rotor and the second rotor is the left rotor; then

TRcoll 1 —1 0 0

⎛ ⎞

⎜ TRlng 0 0 —1 1

⎡ ⎤

⎜

TLcoll

⎟

⎟⎢⎢

1 1 0 0

⎥ ⎛

⎥ cAC0

⎜

TLlng

⎟

=

⎢

0 0 —1 —1

⎥

⎥⎜ cACc

⎜

Tail

⎟⎢0 —1 0 0

⎥ ⎝

⎥ cACs

Telev 0 0 1 0

⎥

⎝ ⎠ cACp

Trud

⎣ ⎦

0 0 0 1
56 Aircraft

with cyclic stick and pedal connected to rotor controls only for helicopter mode.

7–3 Trim

The aircraft trim operation solves for the controls and motion that produce equilibrium in the

specified flight state. In steady flight (including hover, level flight, climb and descent, and turns),

equilibrium implies zero net force and moment on the aircraft. In general, there can be additional

quantities that at equilibrium must equal target values. In practice, the trim solution can deal with a

subset of these quantities. Usually it is at least necessary to achieve equilibrium in the aircraft lift and

drag forces, as well as in yaw moment for torque balance. The basic purpose of the trim solution is to

determine the component states, including aircraft drag and rotor thrust, sufficient to evaluate the aircraft

performance.

Different trim-solution definitions are required for various flight states. Therefore one or more

trim states are defined for the analysis, and the appropriate trim state selected for each flight state of a

performance condition or mission segment. For each trim state, the trim quantities, trim variables, and

targets are specifed. The available trim quantities include:

aircraft total force and moment; aircraft load factor;

propulsion group power; power margin PavPG — PreqPG;

rotor force (lift, vertical, or propulsive);

rotor thrust CT/σ; rotor thrust margin (CT/σ)m,,x — CT/σ;

rotor flapping βc, βs; rotor hub moment, roll and pitch; rotor torque;

wing force; wing lift coefficient CL; wing lift margin CL.,,. — CL;

tail force.

Targets for aircraft total force and total moment (including inertial loads in turns) are always zero. The

available trim variables include:

aircraft controls;

aircraft orientation, θ (pitch), φ (roll);

aircraft horizontal velocity Vh;

aircraft vertical rate of climb Vz; aircraft sideslip angle β;

aircraft angular rate, θ ̇ (pullup), ψ ̇ (turn).

The aircraft orientation variables are the Euler angles of the body axes relative to inertial axes. The

aircraft controls (appropriately scaled) are connected to the component controls.

A Newton–Raphson method is used for trim. The derivative matrix is obtained by numerical

perturbation. A tolerance a and a perturbation Δ are specified.

7–4 Geometry

The aircraft coordinate system has the x-axis forward, y-axis to the right, and z-axis down, measured

from the center of gravity (fig. 7-1). These aircraft axes are body axes (x is not aligned with the wind),

the orientation determined by the convention used for the input geometry. The center of gravity is the

appropriate origin for describing the motion of the aircraft and summing the forces and moments acting

on the aircraft.

Layout of the geometry is typically in terms of station line (SL, positive aft), buttline (BL, positive
WL starboard

forward

SL aft

port

Aircraft 57

axes for description of aircraft coordinate

aircraft geometry system

(arbitrary reference) (origin at CG)

Figure 7-1. Aircraft geometry.

right), and waterline (WL, positive up), measured relative to some arbitrary origin (fig. 7-1). The x-y-z

axes are parallel to the SL-BL-WL directions. One or more locations are defined for each component

of the aircraft. Each component will at least have a location that is the point where component forces

and moments act on the aircraft. Each location is input in fixed or scaled form. The fixed-form input

is SL/BL/WL (dimensional). The scaled-form input is x/L (positive aft), y/L (positive right), and

z/L (positive up), based on a reference length L, from a reference point. The reference length is the

rotor radius or wing span of a designated component, or the fuselage length. The reference point can

optionally be input, or the location (hub) of a designated rotor, or the location (center of action) of a

designated wing component, or the location (center of action) of the fuselage, or the location of the

center of gravity. Fixed input can be used for the entire aircraft, or just for certain components.

From this fixed or scaled input and the current aircraft size, the actual geometry (x, y, z) can be

calculated for each location. There are also options to calculate geometry from other parameters (such

as tiltrotor span from rotor radius and clearance). This calculated geometry has the sign convention of

the aircraft axes (x positive forward, y positive right, z positive down), but has the origin at the reference

point (which may or may not be the center of gravity). All input uses the same sign convention; all

internal calculations use the same sign conventions. Table 7-1 summarizes the conventions.

Table 7-1. Geometry conventions.

layout scaled input calculated motion and loads

origin arbitrary reference point reference point center of gravity

x SL (+ aft) x/L (+ aft) x (+ forward) x (+ forward)

y BL (+ right) y/L (+ right) y (+ right) y (+ right)

z WL (+ up) z/L (+ up) z (+ down) z (+ down)
58 Aircraft

The location of the aircraft center of gravity is specified for a baseline configuration. With tilting

rotors, this location is in helicopter mode. For each flight state the aircraft center of gravity is calculated,

from the baseline location plus any shift due to nacelle tilt, plus an input center-of-gravity increment.

Alternatively, the aircraft center-of-gravity location for the flight state can be input. Any change of the

center-of-gravity position with fuel burn during a mission is not automatically calculated, but could be

accounted for using the flight-state input.

The aircraft operating length and width are calculated from the component positions and dimensions:

^total = xmax — xmin and wtotal = ymax — ymin; where the maximum and minimum dimensions are for the

fuselage and all rotors, wings, and tails. The corresponding footprint area is then Stotal = ^totalwtotal.

7–5 Aircraft Motion

The aircraft velocity and orientation are defined by the following parameters: flight speed V; turn

rate; orientation of the body frame relative to inertial axes (Euler angles); and orientation of the velocity

frame relative to inertial axes (flight path angles). Aircraft conventions are followed for the direction

and orientation of axes: the z-axis is down, the x-axis forward, and the y-axis to the right; and a yawpitch-roll sequence is used for the Euler angles. However, the airframe axes are body axes (fixed to the

airframe, regardless of the flight direction) rather than wind axes (which have the x-axis in the direction

of the flight speed). The orientation of the body frame F relative to inertial axes I is defined by yaw,

pitch, and roll Euler angles, which are rotations about the z, y, and x axes respectively:

CFI = XφFYθF

ZψF

So yaw is positive to the right, pitch is positive nose up, and roll is positive to the right. The flight path

is specified by the velocity V, in the positive x-axis direction of the velocity axes. The orientation of the

velocity axes V relative to inertial axes I is defined by yaw (sideslip) and pitch (climb) angles:

C VI

= YθV ZψV ZψF

Sideslip is positive for the aircraft moving to the right, and climb is positive for the aircraft moving up.

Then

CFV = CFICIV = XφF

YθFZ−ψVY−θV

In straight flight, all these angles and matrices are constant. In turning flight at a constant yaw rate, the

yaw angle is ψF = ψ ̇ Ft; the turn radius is RT = Vh/ψ ̇ F; and the nominal bank angle and load factor are

tan φF = n2 —1 = ψ ̇ FVh/g. Then the forward, sideward, and climb velocities are:

Vf = V cos θV cos ψV = Vh cos ψV

Vs = V cos θV sin ψV = Vh sin ψV

Vc = V sin θV = Vh tan θV

where Vh = V cos θV is the horizontal velocity component. The velocity components in airframe axes

are vAC F == v

FI/F = CFV (V 0 0) T (aircraft velocity relative to the air). The aircraft angular velocity is

φF 1 0 — sin θF φF

ωA

= wFI/F = R θF =

(

ψ ̇ F )

0 cos φF sin φF cos θF θF

(

ψ ̇ F )0 — sin φF cos φF cos θF
Aircraft 59

For steady-state flight, θF = φF = 0; ψF is nonzero in a turn.

Accelerated flight is also considered, in terms of linear acceleration aFAC = v ̇

FI/F = gnL and pitch

rate  ̇θF . The nominal pullup load factor is n = 1 +  ̇θFVh/g. For accelerated flight, the instantaneous

equilibrium of the forces and moments on the aircraft is evaluated for specified acceleration and angular

velocity; the equations of motion are not integrated to define a maneuver. Note that the fuselage and

wing aerodynamic models do not include all roll-moment and yaw-moment terms needed for general

unsteady flight (notably derivatives Lv , Lp , Lr

, Nv , Np , Nr).

The aircraft pitch and roll angles are available for trim of the aircraft. Any motion not selected for

trim will remain fixed at the values specified for the flight state. The pitch and roll angles each can be

zero, constant, or a function of flight speed (piecewise linear input). The flight state input can override

this value of the aircraft motion. The input value is an initial value if the motion is a trim variable.

7–6 Loads and Performance

For each component, the power required and the net forces and moments acting on the aircraft can

be calculated. The aerodynamic forces F and moments M are typically calculated in wind axes and then

resolved into body axes (x , y, z) relative to the origin of the body axes (the aircraft center of gravity).

The power and loads of all components are summed to obtain the aircraft power and loads. Typically

the trim solution drives the net forces and moments on the aircraft to zero.

The aircraft equations of motion, in body axes F with origin at the aircraft center of gravity, are the

equations of force and moment equilibrium:

m(v ̇

FI/F +WFI/F

v

FI/F)

= FF + FFgrav

I

Fω ̇FI/F +WFI/FIF

ω

FI/F = MF

where m = W/g is the aircraft mass; the gravitational force is F

F

grav = mCFIgI = mCFI(0 0 g) T ; and

the moment of inertia matrix is ⎡ ⎤

Ixx —Ixy —Ixz

F

I = ⎣—Iyx Iyy —Iyz 

⎦

—Izx —Izy Izz

For steady flight, ω ̇

FI/F = 

v ̇

FI/F = 0, and ωFI/F = R(00 ψ ̇

F)

T is nonzero only in turns. For accelerated

flight, v ̇

FI/F can be nonzero, and ωFI/F = R(0  ̇θF ψ ̇

F)

T .The equations of motion are thus

m(aFAC + ω^

F

ACvFAC) = F

F

+ F

F

grav

F F F F ωACI ωAC = M

The body axis load factor is n = (CFIgI — (aFAC + ω^FACvFAC)) /g . The aFAC term is absent for steady flight.

The forces and moments are the sum of loads from all components of the aircraft:

F

F

= F us + E F otor + E Force + E Fwing + E F ail + E FFengine + F

F

tank

M

F

= M fus + Mrotor + E Mforce + E Mwi ng + E Mtai l + E MFengine + M

F

tank

Forces and moments in inertial axes are also of interest (FI = CIF F F and MI = C

IFMF). A particular

component can have more than one source of loads; for example, the rotor component produces hub

forces and moments, but also includes hub and pylon drag. The equations of motion are E f = F

F

+

FFgrav — F Finertial = 0 and Em = MF — MFinertial = 0.
60 Aircraft

The component power required Pcomp is evaluated for all components (rotors) of the propulsion

group. The total power required for the propulsion group PreqPG is obtained by adding the transmission

losses and accessory power. The power required for the propulsion group must be distributed to the

engine groups. The fuel flow is calculated from the power required. The fuel flow of the propulsion group

is obtained from the sum over the engine groups. The total fuel flow is the sum from all components of

the aircraft: w ̇ = E w ̇ reqEG + E w ̇ force.

7–7 Aerodynamics

Each component has a position z

F

in aircraft axes F, relative to the reference point; and orientation

of component axes B relative to aircraft axes given by the rotation matrix CBF . It is expected that the

component axes are (roughly) x forward and z down (or in negative lift direction). The aerodynamic

model must be consistent with the convention for component orientation. Acting at the component

are interference velocities v 

F

int (velocity of air, in F axes), from all other components. Then the total

component velocity relative to the air is

^

v F = vFAC +^ω

F

ACΔzF- v

F

int

where Δz

F

 = zF - zFcg. Then v

B = CBF

vF is the velocity in component axes. The aerodynamic

environment is defined in the component axes: velocity magnitude v = Iv

B I, dynamic pressure q = 1/2 ρv2 ,

angle-of-attack α, and sideslip angle β. The angle-of-attack and sideslip angle provide the transformation

between airframe axes and velocity axes:

CBA 

= Yα Z-β

This is the conventional aircraft definition, corresponding to yaw-then-pitch of the airframe axes relative

to the velocity vector. By definition, the velocity is along the x-axis in the A axes, vB = C

BA (v 0 0) T ;

from which the angle-of-attack and sideslip in terms of the components of vB are obtained:

α = tan- 1 v

B

3

/vB

1

β = sin- 1 v

B

2

/Iv

B

I

This definition is not well behaved for v

B

1 = 0 (it gives α = 90sign(v

B

3

)), so for sideward flight a

pitch-then-yaw definition can be useful: CFA = Z-βYα . Then

α = sin- 1v

B

3

/Iv

B

I

β = tan- 1 v

B

2

/vB

1

which gives β = 90sign(v

B

2

 ) for v

B

1 = 0.

From v, q, α, and β, the aerodynamic model calculates the component force and moment, in wind

axes acting at zF :

-D Mx

F

A

= Y

(

M

A

= 

⎝

My

⎠

-L Mz

where D, Y, and L are the drag, side force, and lift; Mx , My , and Mz are the roll, pitch, and yaw moments.

The aerodynamic loads in aircraft axes acting at the center of gravity are then:

F

F = CFBC

BAFA

MF = C

FBCBAMA + Δ^z

F

F

F
Aircraft 61

where Δz

F = zF - zF

cg . In hover and low speed, the download is calculated: F

I

z = k

T (CIFFF), the

downward component of the aerodynamic force in inertial axes. Download can be expressed as a fraction

of the total rotor vertical force, or as a fraction of gross weight. The aerodynamic model also calculates

the interference velocities caused by this component at all other components: v F

int = C

FBvB

int .

Equations for the aerodynamics models are defined for all angles in radians. Input values of angles

will, however, be in degrees.

The aircraft neutral point is calculated from the airframe aerodynamics with all controls set to zero.

The neutral point is here defined as the longitudinal position about which the derivative of the pitch

moment with lift is zero. Hence SLna = SLcg - ΔM/ ΔL, with the change in lift and moment calculated

from the loads at angles of attack of 0 and 5 degree.

7–8 Trailing-Edge Flaps

The lifting surfaces have controls in the form of trailing-edge flaps: flap, flaperon, and aileron for

wings; elevator or rudder for tails. The aerodynamic loads generated by flap deflection δf (radians) are

estimated based on two-dimensional aerodynamic data (as summarized in refs. 1 and 2). Let if = c f

/c

be the ratio of the flap chord to the wing chord. The lift coefficient is cP = cPα (α + τηδ f), where

η == 0. 85 - 0.43δf

 is an empirical correction for viscous effects (ref. 1, eq. 3.54 and fig. 3.36). Thin

airfoil theory gives

τ = 1- f -^in θf == (sin(-πif

))

with θf = cos−1 (2if - 1) (ref. 1, eq. 3.56 and fig. 3.35\; ref. 2, eq. 5.40). The last expression is an

approximation that is a good fit to the thin airfoil theory result for n = 1/2 , and a good approximation

including the effects of real flow for n = 2/3 (ref. 2, fig. 5.18); the last expression with n = 2/3 is used

here. The increase of maximum lift coefficient caused by flap deflection is less than the increase in lift

coefficient, so the stall angle-of-attack is decreased. Approximately

ΔAcjax = (1 - if

)(1 + if

 - 5i

2

f

 + 3i

3

f) OCP

(ref. 1, fig. 3.37). Thin airfoil theory gives the moment-coefficient increment about the quarter chord:

Δcm = -0.85((1- if) sin θf

)

δf = -0.85((1 -if)2 (1 -if)if)δf

(ref. 1, eq. 3.57; ref. 2, eq. 5.41); with the factor of 0 . 85 accounting for real flow effects (ref. 2, fig. 5.19).

The drag increment is estimated using

ΔCD = 0.9 i

1.38

Sf 

sin2

f

δf

for slotted flaps (ref. 1, eq. 3.51). In summary, the section load increments are:

ΔcP = cPα cC Lfηfδf

ΔcPmax = XfΔcP

C

Δcm = f Mfδf

c

The decrease in angle-of-attack for maximum lift is

Δαmax =

ΔcP - ΔcPmax 

-(1 - Xf ) 

ΔcP

- = l

cPα cPα
62 Aircraft

The coefficients

ηf = 0. 85 — 0 .43 IδfI = η0 — η1 IδfI

1 π 2/3

if

L f = ' (sin( 2

i f

))

Xf = (1 — if

)(1 + if

 — 

5i

2

f + 3i

3

f)

Mf = —0.85 

if

((1 — if ) 2 (1 — if)if)

Df =0.9 i

f38

follow from the former equations.

For three-dimensional aerodynamic loads, these two-dimensional coefficients are corrected by

using the three-dimensional lift-curve slope, and multiplying by the ratio of flap span to wing span b f /b.

Then the wing load increments caused by flap deflection, in terms of coefficients based on the wing area,

are:

ΔCL = 

S

f

 CLαLfηfδf

ΔCM = 

Sf Mf δf

ΔCD = 

Sf

 D f sin2δf

where Sf/S is the ratio of flap area to wing area.

7–9 Drag

Each component can contribute drag to the aircraft. A fixed drag can be specified, as a drag area

D/q; or the drag can be scaled, specified as a drag coefficient CD based on an appropriate area S. There

may also be other ways to define a scaled drag value. For fixed drag, the coefficient is CD = (D/q)/S

(the aerodynamic model is formulated in terms of drag coefficient). For scaled drag, the drag area

is D/q = SCD . For all components, the drag (D/q)comp or CDcomp is defined for forward flight or

cruise; typically this is the minimum drag value. For some components, the vertical drag ((D/q)Vcomp or

CDVcomp) or sideward drag ((D/q)Scomp or CDScomp) is defined. For some components, the aerodynamic

model includes drag due to lift, angle-of-attack, or stall.

Table 7-2 summarizes the component contributions to drag, and the corresponding reference areas.

If no reference area is indicated, then the input is only drag area D/q. An appropriate drag reference

area is defined for each component, and either input or calculated. Wetted area is calculated for each

component, even if it is not the reference area. The component wetted areas are summed to obtain the

aircraft wetted area. Some of the weight models also require the wetted area. The component drag

contributions must be consistent. In particular, a rotor with a spinner (such as on a tiltrotor aircraft)

would likely not have hub drag. The pylon is the rotor support and the nacelle is the engine support. The

drag model for a tiltrotor aircraft with tilting engines would use the pylon drag (and no nacelle drag),

since the pylon is connected to the rotor-shaft axes; with non-tilting engines it would use the nacelle

drag as well.

Optionally the aircraft drag can be fixed. The quantity specified is the sum (over all components)

of the drag area D/q (minimum drag, excluding drag due to lift and angle-of-attack), without accounting

for interference effects on dynamic pressure. The input parameter can be D/q; or the drag can be scaled,

specified as a drag coefficient based on the rotor disk area, so D/q = ArefCD (Aref is the reference rotor

ΔCLmax = XfΔCL

Δαmax = — (1 — Xf

) 

ΔCL

CLα
Aircraft 63

disk area); or the drag can be estimated based on the gross weight, D/q = k(WMTO/1000) 2/3

(WMTO is

the maximum takeoff gross weight; units of k are feet2/kilo-pound2 /3 or meter2/Mega-gram2

/

3 ). Based

on historical data, the drag coefficient CD = 0. 02 for old helicopters and CD = 0. 008 for current low-drag

helicopters. Based on historical data, k = 9 for old helicopters, k = 2. 5 for current low-drag helicopters,

k = 1 . 6 for current tiltrotors, and k = 1 . 4 for turboprop aircraft (English units). If the aircraft drag is

input, then the fuselage contingency drag is adjusted so the total aircraft D/q equals the input value.

Optionally the aircraft vertical drag (download fraction) can be fixed. The quantity specified is

the sum over all components of the vertical drag area (D/q) V . The input parameter can be (D/q) V , or

k = (D/q) V/Aref (Aref is reference rotor disk area). Approximating the dynamic pressure in the wake

as q = 1/2ρ(2vh

)

2

= T/Aref , the download fraction is DL/T = q(D/q) V/T = k. If the aircraft vertical

drag is input, then the fuselage contingency vertical drag is adjusted so the total aircraft (D/q) V equals

the input value.

The nominal drag areas of the components and the aircraft are part of the aircraft description and

are used when the aircraft drag is fixed. The nominal drag area is calculated for low-speed helicopter

flight, for high-speed cruise flight, and for vertical flight. An incidence angle i is specified for the rotors,

wings, and nacelles, to be used solely to calculate the nominal helicopter and vertical drag areas. The

convention is that i = 0 if the component does not tilt. Table 7-3 summarizes the contributions to the

nominal drag areas, with D for the drag in normal flow and DV for the drag in vertical flow. While

vertical drag parameters are part of the aerodynamic model for the hub, pylon, and nacelle, aerodynamic

interference at the rotor and at the engine group is not considered, so these terms do not contribute to

download. In the context of download, only the fuselage, wing, tail, and contingency contribute to the

nominal vertical drag.

From the input and the current aircraft size, the drag areas D/q and coefficients CD are calculated.

The aerodynamic analysis is usually in terms of coefficients. If the aircraft drag is fixed for the aircraft

model, then the fuselage contingency drag is set:

(D/q)cont = (D/q)fixed — (D/q)comp

and similarly for fixed vertical drag. Note that this adjustment ignores changes caused by interference

in the dynamic pressure and the velocity direction, which will affect the actual component drag.

The component aerodynamic model calculates the drag, typically from a drag coefficient CD , a

reference area, and the air velocity of the component. The drag force is then D = E qcompSrefCD , where

the dynamic pressure qcomp includes interference. From the aerodynamic forces and moments in wind

axes, the total force and moment in body axes ( FF and MF) are calculated. For reference, the aircraft

total drag and total drag area are

DAC = eT

d FFd

(D/q) AC = DAC/q

where the aircraft velocity (without interference) gives the direction ed = —v FAC / 1v FAC 1 and dynamic

pressure q = 1/2ρ 1vF 1

2

. Here F

F

d

is the component force in body axes that is produced only by the AC

component drag. An overall-skin friction drag coefficient is then CDAC = (D/q) ACwet/SAC, based

on the aircraft wetted area SAC = E Swet and excluding drag terms not associated with skin friction

(specifically landing gear, rotor hub, and contingency drag).
64 Aircraft

Table 7-2. Component contributions to drag.

component drag contribution reference area

fuselage fuselage fuselage wetted area

fuselage vertical fuselage projected area

fittings and fixtures fuselage wetted area

rotor-body interference fuselage wetted area

contingency (aircraft) —

payload increment (flight state) —

landing gear landing gear —

rotor hub, hub vertical rotor disk area

pylon, pylon vertical pylon wetted area

spinner spinner wetted area

wing wing, wing vertical wing planform area

wing-body interference wing planform area

tail tail, tail vertical tail planform area

engine nacelle, nacelle vertical nacelle wetted area

momentum drag —

fuel tank auxiliary tank (flight state) —

Table 7-3. Component contributions to nominal drag area.

component drag contribution cruise helicopter vertical

fuselage fuselage D D DV

fittings and fixtures D D D

rotor-body int D D D

landing gear landing gear D D 0

retractable 0 D 0

rotor hub D D cos2i + DV sin2i 0

pylon D D cos2

i + DV sin2i 0

spinner D D 0

wing wing D D cos2

i + DV sin2i D sin2i + DV cos2 i

wing-body int D D D

tail horizontal tail D D DV cos 2 φ

tail vertical tail D D DV sin 2 φ

engine nacelle D D cos2 i + DV sin2

i 0

contingency D D DV
Aircraft 65

7–10 Performance Indices

The following performance indices are calculated for the aircraft. The aircraft hover figure of merit

is M = W W/2ρAref/P. The aircraft effective drag is De = P/V, hence the effective lift-to-drag ratio

is L/De = WV/P. Isolated rotor performance indices are described in chapter 11.

7–11 Weights

The design gross weight WD is a principal parameter defining the aircraft, usually determined

by the sizing task for the design conditions and missions. The aircraft weight statement defines the

empty weight, fixed useful load, and operating weight for the design configuration. The aircraft weight

statement is the sum of the weight statements for all the aircraft components, the component weight

determined by input or by parametric calculations with technology factors. The definition of the weight

terms is as follows.

gross weight WG = WE + WUL = WO + Wpay + Wfuel

operating weight WO = WE + WFUL

useful load WUL = WFUL + Wpay + Wfuel

where WE is the weight empty; WFUL the fixed useful load; Wpay the payload weight; and Wfuel the

usable fuel weight. The weight empty consists of structure, propulsion group, systems and equipment,

vibration, and contingency weights. If the weight empty is input, then the contingency weight is adjusted

so WE equals the required value. If the design gross weight is input, then the payload or fuel weight

must be fallout.

The structural design gross weight WSD and maximum takeoff weight WMTO can be input, or

specified as an increment d plus a fraction f of a weight W:

J WSD = dSDGW + fSDGWW = 11 dSDGW + fSDGWWD

dSDGW + fSDGW (WD — Wfuel + ffuelWfuel—cap)

^

dWMTO + fWMTOWD

WMTO = dWMTO + fWMTOW = dWMTO + fWMTO (WD — Wfuel + Wfuel—cap)

This convention allows the weights to be input directly ( f = 0) or scaled with WD . For WSD, W is

the design gross weight WD , or WD adjusted for a specified fuel state (input fraction of fuel capacity).

The structural design gross weight is used in the weight estimation. For WMTO, W is the design gross

weight WD , or WD adjusted for maximum fuel capacity. Alternatively, WMTO can be calculated as the

maximum gross weight possible at a designated sizing flight condition. The maximum takeoff weight

is used in the cost model, in the scaled aircraft and hub drag, and in the weight estimation.

The design ultimate load factor nzult at the structural design gross weight WSD is specified, in

particular for use in the component weight estimates. The structural design gross weight WSD and

design ultimate load factor nzult are used for the fuselage, rotor, and wing weight estimations. The

maximum takeoff weight WMTO is used for the cost and drag (scaled aircraft and hub), and for the

weights (system, fuselage, landing gear, and engine group).

The gross weight WG is specified for each flight condition and mission, perhaps in terms of the

design gross weight WD . For a each flight state, the fixed useful load may be different from the design

configuration because of changes in auxiliary-fuel-tank weight or kit weights or increments in crew or
66 Aircraft

furnishings weights. Thus the fixed useful load weight is calculated for the flight state; and from it the

useful load weight and operating weight are calculated. The gross weight, payload weight, and usable

fuel weight (in standard and auxiliary tanks) complete the weight information for the flight state.

For each weight group, fixed (input) weights can be specified; or weight increments added to the

results of the parametric weight model. The parametric weight model includes technology factors.

Weights of individual elements in a group can be fixed by setting the corresponding technology factor

to zero.

For scaled weights of all components, the AFDD weight models are implemented.

7–12 Weight Statement

Aircraft weight information is stored in a data structure that follows SAWE RP8A Group Weight

Statement format (ref. 3), as outlined in figure 7-2. The asterisks designate extensions of RP8A, for

the purposes of this analysis. Typically only the lowest elements of the hierarchy are specified; higher

elements are obtained by summation. Fixed (input) weight elements are identified in the data structure.

A weight statement data structure exists for each component. The aircraft weight statement is the sum

of the structures from all components.

7–13 References

1) McCormick, B.W.: Aerodynamics, Aeronautics, and Flight Mechanics. New York: John Wiley &

Sons, Second Edition, 1995.

2) Kuethe, A.M.; and Chow, C.-Y.: Foundations of Aerodynamics. New York: John Wiley & Sons,

Fifth Edition, 1998.

3) Weight and Balance Data Reporting Forms for Aircraft (including Rotorcraft), Revision A. Society

of Allied Weight Engineers, Recommended Practice Number 8, June 1997.
Aircraft 67

WEIGHT EMPTY

STRUCTURE

wing group

basic structure

secondary structure

fairings (*), fittings (*), fold/tilt (*)

control surfaces

rotor group

blade assembly

hub & hinge

basic (*), fairing/spinner (*), blade fold (*)

empennage group

horizontal tail (*)

vertical tail (*)

tail rotor (*)

blades, hub & hinge, rotor/fan duct & rotor supts

fuselage group

basic (*)

wing & rotor fold/retraction (*)

tail fold/tilt (*)

marinization (*)

pressurization (*)

crashworthiness (*)

alighting gear group

basic (*), retraction (*), crashworthiness (*)

engine section or nacelle group

engine support (*), engine cowling (*), pylon support (*)

air induction group

PROPULSION GROUP

engine system

engine

exhaust system

accessories (*)

propeller/fan installation

blades (*), hub & hinge (*), rotor/fan duct & supports (*)

fuel system

tanks and support

plumbing

drive system

gear boxes

transmission drive

rotor shaft

rotor brake (*)

clutch (*)

gas drive

Figure 7-2a. Weight statement (* indicates extension of RP8A).
68 Aircraft

SYSTEMS AND EQUIPMENT

flight controls group

cockpit controls

automatic flight control system

system controls

fixed wing systems

non-boosted (*), boost mechanisms (*)

rotary wing systems

non-boosted (*), boost mechanisms (*), boosted (*)

conversion systems

non-boosted (*), boost mechanisms (*)

auxiliary power group

instruments group

hydraulic group

fixed wing (*), rotary wing (*), conversion (*)

equipment (*)

pneumatic group

electrical group

avionics group (mission equipment)

armament group

armament provisions (*), armor (*)

furnishings & equipment group

environmental control group

anti-icing group

electrical system (*), anti-ice system (*)

load & handling group

VIBRATION (*)

CONTINGENCY

FIXED USEFUL LOAD

crew

fluids (oil, unusable fuel) (*)

auxiliary fuel tanks

other fixed useful load (*)

furnishings increment (*)

folding kit (*)

wing extension kit (*)

other kit (*)

PAYLOAD

USABLE FUEL

standard tanks (*)

auxiliary tanks (*)

OPERATING WEIGHT = weight empty + fixed useful load

USEFUL LOAD = fixed useful load + payload + usable fuel

GROSS WEIGHT = weight empty + useful load

GROSS WEIGHT = operating weight + payload + usable fuel

Figure 7-2b. Weight statement (* indicates extension of RP8A).
Chapter 8

Systems

The systems component contains weight information (fixed useful load, vibration, contingency,

and systems and equipment).

8–1 Weights

The weight empty consists of structure, propulsion group, systems and equipment, vibration, and

contingency weights. The vibration control weight can be input, or specified as a fraction of weight

empty: Wvib = fvibWE . The contingency weight can be input, or specified as a fraction of weight empty:

Wcont = fcontWE . However, if the weight empty is input, then the contingency weight is adjusted so

WE equals the required value. The weights of all components are evaluated and summed, producing the

aircraft weight empty less vibration and contingency weight, WX . Then:

a) Fixed weight empty: Wvib input or Wvib = fvibWE ; Wcont = WE — (WX + Wvib) .

b) Both fractional: WE = WX/ (1 — fvib — fcont), Wvib = fvibWE , Wcont = fcontWE .

c) Only vibration weight fractional: Wcont input, WE = (WX + Wcont) / (1 — fvib) ,

Wvib = fvibWE .

d) Only contingency weight fractional: Wvib input, WE = (WX + Wvib) /(1 − fcont),

Wcont = fcont WE .

e) Both input: Wvib and Wcont input, WE = WX + Wvib + Wcont.

Finally, the operating weight WO = WE + WFUL is recalculated.

The fixed useful load WFUL consists of crew (Wcrew), trapped fluids (oil and unusable fuel, Wtrap),

auxiliary fuel tanks (Wauxtank), furnishings increment, kits (folding, wing extension, other), and other

fixed useful load (WFULother). Wcrew, Wtrap, and WFULother are input. For a each flight state, the fixed

useful load may be different than the design configuration because of changes in auxiliary-fuel-tank

weight, kit weight, and crew or furnishings weight increments.

Folding weights can be calculated in several weight groups, including wing, rotor, and fuselage.

These weights are the total weights for folding and the impact of folding on the group. A fraction

ffoldkit of these weights can be in a kit, hence optionally removable. Thus of the total folding weight,

the fraction ffoldkit is a kit weight in the fixed useful load of the weight statement, while the remainder

is kept in the component group weight.

Systems and equipment includes the following fixed (input) weights: auxiliary power group, instruments group, pneumatic group, electrical group, avionics group (mission equipment), armament group

(armor and armament provisions), furnishings and equipment group, environmental control group, and

load and handling group. Systems and equipment includes the following scaled weights: flight controls

group, hydraulic group, and anti-icing group.
70 Systems

Flight controls group includes the following fixed (input) weights: cockpit controls and automatic

flight control system. Flight controls group includes the following scaled weights: fixed-wing systems,

rotary-wing systems, and conversion or thrust-vectoring systems. Rotary-wing flight-control weights can

be calculated for the entire aircraft (using rotor parameters such as chord and tip speed for a designated

rotor), an approach that is consistent with parametric weight equations developed for conventional tworotor configurations. Alternatively, rotary-wing flight control weights can be calculated separately for

each rotor and then summed. The fixed-wing flight controls and the conversion controls can be absent.
Chapter 9

Fuselage

There is one fuselage component for the aircraft.

9–1 Geometry

The fuselage length ifus can be input or calculated. The calculated length depends on the longitudinal

positions of all components. Let xmax and xmin be the maximum (forward) and minimum (aft) position

of all rotors, wings, and tails. Then the calculated fuselage length is

ifus = inose + (xmax — xmin) + iaft

The nose length inose (distance forward of hub) and aft length inose (distance aft of hub) are input, or

calculated as inose = fnoseR and iaft = faftR. Typically faft = 0 or negative for the main rotor and tail

rotor configuration, and faft = 0. 75 for the coaxial configuration. The fuselage width wfus is input.

The fuselage wetted area Swet (reference area for drag coefficients) and projected area Sproj (reference area for vertical drag) are input (excluding or including the tail-boom terms); or calculated from

the nose length:

Swet = fwet (2inosehfus + 2inosewfus + 2hfuswfus) + CboomR

Sproj = fproj (inosewfus) + wboomR

using input fuselage height hfus and factors fwet and fproj; or calculated from the fuselage length:

Swet = fwet (2ifushfus + 2ifuswfus + 2hfuswfus) + CboomR

Sproj = fproj (ifuswfus) + wboomR

Using the nose length and the tail-boom area is probably best for a single-main-rotor and tail-rotor

helicopter. Here Cboom is the effective tail-boom circumference (boom wetted area divided by rotor

radius), and wboom is the effective tail-boom width (boom vertical area divided by rotor radius).

The fuselage contribution to the aircraft operating length is xfus + frefifus (forward) and xfus — (1—

fref)ifus (aft). Here fref is the position of the fuselage aerodynamic reference location aft of the nose, as

a fraction of the fuselage length. If the fuselage length is input, then fref is input; if the fuselage length

is calculated, then fref = (xmax + inose — xfus)/ifus.

9–2 Control and Loads

The fuselage has a position zF , where the aerodynamic forces act; and the component axes are

aligned with the aircraft axes, CBF 

= I. The fuselage has no control variables.
72 Fuselage

9–3 Aerodynamics

The aerodynamic velocity of the fuselage relative to the air, including interference, is calculated in

component axes, v

B . The angle-of-attack αfus, sideslip angle βfus (hence CBA), and dynamic pressure

q are calculated from vB . The reference area for the fuselage forward-flight drag is the fuselage wetted

area Swet, which is input or calculated as described previously. The reference area for the fuselage

vertical drag is the fuselage projected area Sproj, which is input or calculated as described previously.

9-3.1 Drag

The drag area or drag coefficient is defined for forward flight, vertical flight, and sideward flight.

In addition, the forward-flight drag area or drag coefficient is defined for fixtures and fittings, and

for rotor-body interference. The effective angle-of-attack is αe = αfus − αDmin, where αDmin is the

angle of minimum drag; in reverse flow ( |αe| > 90), αe ← αe − 180signαe

. For angles of attack less

than a transition angle αt

, the drag coefficient equals the forward-flight (minimum) drag CD0, plus an

angle-of-attack term. Thus if |αe| ≤ αt

CD = CD0 (1+ Kd |αe|

X

d)

and otherwise

CDt = CD0 ( 1 + Kd |αt

|

Xd)

CD = CDt + 

I 

w

r

t 2 π/2 a−

CDV − CDt I sin 1 

|

t

/// \ t /

and similarly for the transition of payload drag (D/q)pay and contingency drag (D/q) cont . Optionally

there might be no angle-of-attack variation at low angles ( Kd = 0), or quadratic variation (Xd = 2). With

an input transition angle, there will be a jump in the slope of the drag coefficient at αt . For a smooth

transition, the transition angle that matches slopes as well as coefficients is found by solving

I 2Xd

1 αt

dd − Xdαt

d−1 + (Sproj /SKt)C

OV

- O CD

= 0

dCD

This calculation of the transition angle is only implemented with quadratic variation, for which

⎛

αt = 1 1+ 1 − a 

(Sproj /Swet)CDV − CD0 ⎠

a KdCD0

with a = (4/π) − 1; αt is, however, required to be between 15 and 45 degree. For sideward flight

(v

B

x = 0) the drag is obtained using φv = tan−1 (−v

B

z

/vB

y ) to interpolate between sideward and vertical

coefficients:

CD = CDS cos 2 φv + !

Sproj CDV sin2 φv

Swet

Then the drag force is

D = qSwet 

(

CD + CDfit + E CDrb) + q 

(

(D/q)pay + (D/q)cont)

including drag coefficient for fixtures and fittings CDfit and rotor-body interference CDrb (summed over

all rotors); drag area of the payload (specified for flight state); and contingency drag area.

9-3.2 Lift and Pitch Moment

The fuselage lift and pitch moment are defined in fixed form (L/q and M/q), or scaled form

(CL and CM , based on the fuselage wetted area and fuselage length). The effective angle-of-attack is
Fuselage 73

αe = αfus − αzl, where αzl is the angle of zero lift; in reverse flow ( |αe | > 90), αe ← αe − 180 signαe . Let

αmax be the angle-of-attack increment (above or below zero lift angle) for maximum lift. If |αe| ≤ αmax

CL = CLααe

CM = CM0 + CMααe

and otherwise

^^ CL = CLααmaxsignαe

π/2 − |αe|

π/2 − |αmax |

CM = (CM0 + CMααmaxsignαe ) π/2 − |αe

|

π/2 − |αmax |

for zero lift and moment at 90 degree angle-of-attack. In sideward flight, these coefficients are zero.

Then L = qSwetCL and M = qSwetifusCM are the lift and pitch moment.

9-3.3 Side Force and Yaw Moment

The fuselage side force and yaw moment are defined in fixed form (Y/q and N/q), or scaled form

(

//I^ 

and

/ 

d 

C/N^^

, based on the fuselage wetted area and fuselage length). The effective sideslip angle is

Ne = Nfus − Nzy, where βzy is the angle of zero side force; in reverse flow ( |βe

| > 90), βe ← βe − 180 signβe .

Let βmax 

by/^

e the sideslip-angle increment (above or below zero side-force angle) for maximum side force.

If |βe | ≤ Nmax

CY = CNββe

//^^ CN = CN0 + CNβOe

and otherwise

CY = CYββmaxsignβe

π/2 − |βe | 

1

/^

G/2

CN = (CN0 + CNβOmaxsignβe )

G/2

/

2 |βmax |

for zero side force and yaw moment at 90 degree sideslip angle. Then Y = qSwet CY and N = qSwetifusCN

are the side force and yaw moment. The roll moment is zero.

9–4 Weights

The fuselage group consists of the basic structure; wing and rotor fold/retraction; tail fold/tilt; and

marinization, pressurization, and crashworthiness structure.
74 Fuselage
Chapter 10

Landing Gear

There is one landing-gear component for the aircraft. The landing gear can be located on the body

or on the wing. The landing gear can be fixed or retractable; a gear retraction speed is specified (CAS),

or the landing-gear state can be specifed in the flight state.

10–1 Geometry

The landing gear has a position zF , where the aerodynamic forces act. The component axes are

aligned with the aircraft axes, CBF 

= I. The landing gear has no control variables. The height of the

bottom of the landing gear above ground level, hLG, is specified in the flight state. The landing-gear

position zF is a distance dLG above the bottom of the gear.

10-1.1 Drag

The drag area is specified for landing gear extended, (D/q) LG . The velocity relative to the air at

z

F

gives the drag direction ed = −v F

/|v F I 

and dynamic pressure q = 1

/2ρ|v

Fl2 (no interference). Then

F

F

= ed q (D/q) LG

is the total drag force.

10–2 Weights

The alighting gear group consists of basic structure, retraction, and crashworthiness structure.
76 Landing Gear
Chapter 11

Rotor

The aircraft can have one or more rotors, or no rotors. In addition to main rotors, the rotor component

can model tail rotors, propellers, proprotors, ducted fans, thrust vectoring rotors, and auxiliary-thrust

rotors. The principal configuration designation (main rotor, tail rotor, or propeller) is identified for

each rotor component, and in particular determines where the weights are put in the weight statement

(summarized in table 11-1). Each configuration can possibly have a separate performance or weight

model, which is separately specified. Antitorque rotors and auxiliary-thrust rotors can be identified, for

special sizing options. Other configuration features are variable diameter, and ducted fan.

Multi-rotor systems (such as coaxial or tandem configuration) are modeled as a set of separate rotors,

in order to accommodate the description of the position, orientation, controls, and loads. Optionally the

location of the center of the rotor system can be specified, and the rotor locations calculated based on

input separation parameters. The performance calculation for twin rotor systems can include the mutual

influence of the induced velocity on the power.

The main rotor size is defined by the radius R or disk loading W/A, thrust-weighted solidity σ,

hover tip speed Vt;p, and blade loading CW/σ = W/ ρAV't;pσ. With more than one main rotor, the disk

loading and blade loading are obtained from an input fraction of design gross weight, W = fWWD.

The air density ρ for CW/σ is obtained from a specified takeoff condition. If the rotor radius is fixed

for the sizing task, three of (R or W/A), CW/σ, Vt;p, σ are input, and the other parameters are derived.

Optionally the radius can be calculated from a specified ratio to the radius of another rotor. If the

sizing task determines the rotor radius (R and W/A), then two of CW/σ, Vt;p, σ are input, and the other

parameter is derived. The radius can be sized for just a subset of the rotors, with fixed radius for the

others.

For antitorque and auxiliary-thrust rotors, three of ( R or W/A), CW/σ, Vt;p, σ are input, and the

other parameters are derived. Optionally the radius can be calculated from a specified ratio to the radius

of another rotor. The disk loading and blade loading are based on fT, where f is an input factor and T

is the maximum thrust from designated design conditions. Optionally the tail-rotor radius can be scaled

with the main-rotor radius: R = fRmr (0.1348 + 0.0071W/A), where f is an input factor and the units of

disk loading W/A are pound/feet'. Figure 11-1 is the basis for this scaling.

Table 11-1. Principal configuration designation.

configuration weight statement weight model performance model

main rotor rotor group rotor rotor

tail rotor empennage group tail rotor rotor

propeller propulsion group rotor, aux thrust rotor
78 Rotor

0.26

O aircraft O O

0.24 equation

0

0.22 OO o

® O o

0

0.20 000

0 0 O ®MDO (b O) 

0 

O O

® Oo

0.18 O 0 0 aoo 0 ®^®

OS

o O o

0.16 0 

% 

00 ®88 0

00 Ocb

O 0

0.14 00 o

Cb

0.12

0. 2. 4. 6. 8. 10. 12. 14. 16.

disk loading (lb/ft2)

Figure 11-1. Tail-rotor radius scaling.

11–1 Drive System

The drive system defines gear ratios for all the components it connects. The gear ratio is the ratio

of the component rotational speed to that of the primary rotor. There is one primary rotor per propulsion

group (for which the reference tip speed is specified); other components are dependent (for which a gear

ratio is specified). There can be more than one drive-system state, in order to model a multiple-speed or

variable-speed transmission. Each drive-system state corresponds to a set of gear ratios.

For the primary rotor, a reference tip speed Vtip-ref is defined for each drive-system state. By

convention, the “hover tip speed” refers to the reference tip speed for drive state #1. If the sizing task

changes the hover tip speed, then the ratios of the reference tip speeds at different engine states are kept

constant. By convention, the gear ratio of the primary rotor is r = 1. For dependent rotors, either the

/

gear ratio is specified (for each drive-system state) or a tip speed is specified and the gear ratio calculated

(

r = Ωdep /Ωprim , Ω = Vtip−ref /R). For the engine group, either the gear ratio is specified (for each drivesystem state) or the gear ratio calculated from the specification engine turbine speed Ωspec = (2π/60)Nspec

and the reference tip speed of the primary rotor (r = Ωspec/Ωprim, Ωprim = Vtip −ref /R). The latter option

means the specification engine turbine speed Nspec corresponds to Vtip-ref for all drive-system states.

To determine the gear ratios, the reference tip speed and radius are used, corresponding to hover.

The flight state specifies the tip speed of the primary rotor and the drive-system state, for each
Rotor 79

propulsion group. The drive-system state defines the gear ratio for dependent rotors and the engine

groups. From the rotor radius the rotational speed of the primary rotor is obtained (Ωprim = Vtip /R);

from the gear ratios, the rotational speed of dependent rotors ( Ωdep = rΩprim) and the engine groups

(N = (60/2π)rengΩprim) are obtained; and from the rotor radius, the tip speed of the dependent rotor

(Vtip = ΩdepR) is obtained. The flight-state specification of the tip speed can be an input value; the

reference tip speed; a function of flight speed or a conversion schedule; or one of several default values.

These relationships between tip speed and rotational speed use the actual radius of the rotors in the flight

state, which for a variable-diameter rotor may not be the same as the reference, hover radius.

An optional conversion schedule is defined in terms of two speeds: hover and helicopter mode for

speeds below VChover, cruise mode for speeds above VCcruise, and conversion mode for speeds between

VChover and VCcruise. The tip speed is Vtip_hover in helicopter and conversion mode, and Vtip_cruise in

airplane mode. Drive-system states are defined for helicopter, cruise, and conversion-mode flight. The

flight state specifies the nacelle tilt angle, tip speeds, control state, and drive-system state, including the

option to obtain any or all of these quantities from the conversion schedule.

Several default values of the tip speed are defined for use by the flight state, including cruise,

maneuver, one-engine inoperative, drive-system limit conditions, and a function of flight speed (piecewise linear input). Optionally these default values can be input as a fraction of the hover tip speed.

Optionally the tip speed can be calculated from an input CT/σ = t0 − μt1 , from μ = V/Vtip, or from

Mat = Mtip (1 + μ)

2 + μ2

z

; from which Vtip = T/ρAσt0 + (Vt1 /2t0 )

2 + (Vt1 /2t0 ), Vtip = V/μ, or

Vtip = (csMat)

2 − V2

z

− V.

The sizing task might change the hover tip speed (reference tip speed for drive-system state #1),

the reference tip speed of a dependent rotor, a rotor radius, or the specification engine turbine speed

Nspec. In such cases the gear ratios and other parameters are recalculated. Note that it is not consistent

to change the reference tip speed of a dependent rotor if the gear ratio is a fixed input.

11–2 Geometry

The rotor rotation direction is described by the parameter r: r = 1 for counter-clockwise rotation,

and r = −1 for clockwise rotation (as viewed from the positive thrust side of the rotor).

The rotor solidity and blade mean chord are related by σ = Nc/πR; usually thrust-weighted values

are used, but geometric values are also required by the analysis. The mean chord is the average of

the chord over the rotor blade span, from root cutout to tip. The thrust-weighted chord is the average

of the chord over the rotor blade span r, from root cutout to tip, weighted by r2 . A general blade

chord distribution is specified as c(r) = crefcˆ(r), where cref is the thrust-weighted chord. Linear taper is

specified in terms of a taper ratio t = ctip/croot, or in terms of the ratio of thrust-weight and geometric

chords, f = σt

/σg = c.75R/c.50R.

The rotor hub is at position z F

hub. Optionally, a component of the position can be calculated,

superseding the location input. The calculated geometry depends on the configuration. For a coaxial

rotor, the rotor separation is s = |k

TCSF (z F

hub1 − 

z F

hub2)/(2R)| (fraction rotor diameter), or the hub

locations are calculated from the input separation s, and the input location midway between the hubs:

0

F _ F

z

hub — zcenter ± C

FS

⎛

0

sR )
80 Rotor

For a tandem rotor, the rotor longitudinal overlap is o = ΔB/ (2R) = 1— B/ (2R) (fraction rotor diameter),

or the hub locations are calculated from the input overlap o, and the input location midway between the

hubs:

xhub = xcenter f R(1 — o)

For a tail rotor, the longitudinal position can be calculated from the main-rotor radius R, tail-rotor radius

Rtr, and tail-otor/main-rotor clearance dtr:

xhubtr = xhubmr — (Rmr + dtr + Rtr)

For a tiltrotor, the lateral position can be calculated from the rotor radius R (cruise value for variablediameter rotor), fuselage/rotor clearance dfus, and fuselage width wfus:

yhub = f (fR + dfus + 1/2wfus)

with the pivot, pylon, and nacelle center-of-gravity lateral positions adjusted to keep the same relative

position to the hub. The calculated clearance between the rotor and fuselage is dfus = I yhub I — (R+ 1/2wfus) .

Alternatively for a tiltrotor, the lateral position can be calculated from the wing span, yhub = f b/2, so

the rotors are at the wing tips; or from a designated wing-panel edge, yhub = f ηp (b/2).

For twin rotors (tandem, side-by-side, or coaxial), the overlap is o = ΔB/(2R) = 1 — B/(2R)

(fraction of diameter; 0 for no overlap and 1 for coaxial), where the hub-to-hub separation is B =

[(xhub1 — xhub2 ) 2 + (yhub1 — yhub2 )

2 ] 1 /2 (B = 2R for no overlap and B = 0 for coaxial). The overlap area

is mA, with A the area of one rotor disk and

m =

7r 

[

cos−1

(B/2R) — (B/2R) 1 — (B/2R)

2

]

The vertical separation is s = I zhub1 — zhub2 I/(2R).

The reference areas for the component drag coefficients are the rotor disk area A = πR

2

(for

hub drag), pylon wetted area Spylon, and spinner wetted area Sspin. The pylon wetted area is input,

or calculated from the drive system (gear box and rotor shaft) weight, or from the drive-system plus

engine-system (engine, exhaust, and accessories) weight:

Spylon = k (w/Nrotor)

2 /3

where w = Wgbrs or w = Wgbrs + E WES, and the units of k are feet2/pound2 /3 or meter2 /kilogram2

/3 .

The pylon area is included in the aircraft wetted area if the pylon drag coefficient is nonzero. The spinner

wetted area is input, or calculated from the spinner frontal area:

Sspin = f (πR2spin)

where Rspin is the spinner radius, which is specified as a fraction of the rotor radius.

The rotor contribution to the aircraft operating length and width is calculated from the locus of

the rotor disk: zdisk = zhub + RCFS(cos ψ sin ψ 0)T . The longitudinal distance from the hub position is

Δx = R(a cos ψ + b sin ψ), so the maximum distance is Δx = fR a2 + b

2 . The lateral distance from the

hub position is Δy = R(c cos ψ + d sin ψ), so the maximum distance is Δy = fR c2 + d

2 .

11–3 Control and Loads

The rotor controls consist of collective, lateral cyclic, longitudinal cyclic, and perhaps shaft incidence (tilt) and cant angles. Rotor cyclic control can be defined in terms of tip-path plane or no-feathering
Rotor 81

plane command. The collective control variable is the rotor thrust amplitude T or CT/σ (in shaft axes),

from which the collective pitch angle can be calculated. This approach eliminates an iteration between

thrust and inflow, and allows thrust limits to be applied directly to the control variable.

The relationship between tip-path plane tilt and hub moment is M = 

N

2

IbΩ

2 (ν2 — 1)β = Khubβ,

where N is the number of blades, Ω the rotor speed, and ν the dimensionless fundamental flap frequency.

The flap moment of inertia Ib is obtained from the input Lock number: γ = ρacR4 /Ib , for SLS density

ρ and lift-curve slope a = 5. 7. The flap frequency and Lock number are specified for hover radius and

rotational speed. The flap frequency and hub stiffness are required for the radius and rotational speed

of the flight state. For a hingeless rotor, the blade-flap spring is Kflap = IbΩ

2 (ν2 —1), obtained from the

hover quantities; then Khub = 

N

2

Kflap and

ν

2

= 1 + 

Kflap

IbΩ2

For an articulated rotor, the hinge offset is e = Rx/ (1 + x), x = 23

(ν

2 —1) from the hover quantities; then

2

3 e/R

ν 

=1+21— e/R

and Khub = 

N

2

IbΩ

2 (ν2 — 1), using Ib from γ (and scaled with R for a variable diameter rotor) and Ω for

the flight state.

Optionally the rotor can have a variable diameter. The rotor diameter is treated as a control, allowing

it to be connected to an aircraft control and thus set for each flight state. The basic variation can be

specified based on the conversion schedule, or input as a function of flight speed (piecewise linear input).

For the conversion schedule, the rotor radius is Rhover for speeds below VChover, Rcruise = fRhover for

speeds above VCcruise, and linear with flight speed in conversion mode. During the diameter change, the

chord, chord radial distribution, and blade weight are assumed fixed; hence solidity scales as σ — 1/R,

blade-flap moment of inertia as Ib — R2 , and Lock number as γ — R2 .

11-3.1 Tip-Path Plane Command

Tip-path plane command is characterized by direct control of the rotor thrust magnitude and the

tip-path plane tilt. This control mode requires calculation of rotor collective and cyclic pitch angles

from the thrust magnitude and flapping.

a) Collective: magnitude of the rotor thrust T or CT/σ (shaft axes).

b) Cyclic: tilt of the tip-path plane, hence tilt of the thrust vector; longitudinal tilt βc

(positive forward) and lateral tilt βs (positive toward retreating side). Alternatively,

the cyclic control can be specified in terms of hub moment or lift offset, if the

blade-flap frequency is greater than 1/rev.

c) Shaft tilt: shaft incidence (tilt) and cant angles, acting at a pivot location.

The relationship between tip-path plane tilt and hub moment is M = Khubβ , and between moment and

lift offset is M = o(TR). Thus the flapping is

^

βs

)

^= 1 rMx

= 

TR ox

 )

βc

Khub —My Khub —oy

for hub moment command or lift offset command, respectively.
82 Rotor

11-3.2 No-Feathering Plane Command

No-feathering plane command is characterized by control of rotor cyclic pitch angles and direct

control of the rotor thrust magnitude. This control mode requires calculation of rotor collective pitch

angle and tip-path plane tilt from the thrust magnitude and cyclic control, including the influence of

inflow.

a) Collective: magnitude of the rotor thrust T or CT/σ (shaft axes).

b) Cyclic: tilt of the no-feathering plane, usually producing tilt of the thrust vector;

longitudinal cyclic pitch angle θs (positive aft) and lateral cyclic pitch angle θc

(positive toward retreating side).

c) Shaft tilt: shaft incidence (tilt) and cant angles, acting at a pivot location.

11-3.3 Aircraft Controls

Each control can be connected to the aircraft controls cAC: c = c0+STcAC , with c0 zero, constant, or

a function of flight speed (piecewise linear input). The factor S can be introduced to automatically scale

the collective matrix: S = a/6 = 1/60 if the collective control variable is CT/σ; S = ρV2tipAblade (a/6)

if the collective control variable is rotor thrust T. For cyclic matrices, S = 1 with no-feathering plane

command, and S = —1 for tip-path plane command.

11-3.4 Rotor Axes and Shaft Tilt

The rotor hub is at position zFhub, where the rotor forces and moments act; the orientation of the rotor

shaft axes relative to the aircraft axes is given by the rotation matrix CSF . The pivot is at position z

F

pivot.

The hub or shaft axes S have origin at the hub node; the z-axis is the shaft, positive in the positive thrust

direction; and the x-axis downstream or up. The rotor orientation is specified by selecting a nominal

direction in body axes (positive or negative x, y, or z-axis) for the positive thrust direction; the other two

axes are then the axes of control. For a main rotor the nominal direction would be the negative z-axis;

for a tail rotor it would be the lateral axis (ry-axis, depending on the direction of rotation of the main

rotor); and for a propeller the nominal direction would be the positive x-axis. This selection defines a

rotation matrix W from F to S axes. The hub and pivot axes have a fixed orientation relative to the body

axes:

hub incidence and cant: CHF 

= Uθh

Vφh

pivot dihedral, pitch, and sweep: CPF 

= XφhYθh

Zψp

where U and V depend on the nominal direction, as described in table 11-2. The shaft control consists

of incidence and cant about the pivot axes, from reference angles iref and cref:

Ccont = Ui−iref Vc−cref

For a tiltrotor aircraft, one of the aircraft controls is the nacelle angle, with the convention αtilt = 0 for

cruise, and αtilt = 90 degree for helicopter mode. The rotor-shaft incidence angle is then connected to

αtilt by defining the matrix Ti appropriately. For the locations and orientation input in helicopter mode,

iref = 90. Thus the orientation of the shaft axes relative to the body axes is:

CSF = WCHFCFPCcontCPF
Rotor 83

or just CSF = WCHF with no shaft control. From the pivot location zFpivot and the hub location for the

reference shaft control z F

hub0, the hub location in general is

F t / FP PF T / F F

zhub = zpivot + (cCcont C ) (zhub0 — zpivot )

Similarly, the pylon location and nacelle center-of-gravity location can be calculated for given shaft

control. The shift in the aircraft center of gravity produced by nacelle tilt is

W(z

F

 — z

F

 ) = W (Z

F

 — Z

F

) = Wmove (C

FPCT CPF — I) (zF — zF

cg cg0 move nac nac0 cont nac0 pivot )

where W is the gross weight and Wmov e the weight moved. Table 11-2 summarizes the geometry options.

Table 11-2. Rotor-shaft axes.

nominal thrust

zS-axis x

S

-axis W

incidence

+ for T

cant

+ for T Uθh

V φh

main rotor —zF up —x F aft Y180 aft right YθXφ

z

F

down —x F aft Z180 aft right Y_θX_φ

propeller x

F

forward —zF up Y90 up right YθZφ

—xF aft —zF up Z180Y_90 up right Y_θZ_φ

tail rotor (r = 1) y

F

right —x F aft Z180X_90 aft up ZθX_φ

tail rotor (r = —1) —yF left —x F aft Z180X90 aft up Z_θXφ

⎡ ⎤ ⎡ ⎤ —1 0 0 0 0 —1

Y180 = ⎣ 0 1 0 ⎦ Z180Y_90 = ⎣ 0 —1 0 ⎦

0 0 —1 —1 0 0

⎡ ⎤ ⎡ ⎤ —1 0 0 —1 0 0

Z180 = ⎣ 0 —1 0 ⎦ Z180X_90 = ⎣ 0 0 1 ⎦

0 0 1 0 1 0

⎡ ⎤ ⎡ ⎤

0 0 —1 —1 0 0

Y90 = ⎣0 1 0 ⎦ Z180X90 = ⎣ 0 0 —1 ⎦

1 0 0 0 —1 0

11-3.5 Hub Loads

The rotor controls give the thrust magnitude and the tip-path plane tilt angles βc and βs, either

directly or from the collective and cyclic pitch. The forces acting on the hub are the thrust T, drag H,

and side force Y (positive in z, x, y-axis directions respectively). The hub pitch and roll moments are

proportional to the flap angles. The hub torque is obtained from the shaft power Pshaft and rotor speed

Ω. The force and moment acting on the hub, in shaft axes, are then:

(H) 0

F

S

= Y + 0

T ( —fBT)

⎛ ⎞ ⎛ ⎞

Mx Khub (rβs

)

M

S

= ⎝ My

⎠ = ⎝Khub (—βc) ⎠

—rQ —rPshaft/Ω
84 Rotor

The force includes a term proportional to the rotor thrust and an input blockage factor fB = ΔT/T > 0.

This term accounts for blockage or download, as an alternative to including the drag of the fuselage

or a lifting surface in the aircraft trim. For example, fB can model the tail-rotor blockage caused by

operation near the vertical tail. The rotor loads in aircraft axes acting at the center of gravity are then:

FF = CFSFS

MF = CFSMS + Δ^z

FF

F

where Δz

F = zF

hub - z Fcg.

The wind axis lift L and drag X are calculated from the net rotor-hub force FF and the rotor velocity

v

F . The velocity relative to the air gives the propulsive-force direction ep = vF/ IvFI (no interference)

and the velocity magnitude V = Iv F I. The drag and lift components of the force are X = -ep TF

F

and

L = I(I - ep e

T

p )FF

I

, respectively. Thus XV = -(v

F

)T FF and L2 = I F FI

2

-IXI 2 . The rotor contribution

to vertical force is the z-axis component of the force in inertial axes, FV = -k

TCIFFF.

11–4 Aerodynamics

The rotor velocity relative to the air is vF = vFAC + ω^

F

ACΔz

F in aircraft axes. The velocities in shaft

axes are ⎛

 )

-μx

v

S

= C

SFvF = ΩR rμy

μz

rα ̇ x

ω

S

= C

SFωF

C = Ω α ̇ y

(

rα ̇ z )

where ΩR is the rotor tip speed. The advance ratio μ, inflow ratio λ, and shaft angle-of-attack α are

defined as

μ = μ

2

x + μ

2

y

λ = λi + μz

α = tan−1

 (μz /μ)

The blade velocity relative to the air has the maximum amplitude (advancing tip velocity) of μat =

^

(1 + μ)

2 + μ2

z , from which the advancing tip Mach number is Mat = Mtipμat , using the tip Mach

number Mtip = (ΩR)/cs . The rotor thrust coefficient is defined as CT = T/ ρA(ΩR)

2

. The dimensionless

ideal induced velocity λi is calculated from μ, μz , and CT ; then the dimensional velocity is vi = ΩR λi

.

The ideal induced power is then Pideal = Tvi . Note that for these inflow velocities, the subscript “i”

denotes “ideal.” The ideal induced velocity could be solved based on the reference velocity vh rather

than the tip speed ΩR, but the advance ratio is required for other purposes as well.

11-4.1 Ideal Inflow

The ideal wake-induced velocity is obtained from the momentum theory result of Glauert:

CTsλ

2

h

λi =

2 Vλ

2 +—μ2 λ2 + μ2

where λ = λi + μz , λ2

h = ICT I/2 (λh is always positive) and s = sign CT . This expression is generalized

to

λi = λh sF(μ/λh, sμz /λh)
Rotor 85

If μ is zero, the equation for λi can be solved analytically. Otherwise, for non-axial flow, the equation

is written as follows:

soh

2

λ = ^ + μz

λ2 + μ2

Using λ instead of λi as the independent variable simplifies implementation of the ducted fan model. A

Newton–Raphson solution for λ gives:

sλ

2

h

λin =

^

λ2

n

+ μ2

λn — μz — L

λn+1 = λn —

f

1 + Lλn

/(λ2

n + μ2

)

A relaxation factor of f = 0. 5 is used to improve convergence. Three or four iterations are usually

sufficient, using

2

λ =

sλh

^ + μz

(sλh + μz )

2 + μ2

to start the solution. To eliminate the singularity of the momentum theory result at ideal autorotation,

the expression

λ = μz

[

0.373μz + 0.598μ

2

 — 0.9911

λ

2

h J

is used when

1.5μ

2

+ (2sμz + 3λh)

2 < λ2

h

The equation λ = μz (aμ

2

z — bλ

2

h + cμ

2 )/λ2

h is an approximation for the induced power measured in the

turbulent-wake and vortex-ring states. Matching this equation to the axial-flow momentum theory result

at μz = —2λh and μz = —λh gives a = V

f

5/6 = 0. 3726780 and b = (4V

f

5 — 3) /6 = 0.9907120. Then

matching to the forward-flight momentum theory result at (μ = λh , μz = —1.5λh) gives c = 0. 5980197.

For axial flow (μ = 0) the solution is:

⎧

2

2

z

+ s 

\ 2

z

+ λ

2

h —λh < sμz

[

0.373μz 1

λ = / λ

2

 — 0.991] —2λh < sμz < —λh

⎪

h

⎪⎪⎪ μz

(

2

2 — s 

\ 2

z

/ —

λh

s

μz < —2λh

Note that λi and vi are the ideal induced velocities; additional factors are required for the wake-induced

velocity or induced power calculations.

11-4.1.1 Ducted Fan

Rotor momentum theory can be extended to the case of a ducted fan. Consider a rotor system with

disk area A, operating at speed V, with an angle α between V and the disk plane. The induced velocity

at the rotor disk is v, and in the far wake w = fWv. The far-wake area is A^ = A/ fA . The axial velocity

at the fan is fVzVz , with fVz accounting for acceleration or deceleration through the duct. The edgewise

velocity at the fan is fVxVx , with fVx = 1 .0 for wing-like behavior, or fVx = 0 for tube-like behavior of

the flow. The total thrust (rotor plus duct) is T, and the rotor thrust is Trotor = fTT. For this model, the

duct aerodynamics are defined by the thrust ratio fT or far-wake area ratio fA , plus the fan velocity ratio
86 Rotor

fV . The mass flux through the rotor disk is m ̇ = ρAU = ρA,,U,,, where U and U,, are respectively the

total velocity magnitudes at the fan and in the far wake:

U2 = (fVxV cos α) 2 +(fVzV sin α + v)

2

U

2

,, = (V cos α)

2 + (V sin α + w) 2

Mass conservation (fA = A/A,, = U,,/U) relates fA and fW . Momentum and energy conservation give

T = mw  ̇ = ρAU,,w/fA = ρAUfWv

P = 2rhw (2V sin α + w) = T 

(

V sin α + 2)

With these expressions, the span of the lifting system in forward flight is assumed equal to the rotor

diameter 2R. Next it is required that the power equals the rotor induced and parasite loss:

P = Trotor (fVzV sin α + v) = T fT (fVzV sin α + v)

In axial flow, this result can be derived from Bernoulli’s equation for the pressure in the wake. In

forward flight, any induced drag on the duct is being neglected. From these two expressions for power,

Vz + fWv/2 = fT (fVzVz + v) is obtained, relating fT and fW . With no duct (fT = fVx = fVz = 1),

the far-wake velocity is always w = 2v, hence fW = 2. With an ideal duct (fA = fVx = fVz = 1), the

far-wake velocity is fW = 1. In hover (with or without a duct), fW = fA = 2 fT and v = 2/fWvh . The

rotor ideal induced power is Pideal = Tw/2 = fD Tv, introducing the duct factor fD = fW/2.

For a ducted fan, the thrust CT is calculated from the total load (rotor plus duct). To define the duct

effectiveness, either the thrust ratio fT = Trotor/Tor the far-wake area ratio fA = A/A,, is specified

(and the fan velocity ratio fV). The wake-induced velocity is obtained from the momentum theory result

for a ducted fan: λ

2

h

= (fWλi/2) (fVxμ)2 + (fVzμz + λi)2

. 

If the thrust ratio fT is specified, this can

be written

fVzμz + λi =

sλ

2

h

/fT

+ 

μz

(fVzμz + λi)2 + (fVxμ) 2

 fT

In this form, λi can be determined using the free-rotor expressions given previously: replacing λ

2

h

, μ z,

μ, λ with respectively λ

2

h

/ fT , μz / fT , fVxμ, fVzμz + λi . Then from λi the velocity and area ratios are

obtained: (fT

μz

fW = 2 − (1 − fTfVz) λi

J

μ2

+ (μz + fWλi)

2

(fVxμ)2 + (fVzμz + λi)

2

If instead the area ratio fA is specified, it is simplest to first solve for the far-wake velocity fWλi :

μz + fWλi =

sλ

2

h

2fA

+ μz

(μz + fWλi)

2 + μ2

In this form, fWλi can be determined using the free-rotor expressions given previously: replacing λ

2

h

, λ

with respectively λ

2

h

2 fA , μz + fWλi . The induced velocity is

1

(fVzμz + λi)

2 = fA

^

μ

2 + (μz + fWλi)2

] − 

(fVxμ)

2

fA =
Rotor 87

The velocity ratio is fW = (fWλi)/λi , and

z + fWλi

/2

fT = 

μ,

fVzμz + λi

is the thrust ratio. However, physical problems and convergence difficulties are encountered with this

approach in descent, if an arbitrary value of fT is permitted. From the expression for fT , fT should

approach 1/ fVz at high rates of climb or descent. To avoid problems with an arbitrary value of fT , it is

assumed that the input value of fT defines the velocity ratio fW = 2 fT in descent. So in descent μz is

not replaced by μz

/ fT .

11-4.1.2 Ground Effect

The wake-induced velocity is reduced when the rotor disk is in the proximity of the ground plane.

Ground effect in hover can be described in terms of the figure of merit M = (T

3/2 /

√

2ρA) /P as a function

of scaled rotor height above the ground, zg /D = zg /2R. Usually the test data are given as the ratio of the

thrust to out-of-ground-effect (OGE) thrust, for constant power: T/T∞ = (M/M∞)

2/3 = κg ≥ 1. The

effect on power at constant thrust is then P = P∞ fg , where fg = κg

3/

2 ≤ 1. Ground effect is generally

negligible at heights above zg

/D = 1 . 5 and at forward speeds above μ = 3λh.

The ground plane is assumed to be perpendicular to the inertial-frame z-axis. The ground normal

(directed downward) is k

F

g = CFI k in airframe axes, or k

S

g = CSFk

F

g in rotor-shaft axes. The height

of the landing gear above ground level, hLG, is specified in the flight state. The height of the rotor hub

above ground level is then

zg = hLG − (k

F

g ) T (zFhub − zFLG ) + dLG

where zFLG is the position of the landing gear in the airframe and dLG is the distance from the bottom of

the gear to the location zLG. From the velocity

S ⎛

μx

v = −rμy

−λ

the angle c between the ground normal and the rotor wake is evaluated: cos c = (k

S

g )

T

v

S

/|v

S

| (c = 0 for

hover, c = 90 degree in forward flight). Note that if the rotor shaft is vertical, then cos c = λ/ μ2 + λ2

(see

ref. 1). The expressions for ground effect in hover are generalized to forward flight by using ( zg / cos c)

in place of zg . No ground-effect correction is applied if the wake is directed upward (cos c ≤ 0), or if

zg/ cos c > 1 . 5D. From zg/D cos c, the ground-effect factor fg = κg

3/

2 is calculated. Then

(λi)IGE = fg (λi) OGE

is the effective ideal induced velocity.

Several empirical ground-effect models are implemented: from Cheeseman and Bennett (ref. 1,

basic model and using blade-element (BE) theory to incorporate influence of thrust); from Law (ref. 2);

from Hayden (ref. 3); and a curve fit of the interpolation from Zbrozek (ref. 4):
88 Rotor

1

1

3/2

(4zg /R)

2

]

1

1+1.5 σaλi 1

3/2

4CT (4zg /R)

2

Cheeseman and Bennett

Cheeseman and Bennett (BE)

fg =

1.0991 − 0.1042/(zg /D)

1 3/2

1 + (CT/σ)(0.2894 − 0.3913/(zg/D))]

rr 0.03794 1

LLL L

(zg/2R)

2

−3/2

0.0544

0.9122+ (

g

/ ) ^

z R CT/σ

Law

Hayden

Zbrozek

These equations break down at small height above the ground, and so are restricted to zg

/D ≥ 0. 15;

however, the database for ground effect extends only to about z/D = 0.3. Also, fg ≤ 1 is required.

Figure 11-2 shows T/T∞ = κg = f9

2/

3 as a function of z/R for these models (CT/σ = 0. 05, 0. 10, 0. 15),

compared with test data from several sources.

11-4.1.3 Inflow Gradient

As a simple approximation to nonuniform induced velocity distribution, a linear variation over the

disk is used: Δλ = λxr cos ψ + λyr sin ψ. There are contributions to Δλ from forward flight and from

hub moments, which influence the relationship between flapping and cyclic. The linear inflow variation

caused by forward flight is Δλf = λi(κxr cos ψ + κyr sin ψ), where λi

is the mean inflow. Typically κx

is positive, and roughly 1 at high speed; and κy is smaller in magnitude and negative. Both κx and κy

must be zero in hover. Based on references 5–8, the following models are considered:

Coleman and Feingold: κx0 = fx

15π

tan χ/2= fx

15π μ

32 32 7

μ2

 + λ2 + |λ|

κy0 = −fy 2μ

White and Blake: κx0 = fx

√

2- sin χ = fx 7

μ

μ2 + λ2

κy0 = −2fyμ

where tan χ = |λ|/μ is the wake angle. Extending these results to include sideward velocity gives

κx = (κx0μx + κy0μy )/μ and κy = (−κx0μy + κy0μx)/μ. For flexibility, the empirical factors fx and fy

have been introduced (values of 1.0 give the baseline model). There is also an inflow variation produced

by any net aerodynamic moment on the rotor disk, which can be evaluated using a differential form of

momentum theory:

Δλm = 

fm

7

μ

(−2CMyr cos ψ + 2CMxr sin ψ) = λxmr cos ψ + λymr sin ψ

2 +=λ=2

including empirical factor fm . Note that the denominator of the hub-moment term is zero for a hovering

rotor at zero thrust; so this inflow contribution should not be used for cases of low speed and low thrust.
1.3

1.2

e

^ 1.1 H

1.0

Rotor 89

• Hayden Cheeseman & Bennett

n Rabbott Cheeseman & Bennett (BE)

0.9 L

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

z/R

Figure 11-2. Ground-effect models (hover).

11-4.2 Rotor Forces

Direct control of the rotor thrust magnitude is used, so the rotor collective pitch angle Bo must be

calculated from the thrust CT/u. If the commanded variable were the collective pitch angle, then it

would be necessary to calculate the rotor thrust, resulting in a more complicated solution procedure;

in particular, an iteration between thrust and inflow would be needed. There may be flight states

where the commanded thrust can not be produced by the rotor, even with stall neglected in the section

aerodynamics. This condition will manifest as an inability to solve for the collective pitch given the

thrust. In this circumstance the trim method should be changed so the required or specified thrust is an

achievable value.

Cyclic control consists of tip-path plane command, requiring calculation of the rotor cyclic pitch

angles from the flapping; or no-feathering plane command, requiring calculation of the tip-path plane tilt

from the cyclic control angles. The longitudinal tip-path plane tilt is βc (positive forward) and the lateral

tilt is βs (positive toward retreating side). The longitudinal cyclic pitch angle is Bs (positive aft), and the

lateral cyclic pitch angle is Bc (positive toward retreating side). Tip-path plane command is appropriate

for main rotors. For rotors with no cyclic pitch, no-feathering plane command must be used.
90 Rotor

The forces acting on the hub are the thrust T, drag H, and side force Y (positive in z, x, yaxis directions, respectively). The aerodynamic analysis is conducted for a clockwise rotating rotor,

with appropriate sign changes for lateral velocity, flapping, and force. The analysis is conducted in

dimensionless form, based on the actual radius and rotational speed of the flight state. The inplane hub

forces are produced by tilt of the thrust vector with the tip-path plane, plus forces in the tip-path plane,

and profile terms (produced by the blade drag coefficient). The orientation of the tip-path axes relative

to the shaft axes is then CPS = Xrβs

Y−βc. Then

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞

CH 0 CHCHo

CY = C

SP 0 + rCYtpp

tpp

⎠ + ⎝rCYo ⎠

CTCT /C33

P 0 0

The inplane forces relative to the tip-path plane can be neglected, or calculated by blade-element theory.

Note that with tip-path plane command and CHtpp and CYtpp neglected, it is not necessary to solve for

the rotor collective and cyclic pitch angles. In general the inplane forces relative to the tip-path plane

are not zero, and may be significant, as for a rotor with large flap stiffness. Figures 11-3a and b show

respectively the tip-path plane tilt and thrust vector tilt with cyclic pitch control (no-feathering plane tilt)

as functions of flap stiffness (frequency), for several rotor thrust values. The difference between tip-path

plane tilt (fig. 11-3a) and thrust vector tilt (fig. 11-3b) is caused by tilt of the thrust vector relative to the

tip-path plane.

The profile inplane forces can be obtained from simplified equations, or calculated by blade-element

theory. The simplified method uses:

( 

CHo

CYo

J

= 8 cdmeanFH 1 

−

AY /1 

J

where the mean drag coefficient cdmean is from the profile power calculation. The function FH accounts for the increase of the blade-section velocity with rotor edgewise and axial speed: CHo =

f 1

2

σcdU(rsin ψ + μ)dr = f 12 σcd (u

2

T + u

2

R + u

2

P)

1/2

(rsin ψ + μ)dr; so (from Harris)

2π

FH = 4 ((r + μ sin)2 +(μcos)2 + μz

)

1/2 (r sin ψ + μ) dr dψ

2π o 

f1

1

2( 1 3 V2 −1 1 (

ILμz

23 311n r√1+ V2+11 +V l\3μ+4μ (1+V2)2)++4μ J L V

J

with V2 = μ

2

+ μ

2

z

.

11-4.3 Blade-Element Theory

Blade-element theory is the basis for the solution for the collective and cyclic pitch angles (or flap

angles) and evaluation of the rotor inplane hub forces. The section aerodynamics are described by lift

varying linearly with angle-of-attack, cj = c^αα (no stall), and a constant mean drag coefficient cdmean

(from the profile power calculation). The analysis is conducted in dimensionless form (based on density

ρ, rotor rotational speed Ω, and blade radius R of the flight state). So in the following σ, ν, and γ are

for the actual R and Ω, and a = 5. 7 is the lift-curve slope used in the Lock number γ. The blade-section

aerodynamic environment is described by the three components of velocity, from which the yaw and
Rotor 91

1.1

1.0 ^ ----------- CT/6 = 0.14

0.9 - - - - - CT/6 = 0.10

0.8 \` ` - CT/6 = 0.06

0.7 \ \ . CT/6 = 0.02

0.6 \^

U

0.5 \ .

0.4

0.3

0.2

0.1

0.0

1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0

flap frequency v (per-rev)

Figure 11-3a. Tip-path plane tilt with cyclic pitch.

1.1

1.0

0.9

-a

0.8

0.7 - - - - - - - - - -

.4 0.6 75U

0.5

0.4 CT/6 = 0.02

0.3

CT/6=

- - - CT/CF=

6 = 0.06

- 0.10

0.2 CT/6 = 0.14

0.1

0.0

1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0

flap frequency v (per-rev)

Figure 11-3b. Thrust-vector tilt with cyclic pitch.
92 Rotor

inflow angles are obtained, and then the angle-of-attack:

U2

= u2 + u

2

T P

uT = r + μx sin ψ + μy cos ψ

cos Λ = U/ u2

T + u

2

P + u

2

R

uR = μx cos ψ − μy sin ψ

, /^ φ = tan−1

uP/uT

uP = λ + r(β + α ̇ x sin O − &y cos ψ) + uRN

α = θ − φ

In reverse flow (|α| > 90), α ← α − 180signα, and then cj = c^αα still (airfoil tables are not used). The

blade pitch consists of collective, cyclic, twist, and pitch-flap coupling terms. The flap motion is rigid

rotation about a hinge with no offset, and only coning and once-per-revolution terms are considered:

θ = θ0 .75 + θtw + θc cos ψ + θs sin ψ − KPβ

β = β0 + βc cos ψ + βs sin ψ

where KP = tan δ3 . The twist is measured relative to 0 . 75R; θtw = θL (r − 0. 75) for linear twist. The

inflow includes gradients caused by edgewise flight and hub moments:

λ = μz + λi(1 + κxr cos ψ + κy r sin ψ) + Δλm

= μz + λi(1+ κxr cos ψ + κy rsin ψ)+ fm

μ

(−2CMyr cos ψ +2CMxr sin ψ)

2 + λ2

From the hub moments ( 

−CMy

CM) 2 

ν2 

γ 

1 ( 

βs

)

the inflow gradient is

σa

Δλm =

μ

fm

+ λ

2 8 I 

νγ/81 (rβc cos ψ + rβs sin ψ) = Kmνγ/81 (rβc cos ψ + rβs sin ψ)

The constant Km is associated with a lift-deficiency function:

= 1 =

C 1 + Km

1

1 + f mσa/ (8 μ

2 + λ2

)

The blade chord is c (r) = crefcˆ(r), where cref is the thrust-weighted chord (chord at 0 . 75R for linear

taper). Yawed flow effects increase the section drag coefficient, hence cd = cdmean / cos Λ. The section

forces in velocity axes and shaft axes are

L = 2 ρU

2

cc, Fz = L cos φ − D sin φ = 2 ρUc(QuT − cduP)

D = 2 ρU

2

ccd

Fx = L sin φ + D cos φ = 2 ρUc(QuP + cduT )

R = 2 ρU

2

ccr = D tan Λ Fr = −βFz + R = −βFz

+ 2 ρUccduR

These equations for the section environment and section forces are applicable to high inflow (large μz),

sideward flight (μy), and reverse flow (uT < 0). The total forces on the rotor hub are

T = N fFz dr

H = N f Fx sin ψ + Fr cos ψ dr

Y = N

f

−Fx cos ψ + Fr sin ψ dr
Rotor 93

with an average over the rotor azimuth implied, along with the integration over the radius. Lift forces

are integrated from the root cutout rroot to the tip-loss factor B. Drag forces are integrated from the root

cutout to the tip.

In coefficient form (forces divided by ρAV2tip) the rotor thrust and inplane forces are:

CT = σ

J

F$z dr

CH = σ

J

F$x sin ψ + F$r cos ψ dr

CY = σ

J

—Fx cos ψ + Fr sin ψ dr

1

Fz = 1 cUˆ (cMT — cduP)

1

Fx = 2ˆcU(cMP + cduT)

1

Fr = —βFz + 2ˆcUcduR

(and the sign of CY is changed for a clockwise rotating rotor). The terms ΔFx = Fzβ ̇ and ΔFr = —Pzβ

produce tilt of the thrust vector with the tip-path plane ( CH = —CTβc and CY = —CTβs), which

are accounted for directly. The section drag coefficient cd produces the profile inplane forces. The

approximation uP = μz is consistent with the simplified method (using the function FH), hence

J

Fxo = 2 ˆcU0cduT

CHo = σ Fxo sin ψ + Fro cos ψ dr = 2σ

J

cUˆ 0cd(rsin ψ + μx) dr

Fro = 12 ˆcU0cduR

CYo = σ 

J 

—Fxo cos ψ + Fro sin ψ dr = — 2 

J 

cUˆ 0cd(r cos ψ + μy ) dr

where U

2

0 = u

2

T + μ

2

z , and cd = cdmean / cos Λ. Using blade-element theory to evaluate CHo and CYo

accounts for the planform (ˆc) and root cutout. Using the function FH implies a rectangular blade and no

root cutout (plus at most a 1% error approximating the exact integration). The remaining terms in the

section forces produce the inplane loads relative to the tip-path plane:

Fxi = Fx — Fzβ ̇ — Fxo = 

1

2

cUc ˆ ^ (uP — uTβ)  ̇

+ 

1

2

cUc ˆ d ((1 — U0

/U)uT + uPβ) ̇

F$

ri = F$r + Fzβ — _Pro =1

2

cUc ˆ d(1 — U0/U)uR

CHtpp = σ 

J 

F$

xi sin ψ + F$ri cos ψ dr

CYtpp = σ 

J 

—F$

xi cos ψ + F$ri sin ψ dr

(including small profile terms from U0 =^ U).

Evaluating these inplane forces requires the collective and cyclic pitch angles and the flapping

motion. The thrust equation must be solved for the rotor collective pitch. The relationship between

cyclic pitch and flapping is defined by the rotor-flap dynamics. The flap motion is rigid rotation about a

central hinge, with a flap frequency ν > 1 for articulated or hingeless rotors. The flapping equation of

motion is /

+ ν

2

β + MY sin ψ + M.x cos ψ = 

a 

J Fz r dr + (ν

2

including precone angle βp ; the Lock number γ = ρacrefR

4 /Ib . This equation is solved for the mean

(coning) and 1/rev (tip-path plane tilt) flap motion:

ν

2β0 = 

a 

J 

Fzrdr + (ν

2

 — 1)βp

(ν

2

 — 1) 

C A) 

= a 

J 

Fzrdr 1 2sin ψ + C

2α ̇ y
94 Rotor

with an average over the rotor azimuth implied. The solution for the coning is largely decoupled by

introducing the thrust:

ν02β0 = 6CT

8 σa 

+ (ν0 − 1)βp + a f F,Z (r − 3/4) dr

A separate flap frequency ν0 is used for coning, in order to model teetering and gimballed rotors. For

an articulated rotor, βp = 0 should be used.

The thrust and flapping equations of motion that must be solved are:

f

6CT

Et = 6

a

_P,Z dr − 

σa

(E./ 8 2cos ψ ν2 − 1 β 16 E = a f F,Zr dr (2 sin 

ψ γ/8

(

β3

/ + γ 

(

α ̇ v

The solution v such that E(v) = 0 is required. For tip-path plane command, the thrust and flapping are

known, so v = (θ0.75 θ, θ3 )

T . For no-feathering plane command, the thrust and cyclic pitch are known, so

v = (θ0.75 β,β3 )

T . Note that since cj = c^aα is used (no stall), these equations are linear in θ. However,

if ∂T/ ∂θ0.75 is small, the solution may not produce a reasonable collective. A Newton–Raphson solution

method is used: from E(v,,,+1) ∼= E(v,,, ) + (dE/dv)(v,,+1 − v,,, ) = 0, the iterative solution is

v,,,+1 = v,,, − C E(v,,)

where C = f (dE/dv)

−1

, including the relaxation factor f . The derivative matrix can be estimated from

δα = δθ − (uT

/U2 )δuP and δU = (uP/U)δuP

, hence

δP,Z = 2ˆcQauT (αδU + Uδα) ∼= 2ˆcciaUuTδα = 2ˆccia

(

UuTδθ − (u

2

T/U)δuP

)

with

δθ = θ0 .75 + θ, cos ψ + θ3 sin ψ − Kp (β, cos ψ + β3 sin ψ)

δuP = δλ + rδβ ̇ + uRδβ

= 

(

K,,,

(

νγ/81 ) r cos ψ − r sin ψ + uR cos ψ

/

β, 

(

+ K,,,,

(

νγ/81 ) r sin ψ + r cos ψ + uR sin ψ

/

β3

For hover δF,Z = 12 ccˆ ^a (r

2

δθ − rδuP), and the derivatives are easily evaluated. Including axial flight gives

δEt = e0θ0 .75 (

δE3

/

= 

(

θ3

θ,

/

−

n J 

(

β3

/

where

2 

−

2 

−

n = 

ν

γ/8

1

(1 + e0

K,,,,) + eoKp = Cγ/ 8 + eoKp

The constants are functions of μ,Z :

a e

0 = 3 f UuT dr = 3 f r r2 + μ

2

,Z dr =(1 + μ

2

,Z )

3/2

−|μ,Z |

3

c^a

a e

o = 4 f UuT

r dr = 4 f r

2 r2 + μ

2

,Z dr = 1 + μz (1 + 2μz ) − 2

μ

4

ln

cPa

1 + 1 + μz

(

|μ,Z |

c « ep = 4 f (u

2

T/U)r

2

 dr = 4 f 

r2 + 2 

dr = 1 + μz (1 − 2μz ) + 32 μ

4

ln I 1 + |μz T—+ μz

μ,Z \

|
Rotor 95

(all equal to 1 for μz = 0). Hence

(θ0.75 )n+1 = (θ0.75 )n − 

f E

t

(

θs

θc

)

n

−

e0

(

c

l( θθ

c

)

n+1 s —

eθ \

Es /

for tip-path plane command; or

(θ0.75 ) n+1 = (θ0 .75 )n − 

f E

t

e0

(

βc

)

n+1 = 

(

βs ) 

+

β

l

e

2

β + n2

p n J 

(

Ec

)

Es n

for no-feathering plane command. Alternatively, the derivative matrix dE/dv can be obtained by numerical perturbation. Convergence of the Newton–Raphson iteration is tested in terms of |E| < a for

each equation, where a is an input tolerance.

11–5 Power

The rotor power consists of induced, profile, and parasite terms: P = Pi + Po + Pp . The parasite

power (including climb/descent power for the aircraft) is obtained from the wind-axis drag force: Pp =

−XV = (v

F)TFF .

The induced power is calculated from the ideal power: Pi = κPideal = κ fDTvideal. The empirical

factor κ accounts for the effects of nonuniform inflow, non-ideal span loading, tip losses, swirl, blockage,

and other phenomena that increase the induced power losses ( κ > 1). For a ducted fan, fD = fW/2 is

introduced. The induced power at zero thrust is zero in this model (or accounted for as a profile power

increment). If κ is deduced from an independent calculation of induced power, nonzero Pi at low thrust

will be reflected in large κ values.

The profile power is calculated from a mean blade drag coefficient: Po = ρA(ΩR)

3CPo , CPo =

(σ/8)cdmeanFP

. The function FP

(μ, μz) accounts for the increase of the blade section velocity with rotor

edgewise and axial speed: CPo = f 1

2 σcdU3dr = f 12 σcd

(u

2

T + u

2R + u2

P ) 3/2 dr; so (from Harris)

2π 1

FP = 4 1 (

(r + μsin + (μ cos ψ)

2 + μz ) 

3/2 dr d 2π 0 0

∼ 1+ V2 1+5V

 3 2 2 4+7V

2 +4V4 9 μ4

1

C 2

+

8

μ

 (1+ V2 )

2

161+ V2

/

(

2

μz

3 4 3 2 2 9 4) f

√

1+ V2 +1 1

 + 2 μzμ+16 μ 

ln

V J

with V2 = μ

2

+ μ

2

z . This expression is exact when μ = 0, and fP ∼ 4V3 for large V.

Two performance methods are implemented, the energy method and the table method. The induced

power factor and mean blade drag coefficient are obtained from equations with the energy method, or

from tables with the table method. Optionally κ and cdmean can be specified for each flight state,

superseding the values from the performance method.
96 Rotor

11-5.1 Energy Performance Method

11-5.1.1Induced Power

The induced power is calculated from the ideal power: Pi = κPideal = κfDTvideal. Reference

values of κ are specified for hover, axial cruise (propeller), and edgewise cruise (helicopter): κhover,

κprop, κedge. Two models are implemented: constant model and standard model. The constant model

uses κ = κhover if μ = μz = 0; or κ = κprop if IμI < 0.1 Iμz

I; or κ = κedge otherwise.

The standard model calculates an axial-flow factor κaxial from κhover, κclimb, and κprop . Let Δ =

CT /σ - (CT/σ)ind. For hover and low-speed axial climb, including a variation with thrust, the inflow

factor is

κh = κhover + kh1Δh + kh2Δ

2

h + (κclimb - κhover) — tan−1 [((Iμz

I/λh)/Maxial)

X8xi8l

]

π

where Iμz

I/λh = Maxial is the midpoint of the transition between hover and climb and Xaxial is large

for a fast transition. Figure 11-4 illustrates κ in hover (with a minimum value). Figure 11-5 shows the

behavior of this function for a helicopter in climb ( Xaxial = 0. 65). A polynomial describes the variation

with axial velocity, scaled so κ = κh at μz = 0 and κ = κp at μz = μzprop, including a variation with

thrust:

κp = κprop + kp 1Δp + kp2Δ

2

p

κaxial = κh + ka1μz + S(ka2μ

2

z + ka3μz

 a)

where S = (κp - (κh +ka1μzprop))/(ka2μzprop+ka3μzpiop) ; S = 0if ka2 = ka3 = 0 (not scaled); κaxial = κh

if μzprop = 0. A polynomial describes the variation with edgewise advance ratio, scaled so κ = κaxial at

μ = 0 and κ = foff κedge at μ = μedge . Thus the induced power factor is

κ = κaxial + ke1μ + S(ke2μ

2 + ke3μX

e)

where S = (foff κedge - (κaxial + ke1μedge))/(ke2μ

2

edge + ke3μ

Xe

edge )

; S = 0 if ke2 = ke3 = 0 (not scaled);

κ = κaxial if μedge = 0. The function foff = 1- ko1 (1 - e

−ko2

ox) accounts for the influence of lift offset,

ox = rMx

/TR = (Khub /TR)βs

. Figure 11-6 illustrates κ in edgewise flight. Minimum and maximum

values of the induced power factor, κmin and κmax, respectively, are also specified.

11-5.1.2 Profile Power

The profile power is calculated from a mean blade drag coefficient: CPo = (σ/8) cdmeanFP . Since the

blade mean lift coefficient is cj = 6CT/σ, the drag coefficient is estimated as a function of blade loading

CT/σ (using thrust-weighted solidity). With separate estimates of the basic, stall, and compressibility

drag, the mean drag coefficient is:

cdmean = χS (cdbasic + cdstall + cdcomp)

where χ is a technology factor. The factor S = (Reref/Re) 0 . 2 accounts for Reynolds number effects on

the drag coefficient; Re is based on the thrust-weighted chord, 0 . 75Vtip , and the flight state; and Reref

corresponds to the input cd information. The following models are implemented for the basic drag:

a) Array model: The basic drag cdbasic is input as a function of CT/σ; the array is linearly interpolated.

b) Equation model: The basic drag cdbasic is a quadratic function of CT/σ, plus an additional term

allowing faster growth at high (sub-stall) angles of attack. Let Δ = ICT/σ - (CT/σ)Dmin I , where
0.03 0.06 0.09 0.12 0.15 0.18

1.30

1.25

1.20

1.15

1.10

1.05

1.00

0.00

Rotor 97

CT/6

Figure 11-4. Induced power factor for rotor in hover.

(CT/σ)Dmin corresponds to the minimum drag and Δsep = ICT/σI

 - 

(CT/σ)sep . Values of the basic

drag equation are specifed for helicopter (hover and edgewise) and propeller (axial climb and cruise)

operation:

cdh = d0hel + d1helΔ + d2helΔ

2

+ dsepΔsep

p

= O p — 

d p roP + d l P Pro 0 + d2P Pro 02 + dse P Osep

X

s

cd

The separation term is present only if Δsep > 0. The helicopter and propeller values are interpolated as

a function of μz

:

cdbasic = cdh + (cdp - cdh) 2 tan−1(IμzI/λh)

π

so Iμz

I/λh = 1 is the midpoint of the transition.

The stall drag increment represents the rise of profile power caused by the occurrence of significant

stall on the rotor disk. Let Δs = ICT/σI - (fs/foff) (CT/σ)s (fs

 is an input factor). The function

foff = 1- do 1 (1- e

−do2 ox ) accounts for the influence of lift offset, ox = rMx

/TR = (Khub /TR)βs

. Then

cdstall = ds1Δ3 

s

1 + ds2Δ

Xs2

s

(zero if Δs

< 0). The blade loading at which the stall affects the entire rotor

power, (CT/σ)s

, is an input function of the velocity ratio V = Vμ

2 + μ2

z

.

The compressibility drag increment depends on the advancing tip Mach number Mat, and the tip

airfoil thickness-to-chord ratio τ. The following models are implemented:

a) Drag divergence model: Let ΔM = Mat - Mdd, where Mdd is the drag divergence Mach number of

the tip section. Then the compressibility increment in the mean drag coefficient is

cdcomp = d m1ΔM + d m2ΔM

X m

(ref. 9). Mdd is a function of the advancing tip lift coefficient, c^ (1,90). The advancing tip lift is estimated

from α(1,90) = (θ.75 +0.25θL + θs-(λ -βc)/(1+ μ)) = 1.6(1 - 2.97μ+2.21μ

2

)(6CT/σa)+0.25θtw (zero

above μ = 0. 6). Then the Korn expression (ref. 10) gives Mdd for small lift coefficient:

Mdd = κA - κIc^I- τ = Mdd0 - κIc^I
4.50

4.00

3.50

3.00 k

2.50

2.00

1.50

1.00

0.00 0.10 0.20 0.30 0.40 0.50

98 Rotor

1.14

1.12

1.10 \^

-_ _--------

1.08

1.06 Maxial = 1.176

1.04 - Maxial = 0.5

1.02 Maxial = 2.0

1.00

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

fez/Ah

Figure 11-5. Induced power factor for rotor in axial flight.

μ

Figure 11-6. Induced power factor for rotor in edgewise flight.

where Mdd0 is the drag divergence Mach number at zero lift, and typically κ = 0. 16.

b) Similarity model: From transonic small-disturbance theory (refs. 11–12), the scaled wave drag must

be a function only of K1 = (M

2

at - 1)/ [M2

atτ(1 + γ)] 2/3

. An approximation for the wave drag increment

is

τ

5/3

τ

5/3 /

Δcd = [M2

at(1 + γ)] 1/3

D(K1)=

[Mat(1 + γ)]

1/

3 

1.774(K1 + 1.674)5 2

(constant for K1 > -0. 2). Integration of Δcd over the rotor disk gives the compressibility increment in

the profile power. Following Harris, the resulting compressibility increment in the mean drag coefficient
Rotor 99

- - transient limit

0.20 _ _

- steady limit

high stall

0.16 - _

low stall

0.12 \

U

0.08

0.04

0.00

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70

μ

Figure 11-7. Stall function.

0.0350

0.0300

0.0250

0.0200

0.0150

0.0100

0.0050

0.0000

0.00

cd mean (low stall)

cd mean (high stall)

- - - - - cdh (with separation)

- - - - - - - - - - - cdh (quadratic)

0.03 0.06 0.09 0.12 0.15 0.18

CT/6

Figure 11-8. Mean drag coefficient for rotor in hover.

is approximately:

cdcomp = 1.52f (K1 + 1) 2 [(1 + μ)τ]

5/2

(1 + γ)

1/2

including the input correction factor f; cdcomp is zero for K1 < - 1, and constant for K1 > -0. 2.

Figure 11-7 shows typical stall functions (CT/σ)s for two rotors with different airfoils, and for

reference typical helicopter rotor steady and transient load limits. Figure 11-8 illustrates the mean drag
100 Rotor

CT/6= 0.14 0.0350

• - • • CT

/6= 0.12

0.0300 CT

/6= 0.10

CT/6= 0.08 0.0250

j

cd comp

0.0200

0.0150

/•

,,

0.0100 - • - - ... .. ^•^

0.0050

0.0000 - - - -

0.00 0.10 0.20 0.30 0.40 0.50

μ

Figure 11-9a. Mean drag coefficient for rotor in forward flight, high stall.

0.0350 /

0.0300 /

0.0250 / /• '

0.0200

0.0150

0.0100 . .

0.0050

0.0000 - - - -

0.00 0.10 0.20 0.30 0.40 0.50

μ

Figure 11-9b. Mean drag coefficient for rotor in forward flight, low stall.

coefficient in hover, showing Cdh without and with the separation term, and the total for the high stall

and low stall cases. Figure 11-9 illustrates the mean drag coefficient in forward flight, showing the

compressibility term Cdcomp, and the growth in profile power with CT/u and μ as the stall drag increment

increases.
Rotor 101

11-5.1.3 Twin Rotors

For twin rotors, the induced power is determined by the induced velocity of the rotor system, not

the individual rotors. The induced power is still obtained using Pi = κPideal = κ fDTvideal for each rotor,

but the ideal induced velocity is calculated for an equivalent thrust CTe based on the thrust and geometry

of both rotors. The profile power calculation is not changed for twin rotors.

In hover, the twin-rotor induced velocity is vi = κtwin T/2ρAp , from the total thrust T and the

projected disk area Ap = (2 — m) A. The overlap fraction m is calculated from the rotor hub separation

t. A correction factor for the twin-rotor ideal power is also included. For a coaxial rotor, typically

κtwin = 0.90. So the ideal inflow is calculated for CTe = (CT1 + CT2 )/(2 — m).

In forward flight, the induced velocity of a coaxial rotor is vi = κtwinT/(2ρAV), from the total

thrust T and a span of 2R. The correction factor for ideal induced power (biplane effect) is κtwin = 0. 88

to 0. 81 for rotor separations of 0. 06D to 0. 12D. The ideal inflow is thus calculated for CTe = CT1 + CT2 .

The induced velocity of side-by-side rotors is vi = κtwinT/(2ρAeV), from the total thrust T and a span of

2R+ t, hence Ae = A(1+t/2R) 2 . The ideal inflow is thus calculated for CTe = (CT1 + CT2 ) / (1 + t/2R)

2

.

The induced velocity of tandem rotors is vF = κtwin (TF/ (2ρAV) + xRTR/ (2ρAV)) for the front rotor and

vR = κtwin (TR/(2ρAV) + xFTF/(2ρAV)) for the rear rotor. For large separation, xR = 0 and xF = 2;

for the coaxial limit xR = xF = 1 is appropriate. Here xR = m and xF = 2 — m is used.

To summarize, the model for twin-rotor ideal induced velocity uses CTe = x1CT1 + x2CT2 and the

correction factor κtwin. In hover, xh = 1/(2 — m); in forward flight of coaxial and tandem rotors, x f = 1

for this rotor and x f = m or x f = 2 — m for the other rotor; in forward flight of side-by-side rotors,

x f = 1/(1 + t/2R)

2 (x = 1/2 if there is no overlap, t/2R > 1). The transition between hover and forward

flight is accomplished using

x = 

x fμ

2 + xhCλ2

h

μ2 + Cλ

2

h

with typically C = 1 to 4. This transition is applied to x for both rotors, and to κtwin.

With a coaxial rotor in hover, the lower rotor acts in the contracted wake of the upper rotor.

Momentum theory gives the ideal induced power for coaxial rotors with large vertical separation (ref.13):

Pu = Tuvu , v2

u = Tu

/2ρA for the upper rotor; and PP = (α ̄s/Vf

τ)TP vP , v2

P = TP /2ρA for the lower rotor.

Here τ = TP /Tu

; α ̄ is the average of the disk loading weighted by the induced velocity, hence a measure

of nonuniform loading on the lower rotor (α ̄ = 1 . 05 to 1 . 10 typically); and the momentum theory solution

is

α ̄s = 1 1

2τ

3/

2 \ + 4(1 + τ)2 ̄ατ — 1)

The optimum solution for equal power of the upper and lower rotors is α ̄sτ = 1, giving τ = TP /Tu = 2/3.

Hence for the coaxial rotor in hover the ideal induced velocity is calculated from CTe = CTu for the

upper rotor and from CTe = (α ̄s/Vf

τ)

2CTP for the lower rotor, with κtwin = 1. Thus xh = 1/(2 — m) = 1/2

and the input hover κtwin is not used, unless the coaxial rotor is modeled as a tandem rotor with zero

longitudinal separation.

11-5.2 Table Performance Method

The induced power is calculated from the ideal power: Pi = κPideal = κfDTvideal. The induced

power factor κ is obtained from an input table (linearly interpolated) that can be a function of edgewise

advance ratio μ or axial velocity ratio μz , and of blade loading CT/σ.
102 Rotor

The profile power is calculated from a mean blade drag coefficient: Po = ρA(ΩR)

3CPo =

ρA(ΩR)

3 σ

8 cdFP . The mean drag coefficient cd , or alternatively cdFP = 8CPo/σ, is obtained from

an input table (linearly interpolated) that can be a function of edgewise advance ratio μ and blade

loading CT/σ.

11–6 Performance Indices

Several performance indices are calculated for each rotor. The induced power factor is κ = Pi

 /Pideal.

The rotor mean drag coefficient is cd = (8CPo/σ)/FP , using the function F(μ, μz ) given previously. The

rotor effective lift-to-drag ratio is a measure of the induced and profile power: L/De = VL/(Pi + Po ).

The hover figure of merit is M = T fDv/P. The propeller propulsive efficiency is η = TV/P. These two

indices can be combined as a momentum efficiency: ηmom = T(V +w/2) /P, where w/2 = fWv/2 = fD v.

11–7 Interference

The rotor can produce aerodynamic interference velocities at the other components (fuselage, wings,

tails). The induced velocity at the rotor disk is κvi, acting opposite the thrust ( z-axis of tip-path plane

axes). So v

P

ind = -kPκvi

, and v F

ind = CFPv P

ind. The total velocity of the rotor disk relative to the air

consists of the aircraft velocity and the induced velocity from this rotor: v

F

total = vF - vFind. The direction

of the wake axis is thus e

P

w = -CPFv

F

total/ vFtotal (for zero total velocity, e

P

w = -kP is used). The angle

of the wake axis from the thrust axis is χ = cos- 1 (kP)Te

P

w .

The interference velocity v F

int at each component is proportional to the induced velocity v F

ind (hence

is in the same direction), with factors accounting for the stage of wake development and the position of

the component relative to the rotor wake. The far-wake velocity is w = fWvi

, and the contracted wake

area is Ac = πR

2

c = A/ fA . The solution for the ideal inflow gives fW and fA . For an open rotor, fW = 2.

For a ducted rotor, the inflow and wake depend on the wake area ratio fA , or on the ratio of the rotor

thrust to total thrust: fT = Trotor/T. The corresponding velocity and area ratios at an arbitrary point on

the wake axis are fw and fa

, related by

μ2 + (μz + fw λi)

2

fa = (fVxμ)

2

 + (fVzμz + λi)

2

Vortex theory for hover gives the variation of the induced velocity with distance z below the rotor disk:

^ ^

1 +

z/R

v = v (0) ^

1+(z/R)

2

With this equation the velocity varies from zero far above the disk to v = 2v (0) far below the disk. To

use this expression in edgewise flow and for ducted rotors, the distance z/R is replaced by ζw /tR, where

ζw is the distance along the wake axis, and the parameter t is introduced to adjust the rate of change (t

small for faster transition to far-wake limit). Hence the velocity inside the wake is fw vi , where

⎧

⎪ ζw /tR

⎪⎪ ^

⎨⎪ 1 + 1 + (ζw /tR)

2

ζw < 0

fw = fW fz =

⎪⎪⎪⎪ 1 + (fW - 1) ζw/tR

⎩^

ζw > 0

1 + (ζw /tR)

2

and the contracted radius is Rc = R/v'fa

.
Rotor 103

The wake is a skewed cylinder, starting at the rotor disk and with the axis oriented by e

P

w . The

interference velocity is required at the position z

F

B on a component. Whether this point is inside or

outside the wake cylinder is determined by finding its distance from the wake axis, in a plane parallel to

the rotor disk. The position relative to the rotor hub is ξ

P

B = CPF (z

F

B — z F

hub); the corresponding point on

the wake axis is ξ

P

A = e

P

w ζw . Requiring ξ

P

B and ξ

P

A have the same z value in the tip-path plane axes gives

ζw = (k

P)TCPF (zF

B — z

F

hub)

(k

P

)

TeP

\\ w

from which fz

, fw , fa, and Rc are evaluated. The distance r from the wake axis is then

r2 = ((iP) T

(ξ

P

B — ξ PA )

) 2

+ 

(

(j

P)T

 (ξ

P

B — ξPA )

) 2

The transition from full velocity inside the wake to zero velocity outside the wake is accomplished in

the distance sRc , using

⎧

⎨ 1 r < Rc

fr = 1 — (r — Rc )/(sRc

)

⎩

0 r > (1 + s)Rc

(s = 0 for an abrupt transition, s large for always in wake).

The interference velocity at the component (at z

F

B) is calculated from the induced velocity v F

ind,

the factors fW fz accounting for axial development of the wake velocity, the factor fr accounting for

immersion in the wake, and an input empirical factor Kint:

vFint = Kint fW fz frft v

F

ind

An additional factor ft for twin rotors is included. Optionally the development along the wake axis

can be a step function (fWfz = 0, 1, fW above the rotor, on the rotor disk, and below the rotor disk,

respectively); nominal (t = 1); or use an input rate parameter t. Optionally the wake immersion can use

the contracted radius Rc or the uncontracted radius R; can be a step function (s = 0, so fr = 1 and 0

inside and outside the wake boundary); can be always immersed ( s = oo so fr = 1 always); or can use

an input transition distance s. Optionally the interference factor Kint can be reduced from an input value

at low speed to zero at high speed, with linear variation over a specified speed range.

To account for the extent of the wing or tail area immersed in the rotor wake, the interference

velocity can be calculated at several of points along the span and averaged. The increment in position

is Δz

F

B = C

FB (0Δy 0)T ;where Δy = (b/2)(—1 + (2i — 1) /N) for i = 1 to N, and b is the wing span.

For twin main rotors (tandem, side-by-side, or coaxial), the performance may be calculated for

the rotor system, but the interference velocity is still calculated separately for each rotor, based on

its disk loading. At the component, the velocities from all rotors are summed, and the total used to

calculate the angle-of-attack and dynamic pressure. This sum must give the interference velocity of

the twin-rotor system, which requires the correction factor ft . Consider differential momentum theory

to estimate the induced velocity of twin rotors in hover. For the first rotor, the thrust and area in the

non-overlap region are (1— m)T1 and (1— m)A, hence the induced velocity is v1 = κ T1 /2ρA; similarly

v2 = κ T2 /2ρA. In the overlap region the thrust and area are mT1 + mT2 and mA, hence the induced

velocity is vm = κ _(T1 + T2 )/2ρA. So for equal thrust, the velocity in the overlap region (everywhere

for the coaxial configuration) is 

v/

2 larger. The factor KT is introduced to adjust the overlap velocity:
104 Rotor

vm = κ(KT/√2) (T1 + T2 )/2ρA. The interference velocities are calculated separately for the two

rotors, with the correction factor ft: vint1 = ftκ T1 /2ρA and vint2 = ftκ T2 /2ρA. The sum vint1 +vint2

must take the required value. Below the non-overlap region, the component is in the wake of only one

of the rotors, so the interference velocity from the other rotor is zero, and thus ft = 1. Below the overlap

region, the component is in the wake of both rotors, and the sum of the interference velocities equals vm

if

fth = 

KT /√2

√τ1 + √τ2

where τn = Tn / (T1 + T2 ) is the thrust ratio. For equal thrusts, fth = KT/2; or fth = 1/√2 for the

nominal velocity. The expression ft = fth cos2 χ+sin2

χ gives the required correction factor, with ft = 1

in edgewise flight. Optionally the correction for twin rotors can be omitted ( ft = 1); nominal (KT = √2);

or use an input velocity factor in overlap region ( KT).

11–8 Drag

The rotor component includes drag forces acting on the hub and spinner (at z F

hub) and on the pylon

(at z F

pylon). The component drag contributions must be consistent. In particular, a rotor with a spinner

(such as on a tiltrotor aircraft) would likely not have hub drag. The pylon is the rotor support and the

nacelle is the engine support. The drag model for a tiltrotor aircraft with tilting engines would use the

pylon drag (and no nacelle drag), since the pylon is connected to the rotor shaft axes; with non-tilting

engines it would use the nacelle drag as well.

The body axes for the drag analysis are rotated about the y-axis relative to the rotor shaft axes:

CBF = CBSCSF , where CBS = Y−θref . The pitch angle θref can be input, or the rotation appropriate for

a helicopter rotor or a propeller can be specified.

a) Consider a helicopter rotor, with the shaft axes oriented z-axis up and x-axis

downstream. It is appropriate that the angle-of-attack is α = 0 for forward flight,

and α = −90 degree for hover, meaning that the body axes are oriented z-axis down

and x-axis forward. Hence θref = 180 degree.

b) Consider a propeller or tiltrotor, with the shaft axes oriented z-axis forward and

x-axis up. It is appropriate that the angle-of-attack is α = 0 in cruise and α = 90

degree for helicopter mode (with a tilting pylon), meaning that the body axes are

oriented z-axis down and x-axis forward. Hence θref = 90 degree.

The aerodynamic velocity relative to the air is calculated in component axes, v

B . The angle-of-attack

α and dynamic pressure q are calculated from v

B . The reference areas for the drag coefficients are the

rotor disk area A = πR

2 (for hub drag), pylon wetted area Spylon, and spinner wetted area Sspin; these

areas are input or calculated as described previously.

The hub drag can be fixed, specified as a drag area D/q; or the drag can be scaled, specified as a

drag coefficient CD based on the rotor disk area A = πR

2 ; or the drag can be estimated based on the

gross weight, using a squared-cubed relation or a square-root relation. Based on historical data, the drag

coefficient CD = 0.004 for typical hubs, CD = 0. 0024 for current low-drag hubs, and CD = 0. 0015 for

faired hubs. For the squared-cubed relation: ( D/q) hub = k(WMTO/1000) 2/3 (WMTO is the maximum

takeoff gross weight; units of k are feet2 /kilopound2

/

3 or meter2 /Megagram2

/3 ). Based on historical data,

k = 1 . 4 for typical hubs, k = 0. 8 for current low-drag hubs, and k = 0. 5 for faired hubs (English units).

For the square-root relation: (D/q) hub = k WMTO/Nrotor (WMTO/Nrotor is the maximum takeoff gross
Rotor 105

weight per lifting rotor; units of k are feet 2/pound1/2 or meter2/kilogram1/2); based on historical data

(ref. 14), k = 0. 074 for single-rotor helicopters, k = 0. 049 for tandem-rotor helicopters (probably a blade

number effect), k = 0. 038 for hingeless rotors, and k = 0. 027 for faired hubs (English units).

The hub vertical drag can be fixed, specified as a drag area D/q; or the drag can be scaled, specified

as a drag coefficient CD based on the rotor disk area A = πR

2

.

The pylon forward-flight drag and vertical drag are specified as drag area or drag coefficient, based

on the pylon wetted area. The spinner drag is specified as drag area or drag coefficient, based on the

spinner wetted area.

The drag coefficient for the hub or pylon at angle-of-attack α is

CD = CD0 + (CDV — CD0 ) l sin αl

Xd

Optionally the variation can be quadratic ( Xd = 2). For sideward flight, CDhub = CD0 for the hub and

CDpylon = CDV for the pylon. Then the total component drag force is

D = qACDhub + qSpylonCDpylon + qSspinCDspin

The force and moment produced by the drag are

F

F

= E edD

MF = E LFF

F

where ΔzF = zF — z Fcg (separate locations are defined for the rotor hub and for the pylon), and ed is the

drag direction. The velocity relative to the air gives ed = —v

F

/lv Fl (no interference).

11–9 Weights

The rotor configuration determines where the weights occur in the weight statement, as summarized

in table 11-3. The rotor group consists of blade assembly, hub and hinge, fairing/spinner, and blade-fold

structure. The tail rotor (in empennage group) or the propeller/fan installation (in propulsion group)

consists of the blade assembly, the hub and hinge, and the rotor/fan duct and rotor support.

There are separate weight models for main rotors, tail rotors, and auxiliary-thrust systems (propellers). The tail-rotor model requires a torque calculated from the drive-system rated power and mainrotor rotational speed: Q = PDSlimit/Ωmr. The auxiliary-thrust model requires the design maximum

thrust of the propeller.

Table 11-3. Principal configuration designation.

configuration weight statement weight model performance model

main rotor rotor group rotor rotor

tail rotor empennage group tail rotor rotor

propeller propulsion group rotor, aux thrust rotor
106 Rotor

11–10 References

1) Cheeseman, I.C., and Bennett, W.E.: The Effect of the Ground on a Helicopter Rotor in Forward

Flight. ARC R&M 3021, September 1955.

2) Law, H.Y.H.: Two Methods of Prediction of Hovering Performance. USAAVSCOM TR 72-4,

February 1972.

3)Hayden, J.S .: The Effect of the Ground on Helicopter Hovering Power Required. American Helicopter

Society 32nd Annual National V/STOL Forum, Washington, D.C., May 1976.

4) Zbrozek, J.: Ground Effect on the Lifting Rotor. ARC R&M 2347, July 1947.

5) Coleman, R.P.; Feingold, A.M.; and Stempin, C.W.: Evaluation of the Induced-Velocity Field of an

Idealized Helicopter Rotor. NACA ARR L5E10, June 1945.

6) Mangler, K.W.; and Squire, H.B.: The Induced Velocity Field of a Rotor. ARC R & M 2642, May

1950.

7) Drees, J.M.: A Theory of Airflow Through Rotors and Its Application to Some Helicopter Problems.

Journal of the Helicopter Association of Great Britain, vol. 3, no. 2, July–September 1949.

8) White, T.; and Blake, B.B.: Improved Method of Predicting Helicopter Control Response and Gust

Sensitivity. Annual National Forum of the American Helicopter Society, May 1979.

9) Gessow, A.; and Crim, A.D.: A Theoretical Estimate of the Effects of Compressibility on the

Performance of a Helicopter Rotor in Various Flight Conditions. NACA TN 3798, October 1956.

10) Mason, W.H. Analytic Models for Technology Integration in Aircraft Design. AIAA Paper No.

90-3262, September 1990.

11) Ashley, H.; and Landahl, M.: Aerodynamics of Wings and Bodies. Addison-Wesley Publishing

Company, Inc., Reading, Massachusetts, 1965.

12) Spreiter, J.R.; and Alksne, A.Y.: Thin Airfoil Theory Based on Approximate Solution of the

Transonic Flow Equation. NACA Report 1359, 1958.

13) Johnson, W.: Influence of Lift Offset on Rotorcraft Performance. American Helicopter Society

Specialist’s Conference on Aeromechanics, San Francisco, California, January 2008.

14) Keys, C.N.; and Rosenstein, H.J.: Summary of Rotor Hub Drag Data. NASA CR 152080, March

1978.
Chapter 12

Force

The force component is an object that can generate a force acting on the aircraft, possibly used for

lift, propulsion, or control. The amplitude of the force can be a fixed value, or it can be connected to an

aircraft control for trim. The direction of the force can be fixed or connected to aircraft control.

12–1 Control and Loads

The control variables are the force amplitude A and the force incidence and yaw angles. The force

orientation is specified by selecting a nominal direction e f0 in body axes (positive or negative x, y, or

z-axis), then applying a yaw angle ψ, and then an incidence or tilt angle i (table 12-1). The control

variables can be connected to the aircraft controls cAC:

A = A0 + TAcAC

ψ = ψ0 + TψcAC

i = i0 + TicAC

with A0, ψ0 and i0 zero, constant, or a function of flight speed (piecewise linear input). The force axes

are C

BF 

= UiVψ , where U and V depend on the nominal direction, as described in table 12-1. The force

direction is e f = C

FBe f0. The force acts at position zF

. The force and moment acting on the aircraft in

body axes are thus:

F

F

= e fA

M

F

= LFF

F

where Δz

F

= z

F — zFcg .

Table 12-1. Force orientation.

nominal (F axes) e f0 incidence, + for force yaw, + for force C

BF 

= UiVψ

x forward i up right YiZψ

—x aft —i up right Y_iZ_ψ

y right j aft up ZiX_ψ

—y left —j aft up Z_iX ψ

z down k aft right Y_iX_ψ

—z up —k aft right YiXψ
108 Force

12–2 Performance and Weights

The force generation requires a fuel flow that is calculated from an input thrust-specific fuel consumption (sfc): w ̇ = A(sfc). Units of sfc are pound/hour/pound, or kilogram/hour/Newton.

The force component weight is identified as either engine-system or propeller/fan installation

weight, both of the propulsion group. The force component weight is calculated from specific weight

and the design maximum force Fmax, plus a fixed increment: W = SFmax + ΔW.
Chapter 13

Wing

The aircraft can have one or more wings, or no wings.

13–1 Geometry

The wing is described by planform area S, span b, mean chord c = S/b, and aspect ratio AR = b

2

/S.

These parameters are for the entire wing. The geometry is specified in terms of two of the following

parameters: S or wing loading W/S, b (perhaps calculated from other geometry), c, AR = b

2

/S. With

more than one wing, the wing loading is obtained from an input fraction of design gross weight,

W = fWWD . Optionally the span can be calculated from a specified ratio to the span of another wing.

Optionally for the tiltrotor configuration, the wing span can be calculated from the fuselage and rotor

geometry: b = 2 (fR + dfus) + wfus, where R is the rotor radius (cruise value for variable-diameter rotor),

dfus the rotor-fuselage clearance, and wfus the fuselage width. Note that the corresponding option for the

rotor-hub position is yhub = ± (fR + dfus + 1/2wfus ) . Optionally the wing span can be calculated from the

rotor-hub position: b = 21 yhub I (regardless of how the rotor position is determined). As implemented,

symmetry is not assumed; rather the radius or hub position of the outermost designated rotors is used.

Optionally the wing span can be calculated from an appropriate specification of all wing-panel widths.

The wing is at position z

F

, where the aerodynamic forces act. The component axes are the aircraft

body axes, CBF = I.

The wing planform is defined in terms of one or more wing panels (fig. 13-1). Symmetry of the

wing is assumed. The number of panels is P, with the panel index p = 1 to P. The span station η is

scaled with the semi-span: y = η(b/2), η = 0 to 1. Each panel is a trapezoid, with a straight aerodynamic

center and linear taper. The aerodynamic-center locus (in wing axes) is defined by sweep Λp

; dihedral

δp ; and offsets (xIp, zIp) at the inboard edge relative to the aerodynamic center of the previous panel. The

wing position zF is the mean aerodynamic center. The offset (x ̄A , z ̄A) of the mean aerodynamic center

from the root-chord aerodynamic center is calculated (so the wing planform can be drawn; typically the

aerodynamic center is drawn as the quarter-chord). Outboard panel edges are at ηEp (input or calculated).

A panel is characterized by span bp (each side), mean chord cp , and area Sp = 2bpcp (both sides). The

taper is defined by inboard and outboard chord ratios, λ = c/cref (where cref is a panel or wing reference

chord, depending on the options for describing the geometry).

The span for each panel (if there are more than two panels) can be a fixed input; a fixed ratio of the

wing span, bp = fbp (b/2); or free. The panel outboard edge (except at the wing tip) can be at a fixed input

position yEp; at a fixed station ηEp , yp = ηEp (b/2); calculated from the fuselage and rotor geometry,

yp = fR + dfus + 1/2wfus (for a designated rotor); calculated from the hub position, yp = |yhub | (for a

designated rotor); or adjusted. An adjusted station is obtained from the last station and the span of this
( xA , zA )

mean aero

center

(wing location

r locus

110 Wing

centerline

inboard outboard

edge edge

f

Ep outboard panel edge, f = y/(b/2)

f

Ip fOp wing station

λIp XOp chord ratio, A = c/cref

cIp cOp chord

Ap sweep (+ aft)

Sp dihedral (+ up)

xIp aero center offset (inboard, + aft)

zIp aero center offset (inboard, + up)

bp span (each side)

cp mean chord

Sp area = 2bpcp

Figure 13-1. Wing geometry (symmetric, only right half-wing shown).

panel, yp = yp- 1 + bp or yp = yp- 1 + fbp (b/2); or from the next station and the span of the next panel,

yp = yp+1 - bp+1 or yp = yp+1 - fb (p+1) (b/2). The specification of panel spans and panel edges must

be consistent, and sufficient to determine the wing geometry. Determining the panel edges requires the

following steps.

a) Calculate the panel edges that are either at fixed values (input, or from width, or from hub

position) or at fixed stations; root and tip edges are known.

b) Working from root to tip, calculate the adjusted panel edge yp if panel span bp or ratio fbp

is fixed, and if previous edge yp- 1 is known.

c) Working from tip to root, calculate the adjusted panel edge yp (if not yet known) if panel

span bp+1 or ratio fb(p+1) is fixed, and if next edge yp+1 is known.

At the end of this process, all edges must be known and the positions yp must be unique and sequential.

If this geometry is being determined for a known span, then there must not be a fixed panel span or span

ratio that has not been used. Alternatively, if the wing span is being calculated from the specification of

all panel widths, then the process must leave one and only one fixed panel span or span ratio that has not

been used. Since the wing span is to be calculated, each panel edge is known in the form yp = c0 + c1 b/2.

Then the unused fixed panel span gives the equation (c0 + c1b/2) O - (c0 + c 1 b/2)I = bp (subscript O
Wing 111

denotes outboard edge, subscript I denotes inboard edge), or the unused fixed panel span ratio gives the

equation (c0 + c1b/2)O — (c0 + c1 b/2)I = fp b/2, which can be solved for the semispan b/2.

To complete the definition of the geometry, one of the following quantities is specified for each

panel: panel area Sp ; ratio of panel area to wing area, fs = Sp /S; panel mean chord cp ; ratio of panel

mean chord to wing mean chord, fc = cp/c; chord ratios λI = cI

/cref and λO = cO /cref (taper); or free.

The total wing area equals the sum of all panel areas:

S =

Sp + S fs +2 bp cp +2c bpfc +2cref bp 2(

λI + λO )

If there is one or more taper specification (and no free), then cref is calculated from this equation for S,

and the mean chord is cp = 1

2

(cI + cO ) = cref 12 (λI + λO), Sp = 2bpcp . If there is one (and only one) free

specification, then Sp is calculated from this equation for S, and the mean chord is cp = Sp / (2bp ), with

cI = 2cp /( 1 + λO /λI), cO = 2cp — cI.

Since the panels have linear taper ( c = cref λ), the mean aerodynamic chord is

f

b/2 f1

Sc ̄A =

c

2 dy = b 

J 

c2ref λ

2

 dη

b/2 0

= b cref 

1 //

3 lλ 1 + λIλp + λp) Δηp = 3 lcI + cIcO + co) 2bp

f

b/2

f

1

S = 

c dy = b J crefλ dη

b/2 0

= b cref 2 (λI + λO ) Δηp =

2 (cI + cO ) 2bp

These expressions are evaluated from panel cI and cO , as calculated using λI and λO , or evaluated using

the ratio λO /λI (cref may not be the same for all panels).

The mean aerodynamic center is the point where there is zero moment due to lift: x ̄ACLS =

x ̄A f cic dy = f xcic dy, with ccj = B(y) the spanwise lift distribution. Thus

f

1

B(η)(xA — xAC (η)) dη = 0

The locus of section aerodynamic centers xAC is described by the panel sweep Λp and the offset xIp at

the inboard end of the panel. These offsets can be a fixed input, a fraction of the root chord, or a fraction

of the panel inboard chord. Assuming elliptical loading ( B = 1 — η2) gives

4 xA = J B (η)xACdη = 

J 

1 — η2

(

xIp + 2 tan Λp η

)

dη

0

_ b 1 1 ηO

—

L

xIp 2

(

η 

1 — η2 + sin-1η

)

— rl2 ) 3/

2 2 tan 

3

(1 —

ηI

where 5Ip = 

E

y=2 

(

xIq + (b/2) tan Λq-1 (ηO(q-1) — ηI(q-1))) — (b/2) tan Λp ηIp . The vertical position of

the mean aerodynamic center is obtained in a similar fashion, from panel dihedral δp and offset zIp at

the inboard edge of the panel. Assuming uniform loading ( B = 1) gives

zA = j

r1 

zACdη = 

J 

(

zIp + 2 tan δp η

)

dη = LzIpη + 2 tan δp

12 η

21

ηO

0 J ηI
112 Wing

Then (x ̄A , z ̄A) is the offset of the mean aerodynamic center from the root-chord aerodynamic center.

Finally,

Λ = tan−1

(^ 

b 2 

tan Λp)

δ = tan−1

(^ 

b 2 

tan δp

)

λ =

2c — 1

croot

are the wing overall sweep, dihedral, and taper.

The wing contribution to the aircraft operating length is xwing + (0. 25c) cos i (forward), xwing —

(0.75c) cos i (aft), and ywing f b/2 (lateral).

13–2 Control and Loads

The control variables are flap δF , flaperon δf

 , aileron δa , and incidence i. The flaperon deflection

can be specified as a fraction of flap deflection, or as an increment relative to the flap deflection, or

the flaperon can be independent of the flap. The flaperon and aileron are the same surface, generating

symmetric and antisymmetric loads respectively, hence with different connections to pilot controls.

With more than one wing panel, each panel can have control variables: flap δFp , flaperon δfp , aileron

δap, and incidence ip . The outboard panel (p > 2) control or incidence can be specified independently,

or in terms of the root panel (p = 1) control or incidence (either fraction or increment).

Each control is described by the ratio of the control-surface chord to the wing-panel chord, if =

c f/cp ; and by the ratio of the control-surface span to wing-panel span, fb = b f/bp , such that the controlsurface area is obtained from the panel area by Sf = ^f fbSp .

13–3 Aerodynamics

The aerodynamic velocity of the wing relative to the air, including interference, is calculated in

component axes, v B

. The angle-of-attack αwing (hence CBA) and dynamic pressure q are calculated from

vB . The reference area for the wing aerodynamic coefficients is the planform area, S. The wetted-area

contribution is twice the exposed area: Swet = 2(S — cwfus ), where wfus is the fuselage width.

The wing vertical drag can be fixed, specified as a drag area ( D/q) V ; or the drag can be scaled,

specified as a drag coefficient CDV based on the wing area; or calculated from an airfoil section drag

coefficient (for —90 degree angle-of-attack) and the wing area immersed in the rotor wake:

1

CDV = S cd90 ( 

r

S — c(wfus + 2dfus) — fd90 bF

cF

(1 — cos δF) — fd90 bf

cf

(1 — cos δf)

where wfus is the fuselage width and dfus the rotor-fuselage clearance. The last two terms account for

the change in wing area due to flap and flaperon deflection, with an effectiveness factor fd90.

From the control-surface deflection and geometry, the increments of lift coefficient, maximum-lift

angle, moment coefficient, and drag coefficient are evaluated: ΔCLf, Δαmax f, ΔCMf, ΔCDf. These

increments are the sum of contributions from flap and flaperon deflection, hence weighted by the controlsurface area. The drag-coefficient increment includes the contribution from aileron deflection.
Wing 113

13-3.1 Lift

The wing lift is defined in terms of lift-curve slope CLα and maximum lift coefficient CLmax (based

on wing planform area). The three-dimensional lift-curve slope is input directly or calculated from the

two-dimensional lift-curve slope:

cjα

CLα = 1+ cjα (1+ τ)/(πAR)

where τ accounts for non-elliptical loading. The effective angle-of-attack is αe = αwing + i — αzl, where

αzl is the angle of zero lift; in reverse flow (lαe l > 90), αe +— αe — 180 signαe . Let αmax = CLmax /CLα be

the angle-of-attack increment (above or below zero lift angle) for maximum lift. Including the change of

maximum lift angle caused by control deflection, Amax = αmax + Δαmax f and Amin = —αmax + Δαmax f .

Then ⎧

⎪ CLααe + ΔCLf Amin < αe < Amax

CL

(CLαAmax + ΔCLf) 

G/2

/

2 lAmI l 

/

αe > Amax

=

CLα G(

Amin

/

+ ΔCLf) 

π/2 — lαe l

/2 — lAmin l

αe < Amin

(for zero lift at 90 degree angle-of-attack). Note that CLαAmax + ΔCLf = CLααmax + ΔCLmaxf . In

sideward flight, CL = 0. Finally, L = qSCL is the lift force.

13-3.2 Pitch Moment

The wing pitch moment coefficient is CM = CMac +ΔCM f . Then M = qScCM is the pitch moment.

13-3.3 Roll Moment

The only wing roll moment considered is that produced by aileron control. Typically the flaperon

and aileron are the same surface, but they are treated separately in this model. The aileron geometry is

specified as for the flaperon and flap, hence includes both sides of the wing. The lift-coefficient increment

ΔCLa is evaluated as for the flaperon, so one-half of this lift acts up (on the right side) and one-half

acts down. The roll moment is then Mx = 2(ΔLa/2)y , where y is the lateral position of the aileron

aerodynamic center, measured from the wing centerline (defined as a fraction of the wing semi-span).

The roll-moment coefficient is Cj = — by 2A CLa. Then Mx = qSbCj is the roll moment.

13-3.4 Drag

The drag area or drag coefficient is defined for forward flight and vertical flight. The effective

angle-of-attack is αe = αwing + i — αDmin, where αDmin is the angle of minimum drag; in reverse flow

( lαe l > 90), αe +— αe — 180 signαe . For angles of attack less than a transition angle αt, the drag coefficient

equals the forward-flight (minimum) drag CD0 plus an angle-of-attack term and the control increment.

If the angle-of-attack is greater than a separation angle αs < αt

, there is an additional drag increase.

Thus if lαe l < αt

, the profile drag is

CDp = CD0 (1+ Kdlαe l

Xd + Ks(lαe l — αs)

Xs)+ ΔCDf

where the separation (Ks) term is present only for lαe l > αs; and otherwise

CDt = CD0 (1+ Kd lαt

l

Xd + Ks(lαt

l — αs)

Xs)+ ΔCDf

CDp = CDt + (CDV — CDt) sin π lαe l — αt

(

2 π/2 — αt

/
114 Wing

Optionally there might be no angle-of-attack variation at low angles (Kd = 0 and/or Ks = 0), or

quadratic variation (Xd = 2), or cubic variation for the separation term (Xs = 3). For sideward flight

(v

B

x = 0) the drag is obtained using φv = tan−1

(—v

B

z /vBy )tointerpolate the vertical coefficient: CD =

CD0 cos2 φv + CDV sin2φ v. The induced drag is obtained from the lift coefficient, aspect ratio, and

Oswald efficiency e:

_ (CL — CL0 )

2

CDi

πeAR

Conventionally the Oswald efficiency e can represent the wing parasite-drag variation with lift, as well

as the induced drag (hence the use of CL0). The wing-body interference is specified as a drag area, or a

drag coefficient based on the wing area. Then

D = qSCD = qS 

(

CDp + CDi + CDwb)

is the drag force. The other forces and moments are zero.

13-3.5 Wing Panels

The wing panels can have separate controls and different incidence angles. Thus the lift, drag,

and moment coefficients are evaluated separately for each panel, based on the panel area Sp and mean

chord cp . The coefficient increments due to control-surface deflection are calculated using the ratio of

the control-surface area to panel area, Sf

/Sp = if fb . The lateral position of the aileron aerodynamic

center is ηabp from the panel inboard edge, so y/(b/2) = ηE(p−1) + ηabp /(b/2) from the wing centerline.

Then the total wing coefficients are:

1

CL = SpCLp S1

CM = SC Sp cpCMp

1

C, = S SpCPp

1

CDp =

S 

SpCDpp

The three-dimensional lift-curve slope CLα is calculated for the entire wing and used for each panel.

The induced drag is calculated for the entire wing from the total CL .

13-3.6 Interference

With more than one wing, the interference velocity at other wings is proportional to the induced

velocity of the wing producing the interference: v F

int = Kintv

F

ind. The induced velocity is obtained

from the induced drag, assumed to act in the kB direction: αind = vind / Jv

B J = CDi/CL = CL/(πeAR),

v

F

ind = C

FBkB JvB Jαind . For tandem wings, typically Kint = 2 for the interference of the front wing on

the aft wing, and Kint = 0 for the interference of the aft wing on the front wing. For biplane wings, the

mutual interference is typically Kint = 0. 7 (upper on lower, and lower on upper). The induced drag is

then

C

2

C

2

C

2

I

CDi = πeAR+ CL αint = π

eAR + CL Kintαind = π

eAR+ CLKint 

πe

CL

J

other wing

where the sum is over all other wings.
Wing 115

The wing interference at the tail produces an angle-of-attack change e = E(CL/CLα), where

E = de/dα is an input factor determined by the aircraft geometry. Then from the velocity vB of the

wing, ⎛ ⎞

F

−ev Bz

FB

vint = C ⎝ 0 ⎠

ev Bx

is the interference velocity at the tail.

13–4 Wing Extensions

The wing can have extensions, defined as wing portions of span bX at each wing tip. For the

tiltrotor configuration in particular, the wing weight depends on the distribution of wing area outboard

(the extension) and inboard of the rotor and nacelle location. Wing extensions are defined as a set of

wing panels at the tip. The extension span and area are the sum of the panel quantities, bX = 

^

ext bp

and SX = 

^

ext Sp . The inboard span and area are then bI = b — 2bX , SI = S — SX . Optionally the wing

extensions can be considered a kit, hence the extensions can be absent for designated flight conditions

or missions. As a kit, the wing-extension weight is considered fixed useful load. With wing extensions

removed, the aerodynamic analysis considers only the remaining wing panels. For the induced drag

and interference, the effective aspect ratio is then reduced by the factor (bI/b) 2 , since the lift and drag

coefficients are still based on total wing area S.

13–5 Weights

The wing group consists of: basic structure (primary structure, consisting of torque box and spars,

plus extensions); fairings (leading edge and trailing edge); fittings (non-structural); fold/tilt structure;

and control surfaces (flaps, ailerons, flaperons, spoilers). There are separate models for a tiltrotor or

tiltwing configuration and for other configurations (including compound helicopter).

The AFDD wing-weight models are based on parameters for the basic wing plus the wing tip

extensions (not the total wing and extensions). The tiltrotor-wing model requires the weight on the wing

tips (both sides), consisting of: rotor group, engine system, drive system (except drive shaft), engine

section or nacelle group, air induction group, rotary-wing and conversion flight controls, hydraulic

group, trapped fluids, and wing extensions.
116 Wing
Chapter 14

Empennage

The aircraft can have one or more tail surfaces, or no tail surface. Each tail is designated as

horizontal or vertical, affecting some parameter definitions.

14–1 Geometry

The tail is described by planform area S, span b, chord c = S/b, and aspect ratio AR = b

2

/S. The

tail volume can be referenced to rotor radius and disk area, V = SB/RA; to wing area and chord for

horizontal tails, V = SB/Swcw ; or to wing area and span for vertical tails, V = SB/Swbw . Here the tail

length is B = |xht − xcg | or B = |xvt − xcg | for horizontal tail or vertical tail, respectively. The geometry

is specified in terms of S or V; and b, or AR, or c. The elevator or rudder is described by the ratio of

control-surface chord to tail chord, c f

/c; and the ratio of control-surface span to tail span, b f/b.

The tail contribution to the aircraft operating length is xtail + 0. 25c (forward), xtail − 0. 75c (aft), and

ytail ± (b/2)C (lateral)- where C = cos φ for a horizontal tail and C = cos(φ − 90) for a vertical tail.

14–2 Control and Loads

The tail is at position zF , where the aerodynamic forces act. The scaled input for tail position can

be referenced to the fuselage length or the rotor radius.

The horizontal tail can have a cant angle φ (positive tilt to left, becomes vertical tail for φ = 90 deg).

Thus the component axes are given by C

BF 

= X−φ . The control variables are elevator δe and incidence

i.

The convention for nominal orientation of the vertical tail is positive lift to the left, so aircraft

sideslip (positive to right) generates positive tail angle-of-attack and positive tail lift. The vertical tail

can have a cant angle φ (positive tilt to right, becomes horizontal tail for φ = 90), so the component axes

are given by CBF = X−90+φ . The control variables are rudder δr and incidence i.

14–3 Aerodynamics

The aerodynamic velocity of the tail relative to the air, including interference, is calculated in

component axes, v

B . The angle-of-attack αtail (hence CBA) and dynamic pressure q are calculated from

v

B . The reference area for the tail aerodynamic coefficients is the planform area, S. The wetted area

contribution is Swet = 2S. From the elevator or rudder deflection and geometry, the increments in lift

coefficient, maximum-lift angle, and drag coefficient are evaluated: ΔCLf, Δαmax f, ΔCDf.
118 Empennage

14-3.1 Lift

The tail lift is defined in terms of lift-curve slope CLα and maximum lift coefficient CLmax (based

on tail planform area). The three-dimensional lift-curve slope is input directly or calculated from the

two-dimensional lift-curve slope:

c^α

CLα = 1+ c^α (1+ τ)/(πAR)

where τ accounts for non-elliptical loading. The effective angle-of-attack is αe = αtail + i — αzl, where

αzl is the angle of zero lift; in reverse flow (Iαe I > 90), αe +— αe — 180 signαe . Let αmax = CLmax /CLα be

the angle-of-attack increment (above or below zero lift angle) for maximum lift. Including the change of

maximum lift angle caused by control deflection, Amax = αmax + Δαmax f and Amin = —αmax + Δαmax f.

Then ⎧

⎪ CLααe + ΔCLf

\1

Amin < αe < Amax

CL

= (CLαAmax + ΔCLf) G/ /2 IAm

I 

I)

αe > Amax

G

(CLαAmin

π/2 — IαeI + ΔCLf)/2 — IAminI αe < Amin

(for zero lift at 90 degree angle-of-attack). Note that CLαAmax + ΔCLf = CLααmax + ΔCLmaxf. 

In

sideward flight (defined by (v

B

x )

2

 + (v

B

z )

2

< (0.05 I vB I ) 2 ), CL = 0. Finally, L = qSCL is the lift force.

14-3.2 Drag

The drag area or drag coefficient is defined for forward flight and vertical flight. The effective

angle-of-attack is αe = αtail + i — αDmin, where αDmin is the angle of minimum drag; in reverse flow

( Iαe I > 90), αe +— αe — 180 signαe . For angles of attack less than a transition angle αt

, the drag coefficient

equals the forward-flight (minimum) drag CD0, plus an angle-of-attack term and the control increment.

Thus if Iαe I < αt , the profile drag is

CDp = CD0 (1+ Kd Iαe I

Xd)+ ΔCDf

and otherwise

CDt = CD0 (1+ Kd Iαt

I

Xd)+ ΔCDf

CDp = CDt + (CDV — CDt) sin 2 Iαe I— αt

2 π/2 — αt /

Optionally there might be no angle-of-attack variation at low angles (Kd = 0), or quadratic variation

(Xd = 2). In sideward flight (defined by ( v

B

x )

2

 + (v

B

z

)

2 < (0.05 IvB I)2), the drag is obtained using

φv = tan−1 (—v

B

z

/vB

y ) to interpolate the vertical coefficient: CDp = CD0 cos2 φv + CDV sin2 φv . The

induced drag is obtained from the lift coefficient, aspect ratio, and Oswald efficiency e:

_ (CL — CL0 )

2

CDi

πeAR

Conventionally the Oswald efficiency e can represent the tail parasite-drag variation with lift, as well as

the induced drag (hence the use of CL0). Then

D = qSCD = qS 

(

CDp + CDi)

is the drag force. The other forces and moments are zero.
Empennage 119

14–4 Weights

The tail weight (empennage group) model depends on the configuration: helicopters and compounds, or tiltrotors and tiltwings. Separate weight models are available for horizontal and vertical

tails.

The AFDD tail weight model depends on the design dive speed at sea level (input or calculated).

The calculated dive speed is Vdi ve = 1.25Vmax, from the maximum speed at the design gross weight and

sea-level standard conditions.
120 Empennage
Chapter 15

Fuel Tank

15–1 Fuel Capacity

The fuel-tank capacity Wfuel—cap (maximum usable fuel weight) is determined from designated

sizing missions. The maximum mission fuel required, Wfuel—miss (excluding reserves and any fuel in

auxiliary tanks), gives

Wfuel—cap = max(ffuel—cap Wfuel—miss, Wfuel—miss + Wreserve)

where ffuel—cap > 1 is an input factor. Alternatively, the fuel-tank capacity Wfuel—cap can be input. The

corresponding volumetric fuel-tank capacity is Vfuel—cap = Wfuel—cap /ρfuel (gallons or liters), where ρfuel

is the fuel density (input as weight per volume).

For missions that are not used to size the fuel tank, the fuel weight may be fallout, or the fuel weight

may be specified (with or without auxiliary tanks). The fuel weight for a flight condition or the start of

a mission is specified as an increment d, plus a fraction f of the fuel-tank capacity, plus auxiliary tanks:

Wfuel = min(dfuel + ffuelWfuel—cap, Wfuel—cap) + E NauxtankWaux—cap

where Waux—cap is the capacity of each auxiliary fuel tank. The fuel capacity of the wing can be estimated

from

Wfuel—wing = ρfuel E fctbtw bw

where ctb is the torque box chord, tw the wing thickness, and bw the wing span; and f is the input fraction

of the wing torque box that is filled by primary fuel tanks, for each wing. This calculation is performed

in order to judge whether fuel tanks outside the wing are needed.

15–2 Geometry

The fuel tank is at position zF , where the inertial forces act.

15–3 Fuel Reserves

Mission fuel reserves can be specified in several ways for each mission. Fuel reserves can be

defined in terms of specific mission segments, for example 200 miles plus 20 minutes at Vbe. Fuel

reserves can be an input fraction of the fuel burned by all (except reserve) mission segments, so Wfuel =

(1 + fres)Wfuel—miss. Fuel reserves can be an input fraction of the fuel capacity, so Wfuel = Wmiss—seg +

fres Wfuel—cap . If more than one criterion for reserve fuel is specified, the maximum reserve is used.
122 Fuel Tank

15–4 Auxiliary Fuel Tank

Auxiliary fuel tanks are defined in one or more sizes. The capacity of each auxiliary fuel tank,

Waux-cap, is an input parameter. The number of auxiliary fuel tanks on the aircraft, Nauxtank for each

size, can be specified for the flight condition or mission segment. Alternatively (if the mission is not

used to size the fuel tank), the number of auxiliary fuel tanks at the start of the mission can be determined

from the mission fuel.

Figure 15-1 describes the process for determining Nauxtank from the fuel weight Wfuel and the aircraft

maximum fuel capacity Wfuel-max = Wfuel-cap + E NauxtankWaux-cap . The fuel-weight adjustment

Δ Wfuel is made if fuel weight is fallout from fixed gross weight and payload, accounting for the operating

weight update when Nauxtank changes. If the auxiliary-tank weight is greater than the increment in fuel

weight needed, then the fallout fuel weight Wfuel = WG - WO - Wpay can not be achieved; in such a

case, the fuel weight is capped at the maximum fuel capacity and the payload weight adjusted instead.

The weight and drag of Nauxtank tanks are included in the performance calculation. Optionally the

number of auxiliary tanks required can be calculated at the beginning of designated mission segments

(based on the aircraft fuel weight at that point), and tanks dropped if no longer needed. The weight of the

auxiliary fuel tanks is an input fraction of the tank capacity: Wauxtank = E fauxtankNauxtankWaux-cap.

15-4.1 Auxiliary-Fuel-Tank Drag

The auxiliary fuel tanks are located at position zF . The drag area for one auxiliary tank is specified,

(D/q)auxtank. The velocity relative to the air gives the drag direction ed = -v

F

/Iv

F

I and dynamic

pressure q = 1/2ρIv

F

I

2

(no interference). Then

F

F

= e Nd q auxtank (D/q)auxtank

is the total drag force, calculated for each auxiliary tank size.

15–5 Weights

The fuel system consists of the tanks (including support) and the plumbing.

The weight of the auxiliary fuel tanks is part of the fixed useful load; it is an input fraction of the

tank capacity: Wauxtank = E fauxtankNauxtankWaux-cap.

The AFDD weight model for the plumbing requires the fuel-flow rate (for all engines), calculated

for the takeoff rating and conditions.
Fuel Tank 123

if 

Wfuel > Wfuel—max

for first auxiliary tank Nauxtank = Nauxtank + 1 ΔWfuel = —fauxtankWaux—cap ΔWfuel—max = Waux—cap

repeat if Wfuel > Wfuel—max

if 

Wfuel < 

Wfuel—max — Waux—cap

Nauxtank = Nauxtank — 

1

Δ

Wfuel—max = 

—Waux—cap

Wfuel = 

Wfuel—max (capped)

else if 

Wfuel < Wfuel—max

for last nonzero 

Nauxtank

Nauxtank = Nauxtank — 

1

ΔWfuel = fauxtankWaux—cap

ΔWfuel—max = 

—

Waux—cap

repeat if 

Wfuel < Wfuel—max

undo last increment Nauxtank = Nauxtank + 1 ΔWfuel = —fauxtankWaux—cap ΔWfuel—max = Waux—cap

Figure 15-1. Outline of 

Nauxtank calculation.
124 Fuel Tank
Chapter 16

Propulsion

The propulsion group is a set of components and engine groups, connected by a drive system. The

engine model describes a particular engine, used in one or more engine group. The components (rotors)

define the power required. The engine groups define the power available.

16–1 Drive System

The drive system defines gear ratios for all the components it connects. The gear ratio is the ratio

of the component rotational speed to that of the primary rotor. There is one primary rotor per propulsion

group (for which the reference tip speed is specified); other components are dependent (for which a gear

ratio is specified). There can be more than one drive-system state, in order to model a multiple-speed or

variable-speed transmission. Each drive-system state corresponds to a set of gear ratios.

For the primary rotor, a reference tip speed Vtip-ref is defined for each drive-system state. By

convention, the “hover tip speed” refers to the reference tip speed for drive state #1. If the sizing task

changes the hover tip speed, then the ratios of the reference tip speeds at different engine states are kept

constant. By convention, the gear ratio of the primary rotor is r = 1. For dependent rotors, either the

/

gear ratio is specified (for each drive-system state) or a tip speed is specified and the gear ratio calculated

(

r = Ωdep /Ωprim , Ω = Vtip-ref /R). For the engine group, either the gear ratio is specified (for each drivesystem state) or the gear ratio calculated from the specification engine turbine speed Ωspec = (2π/60)Nspec

and the reference tip speed of the primary rotor (r = Ωspec/Ωprim, Ωprim = Vtip -ref /R). The latter option

means the specification engine turbine speed Nspec corresponds to Vtip-ref for all drive-system states.

To determine the gear ratios, the reference tip speed and radius are used, corresponding to hover.

The flight state specifies the tip speed of the primary rotor and the drive-system state, for each

propulsion group. The drive-system state defines the gear ratio for dependent rotors and the engine

groups. From the rotor radius the rotational speed of the primary rotor is obtained (Ωprim = Vtip /R);

from the gear ratios, the rotational speed of dependent rotors ( Ωdep = rΩprim) and the engine groups

(N = (60/2π)rengΩprim) are obtained; and from the rotor radius, the tip speed of the dependent rotor

(Vtip = ΩdepR) is obtained. The flight-state specification of the tip speed can be an input value; the

reference tip speed; a function of flight speed or a conversion schedule; or one of several default values.

These relationships between tip speed and rotational speed use the actual radius of the rotors in the flight

state, which for a variable-diameter rotor may not be the same as the reference, hover radius.

An optional conversion schedule is defined in terms of two speeds: hover and helicopter mode for

speeds below VChover, cruise mode for speeds above VCcruise, and conversion mode for speeds between

VChover and VCcruise. The tip speed is Vtip-hover in helicopter and conversion mode, and Vtip-cruise in

airplane mode. Drive-system states are defined for helicopter, cruise, and conversion-mode flight. The
126 Propulsion

flight state specifies the nacelle tilt angle, tip speeds, control state, and drive-system state, including the

option to obtain any or all of these quantities from the conversion schedule.

Several default values of the tip speed are defined for use by the flight state, including cruise,

maneuver, one-engine inoperative, drive-system limit conditions, and a function of flight speed (piecewise linear input). Optionally these default values can be input as a fraction of the hover tip speed.

Optionally the tip speed can be calculated from an input CT/σ = t0 — μt1 , from μ = V/Vtip , or from

Mat = Mtip (1 + μ)

2 + μ2

z

; from which Vtip = T/ρAσt0 + (Vt1 /2t0 )

2 + (Vt1 /2t0 ), Vtip = V/μ, or

Vtip = (csMat)

2 — V2

z — V.

The sizing task might change the hover tip speed (reference tip speed for drive-system state #1),

the reference tip speed of a dependent rotor, a rotor radius, or the specification engine turbine speed

Nspec. In such cases thegear ratios and other parameters are recalculated. Note that it is not consistent

to change the reference tip speed of a dependent rotor if the gear ratio is a fixed input.

16–2 Power Required

The component power required Pcomp is evaluated for a specified flight condition as the sum of

the power required by all the components of the propulsion group. The total power required for the

propulsion group is obtained by adding the transmission losses and accessory power:

PreqPG = Pcomp + Pxmsn + Pacc

The transmission losses are calculated as an input fraction 2xmsn of the component power plus windage

loss:

Pxmsn = txmsnfxmsmPcomp + Pwindage (Ωprim/Ωref)

The factor fxmsm can equal 1 or can include a function of the drive-shaft rating (increasing the losses at

low power): ⎧

P

21 1 Xlimit Q< ⎨⎪ 4

fxmsmPcomp = 

(

3 — 43 Q) Pcomp

1

4 <Q< 1

⎩

⎪

Pcomp 1 < Q

where Q = Pcomp /PXlimit, PXlimit = rPDSlimit, and r = N/Nspec = Ωprim/Ωref. Accessory losses are

calculated as an input constant, a fraction of power required (such as environmental control unit (ECU)

losses), infrared suppression (IRS) fan loss (if IRS system is on), and terms that scale with air density

and rotor speed:

Pacc = Pacc0 + tacc (Pcomp + Pxmsn) + 4RfanσNengPeng + Paccdσ + Paccnσ(Ωprim/Ωref)

where σ = ρ/ρ0 is the density ratio.

The power required for the propulsion group must be distributed to the engine groups. With only

one engine group, PreqEG = PreqPG. An engine group power can be fixed at a fraction fP of the

engine power available: PreqEG = fP (Neng — Ninop ) Peng at the specified rating (each operating engine

at fPPeng). The power required for the remaining (perhaps all) engine groups is distributed proportional

to the engine rated power:

PreqEG = 

(

PreqPG — E PreqEG)

(Neng — Ninop )Peng

/

fixed

r

notfixed (Neng — Ninop)Peng
Propulsion 127

If the sum of the fixed PreqEG exceeds the propulsion group power required, or if the power is fixed for

all engine groups, then each is reduced by the ratio PreqPG/Efixed PreqEG. The result is that the power

is distributed proportional to fP times the engine rated power. The fuel flow of the propulsion group is

obtained from the sum over the engine groups: w ̇ reqPG = E w ̇ reqEG.

16–3 Geometry

The length of the drive system BDS can be input or calculated. The calculated length is the sum of

the longitudinal, lateral, and vertical distances from the primary rotor hub to the other hub locations for

all rotors in the propulsion group: BDS = f 

E

(I ΔxI + I ΔyI + I Δz I ), where f is an input factor.

16–4 Drive-System Rating

The drive-system rating is defined as a power limit, PDSlimit. The rating is properly a torque limit,

QDSlimit = PDSlimit/Ω, but is expressed as a power limit for clarity. The drive-system rating can be

specified as follows:

a) Input PDSlimit.

b) From the engine takeoff power rating, PDSlimit = flimit E NengPeng (summed over

all engine groups).

c) From the power available at the transmission sizing flight conditions, PDSlimit =

flimit (Ωref/Ωprim) E NengPav (largest of all conditions).

d) From the power required at the transmission sizing flight conditions, PDSlimit =

flimit (Ωref /Ωprim) E NengPreq (largest of all conditions).

with flimit an input factor. The drive-system rating is a limit on the entire propulsion system. To account

for differences in the distribution of power through the drive system, limits are also used for the torque

of each rotor shaft (PRSlimit) and of each engine group (PESlimit). The engine-shaft rating is calculated

as for the drive-system rating, without the sum over engine groups. The rotor-shaft rating is either input

or calculated from the rotor power required at the transmission sizing flight conditions. The power limit

is associated with a reference rotational speed, and when applied the limit is scaled with the rotational

speed of the flight state. The rotation speed for the drive-system rating PDSlimit is the hover speed of the

primary rotor of the propulsion group (for the first drive state). The rotation speed for the engine-shaft

rating PESlimit is the corresponding engine turbine speed. The rotation speed for the rotor-shaft rating

PRSlimit is the corresponding speed of that rotor.

16–5 Weights

The drive system consists of gear boxes and rotor shafts, drive shafts, rotor brakes, clutches, and

gas drive. The drive-system weight depends on the rotor and engine rotational speeds, evaluated for

the propulsion group primary rotor and a specified engine group, at a specified drive-system state (gear

ratio).

The AFDD drive-system weight model depends on fQ , the second (main or tail) rotor rated torque

as a fraction of the total drive-system rated torque; and on fP , the second (main or tail) rotor rated power

as a fraction of the total drive-system rated power. These parameters are related by the rotational speeds

of the two rotors: fP = fQΩother/Ωmain. Typically fP = fQ = 0. 6 for twin rotors (tandem, coaxial, and
128 Propulsion

tiltrotor configurations). For the single-main-rotor and tail-rotor configuration, typically fQ = 0.03 and

fP = 0.15 (0.18 with a two-bladed teetering main rotor).
Chapter 17

Engine Group

The engine group consists of one or more engines of a specific type. For each engine type an engine

model is defined. The Referred Parameter Turboshaft Engine Model (RPTEM) is implemented.

The engine size is described by the power Peng, which is the sea-level static power available per

engine at a specified takeoff rating. The number of engines Neng is specified for each engine group.

If the sizing task determines the engine power for a propulsion group, the power Peng of at least one

engine group is found (including the first engine group). The total power required is PPG = r E NengPeng,

where r = max(PreqPG/PavPG). The sized power is Psized = PPG — Efixed NengPeng. Then the sized

engine power is Peng = fnPsized /Neng for the n-th engine group (with f1 = 

E

no1,sized fn for the first

group).

17–1 Engine Performance

17-1.1 Power Available

Given the flight condition and engine rating, the power available Pa is calculated (from the specific

power SPa and mass flow m ̇ a

). The flight-condition information includes the altitude, temperature,

flight speed, and primary rotor speed; a power fraction fP ; and the states of the engine, drive system,

and IR suppressor. The engine turbine speed is N = (60/2π)rengΩprim, where Ωprim is the current rotor

speed and reng is the gear ratio (depending on the drive-system state). If the reference primary rotor

speed Ωprim corresponds to the specification turbine speed Nspec, then reng = Ωspec/Ωprim; alternatively,

the engine gear ratio can be a fixed input.

In the engine model, installation losses Ploss are subtracted from Pa (Pav = Pa — Ploss), and then

the mechanical limit applied: Pav = min(Pav , PmechR).

The engine model gives the performance of a single engine. The power available of the engine

group is obtained by multiplying the single-engine power by the number of engines operational (total

number of engines less inoperable engines):

PavEG = (Neng — Ninop )Pav

The propulsion group power available is obtained from the sum over the engine groups: PavPG =

E fPPavEG, including a specified power fraction fP .

The drive-system rating at the flight condition is rPDSlimit, where r = Ωprim/Ωref. Optionally

this limit is applied to the propulsion group power: PavPG = min(PavPG, rPDSlimit) . Similarly the

engine-shaft limit at the flight condition is optionally applied to the engine group power.
130 Engine Group

17-1.2 Performance at Power Required

The engine performance (mass flow, fuel flow, and gross jet thrust) is calculated for a specifed power

required Pq , flight condition, and engine rating. The flight condition includes the altitude, temperature,

flight speed, and primary rotor speed; and engine, drive system, and IRS states. The engine turbine

speed is N = rengΩprim.

The engine model deals with a single engine. The power required of a single engine is obtained

by dividing the engine group power by the number of engines operational (total number of engines less

inoperable engines):

Preq = PreqEG/(Neng — Ninop)

In the engine model, installation losses Ploss are added to Preq (Pq = Preq + Ploss).

The engine model gives the performance of a single engine. The performance of the engine group

is obtained by multiplying the single-engine characteristics by the number of engines operational (total

number of engines less inoperable engines):

m ̇ reqEG = (Neng — Ninop)m ̇ req

w ̇ reqEG = (Neng — Ninop )w ̇ reqKffd

FNEG = (Neng — Ninop )FN

DauxEG = (Neng — Ninop )Daux

The fuel flow has also been multiplied by a factor accounting for deterioration of the engine efficiency.

17-1.3 Installation

The difference between installed and uninstalled power is the inlet and exhaust losses Ploss: Pav =

Pa — Ploss and Preq = Pq — Ploss. The inlet ram recovery efficiency ηd is included in the engine-model

calculations. The inlet and exhaust losses are modeled as fractions of power available or power required:

Ploss = (tin +tex) Pa or Ploss = (tin +tex) Pq . Installation effects on the jet thrust are included in the engine

model. The momentum drag of the auxiliary-air flow is a function of the mass flow m ̇ aux = fauxm ̇ req:

Daux = (1 — ηaux)m ̇ auxV = (1 — ηaux)fauxm ̇ reqV

where ηaux is the ram recovery efficiency. Exhaust losses (tex) and auxiliary-air-flow parameters (ηaux,

faux) are defined for IRS on and off. Inlet particle-separator loss is added to the inlet losses (tin).

17–2 Control and Loads

The engine orientation is specified by selecting a nominal direction e f0 in body axes (positive or

negative x, y, or z-axis; usually thrust forward, hence positive x-axis); and then applying a yaw angle

ψ, then an incidence or tilt angle i (table 17-1). The yaw and incidence angles can be connected to the

aircraft controls cAC:

ψ = ψ0 + TψcAC

i = i0 + TicAC

with ψ0 and i0 zero, constant, or a function of flight speed (piecewise linear input). Hence the incidence

and yaw angles can be fixed orientation or can be control variables. Optionally the lateral position of the

engine group can be set equal to that of a designed rotor (useful for tiltrotors when the rotor hub lateral

position is calculated from the clearance or wing geometry).
Engine Group 131

The engine group produces a jet thrust FN , acting in the direction of the engine; a momentum drag

Daux, acting in the wind direction; and a nacelle drag Dnac, acting in the wind direction. The engine

group is at location zF . The force and moment acting on the aircraft in body axes are thus:

F

F

= e fFN + edDaux + edDnac

MF = LFF

F

where Δz

F

= zF — zFcg, e f is the engine-thrust direction and ed is the drag direction. The velocity relative

to the air gives ed = —v

F/lvFl (no interference). The engine axes are CBF 

= UiVψ , where U and V

depend on the nominal direction, as described in table 17-1. The engine direction is e f = C

FBe f0 .

For a tiltrotor aircraft, one of the aircraft controls is the nacelle angle, with the convention αtilt = 0

for cruise and αtilt = 90 degree for helicopter mode. The engine incidence angle is then connected to

αtilt by defining the matrix Ti appropriately. If the engine nominal direction is defined for airplane mode

(+x), then i = αtilt should be used; if the engine nominal direction is defined for helicopter mode ( —z),

then i = αtilt — 90 should be used.

Table 17-1. Engine orientation.

nominal (F axes) e f0 incidence, + for force yaw, + for force C

BF 

= UiVψ

x forward i up right YiZψ

—x aft —i up right Y_iZ_ψ

y right j aft up ZiX_ψ

—y left —j aft up Z_iXψ

z down k aft right Y_iX_ψ

—z up — k aft right YiXψ

17–3 Nacelle Drag

The engine group includes a nacelle, which contributes to the aircraft drag. The component drag

contributions must be consistent. The pylon is the rotor support and the nacelle is the engine support.

The drag model for a tiltrotor aircraft with tilting engines would use the pylon drag (and no nacelle drag),

since the pylon is connected to the rotor-shaft axes; with non-tilting engines it would use the nacelle

drag as well.

The nacelle drag acts at the engine location zF . The nacelle axes are the engine axes, hence CBF is

calculated as described previously (see table 17-1). For the engine nominal direction forward ( +x-axis),

the nacelle z-axis is downward and the x-axis is forward; zero incidence angle corresponds to zero

angle-of-attack; and 90 degree incidence angle corresponds to 90 degree angle-of-attack (vertical drag).

The velocity, angle-of-attack, and dynamic pressure are calculated at the nacelle (without interference).

The reference area for the nacelle drag coefficient is the nacelle wetted area. The wetted area per engine

is input, or calculated either from the engine system (engine, exhaust, and accessories) weight or from

the engine-system plus drive-system weight:

Swet = k

(

w/Neng)2

/3

where w = WES or w = WES + Wgbrs/NEG and the units of k are feet2 /pound2/3 or meter2 /kilogram2

/3 .

The reference area is then Snac = NengSwet. The nacelle area is included in the aircraft wetted area if
132 Engine Group

the drag coefficient is nonzero. The drag area or drag coefficient is defined for forward flight and for

vertical flight. The drag coefficient at angle-of-attack α is

CD = CD0 + (CDV — CD0 ) | sin α|

Xd

typically using Xd = 2. In sideward flight, CD = CD0

is used. The nacelle drag is Dnac = qSnacCD .

17–4 Engine Scaling

The parameters of the engine model can be defined for a specific engine, but it is also necessary to

scale the parameters as part of the aircraft sizing task, in order to define an engine for a specified power.

In addition, advanced technology must be represented in the model. Scaling and advanced technology

are handled in terms of specifc power and specific fuel consumption (at SLS static conditions, MCP, and

Nspec). Figures 17-1 through 17-3 present historical data for engine specific fuel consumption, weight,

and specific power, respectively; the variation at a given power reflects technology insertion with time.

The engine model includes reference values of the engine performance parameters: P0R, SP0R,

PmechR , sfc0C , SF0C , Nspec, and Nopt0C . Mass flow and fuel flow are obtained from m ̇ 0R = P0R/ SP0R and

w ̇ 0C = sfc0CP0C . The reference power at each engine rating R defines a ratio to MCP: rp0R = P0R/P0C .

Similarly for specifc power and mechanical limits: rs0R = SP0R/ SP0C and rm0R = PmechR/P0C . These

ratios are kept fixed when the engine is scaled.

The engine size is specified as takeoff power Pto = Peng: power at rating R, for SLS static conditions

and specification turbine speed Nspec. Hence the MCP is P0C = Pto/rp0R, and the power at all other

ratings follows.

The actual (perhaps scaled) values of the performance parameters are available for the engine group:

P0R, SP0R, PmechR , sfc0C , Fg0C, Nspec, Nopt0C .

17–5 Weights

The component weight consists of engine system, engine section or nacelle group, and air induction

group. The engine system consists of engine, exhaust system, and accessories. The engine section or

nacelle group consists of engine support, engine cowling, and pylon support. These weights are for the

engine group, consisting of Neng engines.
Engine Group 133

0.3

0. 1000. 2000. 3000. 4000. 5000. 6000. 7000.

takeoff power (hp)

Figure 17-1. Historical data for turboshaft-engine specific fuel consumption.

0. 1000. 2000. 3000. 4000. 5000. 6000. 7000.

takeoff power (hp)

Figure 17-2. Historical data for turboshaft-engine weight.
200.

134 Engine Group

250.

0.

0. 1000. 2000. 3000. 4000. 5000. 6000. 7000.

takeoff power (hp)

Figure 17-3. Historical data for turboshaft-engine specific power.
Chapter 18

Referred Parameter Turboshaft Engine Model

Aircraft gas-turbine-engine performance capabilities are formally specified by computer programs

known as engine decks, which are created by engine manufacturers in an industry-standard format.

Engine decks are typically based on thermodynamic cycle analysis using real engine-component performance maps. The most important performance maps for turboshaft engines are compressor, gasgenerator turbine, and power turbine. These component performance maps are critical to obtaining

realistic off-design engine performance. Design and analysis codes calculate aircraft performance for

a very wide range of operating conditions. Thus engine performance must be realistic even far from

the engine design point. A simple thermodynamic cycle analysis that assumes design-point component

efficiencies everywhere is not realistic for such an application. Rather than developing models for component performance, the approach taken is to use a model for the total engine performance. The engine

is not being designed.

The Referred Parameter Turboshaft Engine Model (RPTEM) is based on curve-fits of engine

performance data for existing or projected engines over a range of operating conditions. The curve-fits

are typically obtained by exercising an engine deck. The use of referred parameters tends to collapse

the data, and provides a basis for scaling the engine. The operating condition is described by pressure

altitude, ambient air temperature, flight Mach number, power-turbine speed, exhaust-nozzle area, and

either engine rating or engine power required. These curve-fits, typically based on real engines, are

scaled to the required size and adjusted to the appropriate technology level to represent a notional

engine. Engine size is represented by mass flow. Engine technology is represented by specific power

available and specific fuel consumption at maximum continuous power (MCP), sea level/standard day

(SLS), static (zero airspeed) conditions. Engine installation effects (inlet and exhaust losses) are also

modeled.

The use of referred parameters to curve-fit engine performance data was suggested by David

Woodley from Boeing during the Joint Vertical Experimental (JVX) program (1983). The RPTEM

was developed and documented by Michael P. Scully and Henry Lee of U.S. Army Aeroflightdynamics

Directorate (AFDD), with a subsequent implementation written by Sam Ferguson (1995).

18–1 Operating Environment

The operating condition and atmosphere give the standard conditions (temperature Tstd and pressure

pstd) for a specifed pressure altitude; the sea-level standard conditions (temperature T0 and pressure p0);

and the operating temperature T and pressure p = pstd. Here the temperatures are O R or °K. The engine

characteristics depend on the temperature ratio θ = T/T0 and pressure ratio δ = p/p0 .

The flight Mach number M = V/cs = V/cs0√θ- is obtained from the aircraft speed V.
136 Referred Parameter Turboshaft Engine Model

The inlet ram air temperature ratio and pressure ratio are obtained then from M and the inlet ram

recovery efficiency ηd :

θM = 1 + γ 2 1M

2

= (1 + 0.2M

2

)

γ

−

γ−1

δM = 1 + γ 2 1 ηdM2 = (1 + 0.2ηdM

2

) 3.5

where the ratio of specific heats γ = 1 . 4.

18–2 Engine Ratings

The power available from a turboshaft engine depends on the engine rating. Each engine rating

has specific operating limitations, most importantly an operating time limit intended to avoid damage

to the engine. Typical engine ratings are given in table 18-1. Engine power is generally specified in

terms of SLS static MCP. Takeoff typically uses MRP. CRP or ERP are restricted to use in one-engine

inoperative (OEI) emergencies.

Table 18-1. Typical engine ratings.

rating description time limit

MCP maximum continuous power ∞

IRP intermediate rated power 30 min

MRP maximum rated power 10 min

CRP contingency rated power 2.5 min

ERP emergency rated power 1.0 min

18–3 Performance Characteristics

The engine performance is described by: the power available Pa

, at each engine rating and the

specification engine-turbine speed Nspec; the mass flow m ̇ and fuel flow w ̇ required to produce power

required Pq at engine-turbine speed N; and the gross jet thrust F at a given power required Pq . Then the

specific power is SP = P/m,  ̇ and the specific fuel consumption is sfc = w/P.  ̇

The reference performance is at sea-level-standard static conditions (subscript 0), and MCP (subscript C). Referred or corrected engine parameters are used in the model:

power

δ^

fuel flow

δ^

mass flow rh

thrust

δ S

specific power

P

θ

m

turbine speed √N

θ

For each rating R, the performance is characterized by the following quantities for sea-level-standard

static conditions: power P0R, specific power SP0R, and mechanical power limit PmechR. The mass flow
Referred Parameter Turboshaft Engine Model 137

is then m ̇ 0R = P0R/SP0R. The gross jet thrust Fg0C is given at MCP. These characteristics are at the

specification turbine speed Nspec.

The installed power required Preq and power available Pav > Preq are measured at the engine output

shaft. In addition to shaft power, the engine exhaust produces a net jet thrust FN , from mass flow that

goes through the engine core. The fuel flow and mass flow are the total required to produce the shaft

power and jet thrust. The forces produced by mass flow that does not go through the engine core (such

as IR suppressor or cooling air) are treated as momentum drag Daux.

The difference between net and gross jet thrust is the momentum drag: Fn = Fg — m ̇ reqV =

m ̇ req (Vj — V), where Vj is the engine-jet-exhaust velocity. Note that traditional units for mass flow are

pound/sec (pps), while this equation requires slug/sec (mreq/g  ̇ replaces m ̇ req).

The uninstalled power required is Pq , the power available Pa , the gross jet thrust Fg , and net jet thrust

Fn . The engine model calculates Pa as a function of flight condition and engine rating; or calculates

engine mass flow, fuel flow, and jet thrust at Pq .

18–4 Installation

The difference between installed and uninstalled power is the inlet and exhaust losses Ploss:

Pav = Pa — Ploss

Preq = Pq — Ploss

The inlet ram-recovery efficiency ηd (through δM) is included in the engine model calculations. The inlet

and exhaust losses are modeled as fractions of power available or power required: Ploss = (tin + tex )Pa

or Ploss = (tin + tex) Pq . So

Pav = Pa (1— tin — tex)

Pq = Preq/(1 — tin — tex)

The engine model gives uninstalled power and the gross thrust Fg for a nominal exhaust nozzle area. The

gross jet thrust FG and exhaust power loss (tex) are both functions of the exhaust-nozzle area. Smaller

exhaust-nozzle areas increase exhaust losses and increase gross thrust. Thus the ratio of installed to

uninstalled thrust is approximated by a function of the exhaust power loss:

FG/ Fg = Kfgr = Kfgr0 + Kfgr1tex + Kfgr2t

2

ex + Kfgr3t

3

ex

so the net installed jet thrust is

FN = KfgrFg — m ̇ reqV

The momentum drag of the auxiliary air flow is a function of the mass flow m ̇ aux = fauxm ̇ req :

Daux = (1 — ηaux)m ̇ auxV = (1 — ηaux)fauxm ̇ reqV

where ηaux is the ram-recovery efficiency. Exhaust losses (tex) and auxiliary air-flow parameters (ηaux,

faux) are defined for IR suppressor on and off. Inlet particle separator loss is added to the inlet losses

(tin).

18–5 Power-Turbine Speed

The shaft power available is a function of the gas power available PG and the power-turbine

efficiency ηt: Pa = ηt PG . Generally the power-turbine speed N has a significant effect on ηt , but almost
138 Referred Parameter Turboshaft Engine Model

no effect on PG . The model used for the efficiency variation is ηt = 1— I(N/Nopt) — 1 IXNη , where Nopt

is the speed for peak efficiency, hence

P(N) ηt

(N) 1 — I(N/Nopt) — 1 I

XNη

P(Nspec) 

=

ηt(Nspec) 

=

1 — I (Nspec /Nopt) — 1 I

XNη

Two approximations for the optimum turbine speed are used. The first is a linear expression in p =

P/(P0Rδ0θ):

• ^ KNoptB

Nopt = Nspec v V 

(

KNoptA θM + δM p

and the second is a cubic function of p = P/(P0Cδ0θ-) :

Nopt = Nopt0C0θ 

(

KNopt0 + KNopt1p + KNopt2p

2

+ KNopt3p

3 ) [θM]

XNopt

The second expression is based on a larger data sample. For power-available calculations, P = Pa

(Nspec );

for power-required calculations, P = Pq

(N).

18–6 Power Available

Given the flight condition and engine rating, the power available Pa is calculated as follows. The

specific power and referred mass flow (at Nspec , relative to SP0 and m ̇ 0 for this rating) are approximated

by functions of the ambient temperature ratio θ and inlet ram air ratios:

SPa(Nspec) = SP0 θ Kspa Lf l

ISM

Xspa

BM]

ma(Nspec) = m0 (δ/0θ-

/

1 eKmfa 

I r 

δM θM

] 

Xmfa

where the static lapse rate (Kspa , Km fa) and ram air exponents (Xspa , Xm fa) are piecewise linear

functions of θ. The power available is then

Pa(Nspec) = P0

 SPa(Nspec) ma(Nspec) 

= P0

(δ0θ 

/ 1 

K pae

Kmfa f δM θM] 

Xsa+Xmf a

0 m0 \ L

This expression for m ̇ a is used only to calculate Pa ; elsewhere the m ̇ q expression for performance at

power required is used to obtain the mass flow at a power Pq

. Finally

Pa (N) = Pa (Nspec) 1 — I (N/Nopt) — 1 I

XNη

1 — I (Nspec /Nopt) — 1 I

XN

is the power available at turbine speed N. Installation losses Ploss are subtracted from Pa (Pav

Pa — Ploss), and then the mechanical limit is applied: Pav = min(Pav , PmechR).

18-6.1 Piecewise Linear Parameters

Several parameters K are input as piecewise linear functions of the temperature ratio θ. One format

of the input is a set of I regions, with the function in the i-th region given by K = K0i + K1iθ. The break

point between the i and i — 1 regions is

θbi = —

K0i — K0(i−1)

K1i — K1(i−1)

Kbi = K0i + K1iθb
Referred Parameter Turboshaft Engine Model 139

for i = 2 to I. Another format is a set of I + 1 break points, with the values at the i-th point (θbi, Kbi).

Then the coefficients between i and i + 1 are

K0i = 

Kbiθb(i+1) − Kb(i+1)θbi

θb(i+1) − θbi

K1i = 

−Kbi + Kb(i+1)

θb(i+1) − θbi

for i = 1 to I. The interpolation is performed using K = K0i + K1iθ for θ in the range θbi to θb(i+1).

Outside the defined regions, the linear expression is continued; hence θbi is used only for i = 2 to I.

18–7 Performance at Power Required

The engine performance (mass flow, fuel flow, and gross jet thrust) is calculated for a specified power

required Pq (which might equal the power available), flight condition, and engine rating. Installation

losses Ploss are added to Preq (Pq = Preq + Ploss). The referred quantities (relative to SLS static MCP

quantities) are approximated by cubic functions of q = Pq (Nspec )/(P0Cδ

√

θ) :

w ̇ req = w ̇ 0C

(

δ√θ-

)

(

Kffq0 + Kffq1q + Kffq2q

2

+ Kffq3q

3

) [θM] Xffq

(

rhreq = rh0C I δ/

√

θ

\

1 (Kmfq0 + Kmfq1q + Kmfq2q

2 + Kmfq3 q3) [θM]

Xmfq

Fg = Fg0C (δ) 

(

K/

fgq0 + Kfgq1q + Kfgq2q

2 + Kfgq3q3

) [θM]

Xfgq

at Nspec , with w ̇ 0C = sfc0CP0C . The mass flow and fuel flow are primarily functions of the gas power PG ,

and are assumed to be independent of ηt , hence independent of turbine speed. However, these equations

are functions of Pq (Nspec), obtained from Pq (N) using

Pq (Nspec) = Pq (N) 1 

−

1

I ( (N

/N opt)

t) 

|

1 |XNη

Then the installed net jet thrust FN and momentum drag Daux are calculated.

18–8 Scaling

The parameters of the engine model can be defined for a specific engine, but it is also necessary to

scale the parameters as part of the aircraft sizing task, in order to define an engine for a specified power.

In addition, advanced technology must be represented in the model. Scaling and advanced technology

are handled in terms of specific power and specific fuel consumption (at SLS static conditions, MCP,

and Nspec).

The engine model includes reference values of the engine performance parameters: P0R, SP0R,

PmechR , sfc0C , SF0C , Nspec, Nopt0C . Mass flow and fuel flow are obtained from m ̇ 0R = P0R/SP0R and

w ̇ 0C = sfc0CP0C . The reference power at each engine rating R defines a ratio to MCP: rp0R = P0R/P0C .

Similarly for specific power and mechanical limits: rs0R = SP0R/SP0C , rm0R = PmechR/P0C . These

ratios are kept fixed when the engine is scaled.

The engine size is specified as takeoff power Pto = Peng, which is the power at rating R for SLS

static conditions and specification turbine speed Nspec. Hence the MCP is P0C = Pto/rp0R, and the

power at all other ratings follows.
140 Referred Parameter Turboshaft Engine Model

The engine technology parameters SP0C and sfc0C are assumed to vary linearly with mass flow m ̇ 0C

up to a limit m ̇ lim, and constant thereafter at SPlim and sfclim . The mass flow at the reference condition

is m ̇ ref = Pref/SPref, with the reference values SPref and sfcref. The functions are defined by the limit

values, and by the intercept values SPzero and sfczero at m ̇ 0C = 0. If m ̇ ref > m ̇ lim, the limit values

are set to the reference values. If m ̇ ref < m ̇ lim, the intercept values are projected from the reference

values: SPzero = SPref — m ̇ refx , x = (SPlim — SPref)/(m ̇ lim — m ̇ ref); and similarly for sfczero. Then for

m ̇ 0C < m ̇ lim

SP0C = SPzero + Ksp1m ̇ 0C = Ksp0 + Ksp 1m ̇ 0C

sfc0C = sfczero + Ksfc1m ̇ 0C = Ksfc0 + Ksfc1m ̇ 0C

and for m ̇ 0C > m ̇ lim

SP0C = SPlim

sfc0C = sfclim

From the limit and intercept values, the slopes are Ksp1 = (SPlim — SPzero )/m ̇ lim and Ksfc1 = (sfclim —

sfczero)/?hlim. Usually the effect of size gives Ksp2 > 0 and Ksfc2 < 0. The power at the limit is

Plim = SPlimm ̇ lim. Using m ̇ 0C = P0C/ SP0C , the specific power equation can be solved for the mass flow

given the power:

⎧

⎨ P0C/SPlim

P0C > Plim or Ksp1 = 0

m ̇ 0C = ^

⎩ (K2 1 ) 2 + P0

CK1

 — 

K2 1 otherwise

From this mass flow, SP0C and sfc0C are calculated, hence the fuel flow w ̇ 0C = sfc0CP0C . The specific

thrust available at MCP is assumed to be constant, and the specification power turbine speed decreases

with the mass flow:

Fg0C = SF0C m ̇ 0C

Nspec = 

(

Nspec — KNs2 / m0C) ref + KNs2 / m0C = KNs1 + KNs2 / m ̇ 0C

Rt

ecJ

1

Nopt0C = Nspec

ref

Then the power and specific power at all ratings R are obtained from the ratios: P0R = rp0RP0C,

SP0R = rs0RSP0C , PmechR = rm0RP0C.

18–9 Engine Speed

The model as described in the previous sections may not adequately account for variation of engine

performance with engine speed, so it is also possible to define the parameters corresponding to a set

of engine-speed ratios r = N/Nspec. Then the engine-performance and power-available quantities are

linearly interpolated to obtain the values at the required engine speed N. If this option is used, then the

correction based on P(N) /P(Nspec ) = ηt(N)/ηt(Nspec) is not applied.

18–10 Weight

The engine weight can be a fixed input value, calculated as a function of power, or scaled with

engine mass flow. As a function of power, the weight of one engine is:

Wone eng = K0eng + K1engP + K2engP

Xeng
Referred Parameter Turboshaft Engine Model 141

where P is the installed takeoff power (SLS static, specified rating) per engine. A constant weight per

power W/P is given by using only K1eng . Alternatively, the specific weight SW = P/W can be scaled

with the mass flow m ̇ 0C . The scaling is determined from the specifc weight SWref at the reference mass

flow m ̇ ref; and either the limit SWlim at m ̇ lim (for m ̇ ref < m ̇ lim) or the intercept SWzero at m ̇ = 0 (for

m ̇ ref > m ̇ lim). Then

/

SWzero + Ksw1m ̇ 0C = Ksw0 + Ksw1m ̇ 0C m ̇ 0C < m ̇ lim

SW =

SWlim m ̇ 0C >_ m ̇ lim

and Wone eng = P/SW.

18–11 Units

In this engine model, only the reference values and scaling constants are dimensional. Conventional

English units and SI units are shown in table 18-2. Units of specific power and specific fuel consumption

follow from these conventions.

Table 18-2. Conventional units.

power P mass flow m ̇ fuel flow w ̇ force F turbine speed N

English: horsepower pound/sec pound/hour pound rpm

SI: kiloWatt kilogram/sec kilogram/hour Newton rpm

18–12 Typical Parameters

Typical values of the principal parameters describing the engine and its performance are given in

table 18-3 for several generic engine sizes. These values represent good current technology. Advanced

technology can be introduced by reducing the specific fuel consumption and weight, and increasing the

specific power. Typical ratios of the power, specific power, and mass flow to the values at MCP are given

in table 18-4 for several ratings. Figure 18-1 shows typical performance characteristics: fuel flow w, ̇

mass flow m,  ̇ and net jet thrust Fg variation with power P and speed (referred quantities, normalized, at

Nspec). Figure 18-2 shows typical variation of the power available with engine-turbine speed. Figures

18-3 to 18-8 show typical power-available characteristics: specific power SP, mass flow m,  ̇ and power P

variation with temperature ratio θ, for static and 200 knots conditions and several engine ratings (referred

quantities, normalized, at Nspec).
142 Referred Parameter Turboshaft Engine Model

Table 18-3. Typical engine-performance parameters.

power (MCP) P0C 500 1000 2000 4000 8000 16000 hp

specific power SP0C 116 125 134 143 153 164 hp/lb/sec

mechanical limit Pmech 750 1500 3000 6000 12000 24000 hp

specific fuel cons. sfc0C 0.54 0.48 0.44 0.40 0.38 0.35 lb/hp-hr

specific jet thrust SF0C 5.9 7.3 8.9 11.0 13.6 16.7 lb/lb/sec

gross jet thrust Fg0C 25 58 134 308 707 1625 lb

mass flow m ̇ 0C 4.3 8.0 15.0 27.9 52.1 97.3 lb/sec

turbine speed Nspec 35600 26100 19100 14000 10200 7500 rpm

weight W/Pto 0.34 0.23 0.18 0.16 0.16 0.15 lb/hp

Table 18-4. Typical parameter ratios for various ratings (percent).

IRP MRP CRP Pmech

power P0 /P0C 120 127 133 150

specific power SP0 /SP0C 117 123 128

mass flow m ̇ 0

/m ̇ 0C 102.6 103.2 103.9
w

·

 / w·

0C static w

·

 / w·

0C 200 knots

m

·

 / m·

0C static m

·

 / m·

0C 200 knots

• ' ' ' ' ' ' • Fg/Fg0C static - - - - - - - - Fg/Fg0C 200 knots

1.5

1.0

0.5

Referred Parameter Turboshaft Engine Model 143

2.0

0.0

0.0 0.5 1.0 1.5 2.0

P(Nspec)/P0C

Figure 18-1. Fuel flow, mass flow, and net jet thrust variation with power.

1.2

1.1

1.0

w

z 0.9

z

a

0.8

0.7

0.6

0.4 0.6 0.8 1.0

N/Nspec

Figure 18-2. Power variation with turbine speed.

1.2
144 Referred Parameter Turboshaft Engine Model

1.8

1.6

1.4

a° 1.2

a

`n 1.0

0.8

0.6

0.4

0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15

temperature ratio 0

Figure 18-3. Specific power variation with temperature ratio, static.

1.8

1.6

1.4

e 1.2

a

`n 1.0

0.8

0.6

0.4

0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15

temperature ratio 0

Figure 18-4. Specific power variation with temperature ratio, 200 knots.
Referred Parameter Turboshaft Engine Model 145

1.20

.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15

temperature ratio 0

Figure 18-5. Mass-flow variation with temperature ratio, static.

1.20

1.15

1.10

1.05

1.00

0.95

0.90

0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15

temperature ratio θ

Figure 18-6. Mass-ßow variation with temperature ratio, 200 knots.

1.15

1.10

1.05

1.00

0.95

0.90

0
146 Referred Parameter Turboshaft Engine Model

1.8

1.6

1.4

0 1.2

a

1.0

0.8

0.6

0.4

0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15

temperature ratio 0

Figure 18-7. Power variation with temperature ratio, static.

1.8

1.6

1.4

0 1.2

a

1.0

0.8

0.6

0.4

0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15

temperature ratio 0

Figure 18-8. Power variation with temperature ratio, 200 knots.
Chapter 19

AFDD Weight Models

This chapter presents the rotorcraft weight models developed by the U.S. Army Aeroflightdynamics

Directorate (AFDD). For some weight groups several models are available, designated AFDDnn. The

weights are estimated from parametric equations, based on the weights of existing turbine-powered

helicopters and tiltrotors. The figures of this chapter compare the weights calculated from these equations

with the actual weights. The results of these equations are the weight in pounds, and the units of the

parameters are noted in the tables. Technology factors χ are included in the weight equations. Typically

the input includes a weight increment that can be added to the results of the weight model. Weights

of individual elements in a group can be fixed by setting the corresponding technology factor to zero,

hence using only the input increment.

The weight models are implemented as part of the aircraft components. The weights are entered

into the weight statement data structure (extended RP8A format) for each component, reflected in the

organization of this chapter.

19–1 Wing Group

The wing group consists of: basic structure (primary structure, consisting of torque box and spars,

plus extensions); fairings (leading edge and trailing edge); fittings (non-structural); fold/tilt structure;

and control surfaces (flaps, ailerons, flaperons, and spoilers). There are separate models for a tiltrotor

or tiltwing configuration and for other configurations (including compound helicopter).

19-1.1 Tiltrotor or Tiltwing Wing

Wing weight equations for a tiltrotor or tiltwing aircraft are based on methodology developed by

Chappell and Peyran (refs. 1 and 2). The wing is sized primarily to meet torsional stiffness requirements.

The primary structure weight is calculated from torque box and spar weights:

Wbox = Atbρtbbw /etb

Wspar = CtAspρsp bw /esp

wprim = (Wbox + Wspar) funits

Wprim = χprimwprim

A consistent mass-length-time system is used in the equations for Wbox and Wspar, which therefore have

units of slug or kilogram. The primary structure weight Wprim however has units of pound or kilogram,

hence a conversion factor funits = g is required for English units. The wing fairing (leading edge and

trailing edge), control surface (flaps, ailerons, flaperons, and spoilers), fittings (non-structural), and
148 AFDD Weight Models

fold/tilt weights are:

wfair = Sfair Ufair Wfair = χfairwfair

wflap = SflapUflap Wflap = χflapwflap

ffit

wfit =

1 — At (wprim + wfair + wflap) Wfit = χfitwfit

wfold = ffold (Wprim + Wfair + Wflap + Wfit + Wtip)

Wfold = χfoldwfold

The control surface area Sflap for a tiltrotor wing is the sum of the flap and flaperon areas. The fairing

area is

Sfair = (bw — wattach) cw ( 1 — wtb) — Sflap

The wing extension weight is:

wext = Sext Uext Wext = χextwext

wefold = fefold Wext Wefold = χefoldwefold

and these terms are added to Wprim and Wfold. The tiltrotor-wing weight (and wing folding weight in

fuselage group) depends on the weight on the wing tips, Wtip, which is the sum of rotor group, engine

section or nacelle group, air induction group, engine system, drive system (except drive shaft), rotary

wing and conversion flight controls, hydraulic group, trapped fluids, and wing extensions. The weight

on wing tip is used as the fraction ftip = Wtip /WSD ; the mass on the wing tip is Mtip (slug or kg).

To estimate the wing weights, the required stiffness is scaled with input frequencies (per rev) of the

wing primary bending and torsion modes. First the torque box is sized to meet the torsional stiffness

(frequency) requirement. Next spar-cap area is added as required to meet the chord and beam bendingfrequency requirements. Finally spar-cap area is added if necessary for a jump takeoff condition. Wing

section form factors, relating typical airfoil and torque-box geometry to ideal shapes, are input or

calculated from the thickness-to-chord ratio and the torque-box-chord to wing-chord ratio:

FB = 0.073 sin(2π(τw — 0 . 151)/0. 1365) + 0.14598τw

+ 0. 610 sin(2π(wtb +0. 080)/2 . 1560) — (0. 4126 — 1.6309τw) (wtb — 0. 131) + 0. 0081

FC = 0.640424w

2

tb — 0. 89717wtb + 0.4615τw + 0. 655317

FT = ((0.27 — τw )/0.12)0.12739 (—0 .96 + V3.32 + 94.6788wtb — (wtb/0.08344) 2

)

— 2.7545w

2

tb + 5. 1799wtb — 0. 2683

FVH = 0. 25 sin(5 . 236wtb) + 0. 325

for beam bending, chord bending, torsion, and spar cap vertical/horizontal bending. The ideal shape

for torsional stiffness is a tube of radius tw , so the torsional stiffness J = FTAtbt

2

w /4. The ideal shape

for chord bending is two caps ctb apart, so ICtb = FCAtbc

2

tb/4. The ideal shape for beam bending is two

caps tw apart, so IBsp = FVHAsp t

2

w /4 and IBtb = FBAtbt

2

w /4. The torque-box cross-sectional area is

obtained from the wing-torsion frequency;

GJ = (ωTΩ) 2 1

2

bw

12 Mtipr2pylon

Atb = 4GJ/(GtbFTt

2

w )
AFDD Weight Models 149

The spar-cap cross-sectional area (in addition to torque-box material) is obtained from beam and chord

bending frequencies:

2 1 3 1

EIC = (ωCΩ) 24 bw 2 Mtip fmode

2 1 3 1

EIB = (ωBΩ) 24 b 2 Mtip fmode

EICtb = EtbFCAtbc

2

tb/4

EICsp = EIC — EICtb

ACsp = EICsp / (Espc

2

tb/4)

EIBtb = EtbFBAtbt

2

w /4

EIVH = EspFVHACsp t

2

w /4

EIBsp = EIB — EIBtb — EIVH

ABsp = EIBsp /(Esp t

2

w /4)

EIsp = EIVH + EIBsp

Asp = ACsp + ABsp

where EICsp and EIBsp are replaced by zero if negative (no additional spar material required). The factor

fmode = 1 — ftip is a mode-shape correction for fuselage motion. Next the primary structure, fairing,

flap, and fitting weights are calculated as before; and the sum Wwing = Wprim + Wfair + Wflap + Wfit.

Additional spar-cap material for a jump takeoff condition is obtained from the ultimate applied bending

moment at the wing root:

MU = Tcaptw (0.75(1 — ftip) — 0.375(tw/bw )(Wwing/WSD))

where Tc ap is the maximum thrust capability of one rotor, equal to the greater of njumpWSD/Nrotor or

(CT/σ)ρAbV

2

tip (from an input CT/σ at the jump takeoff condition, SLS and hover rotor speed). The

bending-moment capacity of the wing is

Mtb = 2EIBtbEU/tw

Msp = 2CmEIsp ^U/tw

Then the additional cross-section area is obtained from the moment deficit:

ΔM = MU — (Mtb + Msp)

ΔAsp = 2ΔM/(^UEsptw )

ΔWspar = CjΔAsp ρsp bw /esp

where ΔM is replaced by zero if negative (no additional spar material required). If ΔWspar is positive, it

is added to Wspar, and the primary structure, fairing, flap, and fitting weights are recalculated. Parameters

are defined in table 19-1, including units as used in these equations. Here a consistent mass-length-time

system is used, producing Wbox and Wspar in slug or kilogram. Typically the input uses conventional

English units for density (pound/inch3 ) and modulus (pound/inch2 ).
150 AFDD Weight Models

Table 19-1. Parameters for tiltrotor-wing weight.

parameter definition units

Wbox, Wspar wing torque box and spar weights slug or kg

WSD structural design gross weight lb

(CT/σ)jump rotor maximum thrust capability (jump takeoff)

njump load factor at WSD (jump takeoff)

Nrotor number of rotors

bw wing span (length of torque box) ft or m

tw = bw — wfus wing length (span less fuselage width) ft or m

cw wing chord ft or m

wtb torque box chord to wing chord ratio

ctb = wtbcw torque box chord ft or m

τw wing airfoil thickness-to-chord ratio

tw = τwcw wing thickness ft or m

rpylon pylon radius of gyration (pitch inertia = r

2

py lon Mtip) ft or m

Ω rotor speed for wing weight design condition rad/sec

ωT wing torsion mode frequency (fraction rotor speed) per rev

ωB wing beam bending mode frequency (fraction rotor speed) per rev

ωC wing chord bending mode frequency (fraction rotor speed) per rev

ρtb density of torque box material slug/ft3 or kg/m3

ρsp density of spar cap material slug/ft3 or kg/m3

Gtb torque box shear modulus lb/ft2 or N/m2

Etb torque box modulus lb/ft2 or N/m2

Esp spar modulus lb/ft2 or N/m2

EU ultimate strain allowable (minimum of spar and torque box)

Ct weight correction for spar taper (equivalent stiffness)

Cj weight correction for spar taper (equivalent strength)

Cm strength correction for spar taper (equivalent stiffness)

etb structural efficiency factor, torque box

esp structural efficiency factor, spars

Ufair unit weight of leading and trailing edge fairings lb/ft2 or kg/m2

Uflap unit weight of control surfaces lb/ft2 or kg/m2

Sfair area of leading and trailing edge fairings ft2 or m2

Sflap area of control surfaces ft2 or m2

ffit wing fittings and brackets (fraction total weight excluding fold)

ffold wing fold/tilt (fraction total weight excluding fold, including weight on tips)

wattach width of wing structural attachments to body ft or m

Uext unit weight of wing extension lb/ft2 or kg/m2

Sext area of wing extensions (span bext times mean chord cext) ft2 or m2

fefold wing extension fold/tilt (fraction extension weight)
AFDD Weight Models 151

19-1.2 Aircraft Wing

There are two models intended for the wing of a compound helicopter: area method and parametric

method. For the area method (based on weight per unit area), the total wing weight excluding folding

is:

wwing = Sw Uw

fprim = 1 — ffair — fflap — ffit

Typically Uw = 5 to 9 lb/ft2 . For the parametric method (AFDD93), the total wing weight including

folding is:

WSD 0.847

wwing = 5.66411fLGloc n

0.39579S0

w.

21754 A0

w.

50016

1000 cos Λw ) z

((1 + λw )/τw )

0.09359 (1 — bfold ) -0. 14356

fprim = 1 — ffair — fflap — ffit — ffold

where fLGloc = 1. 7247 if the landing gear is on the wing, and 1 . 0 otherwise. Based on 25 fixed-wing

aircraft, the average error of the aircraft-wing equation is 3.4% (fig. 19-1). Then the primary structure,

secondary structure, and control surface weights are:

Wprim = χprim fprimwwing

Wfair = χfairffairwwing

Wflap = χflap fflapwwing

Wfit = χfitffitwwing

Wfold = χfold ffoldwwing

The wing extension weight is:

wext = Sext Uext Wext = χextwext

wefold = fefoldWext Wefold = χefoldwefold

and these terms are added to Wprim and Wfold . Parameters are defined in table 19-2, including units as

used in these equations.

Table 19-2. Parameters for aircraft-wing weight.

parameter definition units

Sw wing planform area (theoretical) ft2

Uw unit weight of wing planform lb/ft2

WSD structural design gross weight lb

nz design ultimate flight load factor at WSD g

Λw wing sweep angle deg

Aw wing aspect ratio

λw wing taper ratio (tip chord/root chord)

τw wing airfoil thickness-to-chord ratio

bfold fraction wing span that folds (0 to 1)

ffair fairings (fraction total wing weight)

fflap control surfaces (fraction total wing weight)

ffit fittings (fraction total wing weight)

ffold fold/tilt (fraction total wing weight)

Uext unit weight of wing extension lb/ft2

Sext area of wing extensions (span bext times mean chord cext) ft2

fefold wing extension fold/tilt (fraction extension weight)
152 AFDD Weight Models

19–2 Rotor Group

The rotor group consists of: blades, hub and hinge, spinner, and blade fold structure. The blade

and hub-hinge weights for the AFDD82 model are:

wblade = 0.02606NrotorN

0.6592R1.3371^0.9959V0.6682ν blade tip blade2.5279 Wblade = Xblade blade w

whub = 0.003722NrotorN

0 .2807R1.5377V0.4290ν blade tip

hub

hub ( blade rotor

414 W/N ) 0 .5505 Whub = Xhub hub w

Based on 37 aircraft, the average error of the blade equation is 7.7% (fig. 19-2). Based on 35 aircraft,

the average error of the hub equation is 10.2% (fig. 19-3). The blade and hub-hinge weights for the

AFDD00 model are:

w = 0. 0024419 N

N0 .53479R

1.74231^0.77291V0 .87562ν blade (tilt rotor blade tip blade2.51048 Wblade = Xblade blade w

whub = 0.18370NrotorN

0.16383R0.19937V0 .06171ν blade tip hub0.46203 

( blade rotor W /N ) 1.02958 Whub = Xhub hub w

where ftilt = 1 . 17940 for tilting rotors; 1. 0 otherwise. Based on 51 aircraft, the average error of the

blade equation is 7.9% (fig. 19-4). Based on 51 aircraft, the average error of the hub equation is 9.2%

(fig. 19-5). For teetering and gimballed rotors, the flap frequency ν should be the coning frequency. The

fairing/spinner and blade fold weights are:

Wspin = χspin 7.386NrotorD2spin

Wfold = χfold ffold Wblade

The blade weight is for all blades of the rotors. If the weight is evaluated separately for each rotor, then

Nrotor = 1 should be used in the equations. Typically ffold = 0. 04 for manual fold, and ffold = 0. 28 for

automatic fold. Parameters are defined in table 19-3, including units as used in these equations.

Table 19-3. Parameters for rotor weight.

parameter definition units

Nrotor number of rotors

Nblade number of blades per rotor

R rotor radius in hover ft

c rotor mean geometric blade chord ft

Vtip rotor hover tip velocity ft/sec

νblade, νhub flap natural frequency (for weight estimate) per rev

Dspin spinner diameter ft

ffold blade fold weight (fraction total blade weight)

For lift-offset rotors, the blade and hub weights can be calculated based on the methodology of

reference 3. The blade and hub-hinge weights are:

wblade = Nrotor 0. 000041466 w R3

/(2ht2

.2R

) Wblade = χbladewblade

whub = Nrotor (0.17153 w RNblade Whub = χhubwhub

+ 0.000010543 (Wblade /Nrotor) Vt2ipt.2R/R

+ 0.081304 w R2 2hL/t.2R

)
AFDD Weight Models 153

where w = nzWSD/1000, and

(t .2R

0.8 + 0.2λ1 = τ.2Rc 0.5 + 0.5λ J

is the blade thickness at 20%R. These equations were developed for the coaxial rotor configuration

(Nrotor = 2). The blade-weight estimate is based on the stiffness required for tip clearance of the two

rotors. The first two terms in the hub-weight equation account for the structure required to react the hub

moment and the centrifugal force; the last term accounts for the weight of the main-rotor upper shaft.

Parameters are defined in table 19-4, including units as used in these equations.

Table 19-4. Parameters for lift-offset rotor weight.

parameter definition units

Nrotor number of rotors

WSD structural design gross weight lb

nz design ultimate flight load factor at WSD g

λ blade taper ratio (tip chord/root chord)

τ.2R blade airfoil thickness-to-chord ratio (at 20% R)

c blade mean chord

Nblade number of blades per rotor

h coaxial rotor separation (fraction rotor diameter)

L lift offset (Mroll/TR)

Vtip rotor hover tip velocity ft/sec

19–3 Empennage Group

The empennage group consists of: horizontal tail, vertical tail, tail rotor, and auxiliary thruster. The

weight model depends on the aircraft configuration. The helicopter or compound model is AFDD82.

The horizontal tail weight is:

tiltrotor or tiltwing Wht = χhtSht(0.00395ShtVdive -0.4885)

helicopter or compound Wht = χht0.7176Sht ht

1.1881A0ht

Based on 13 aircraft, the average error of the helicopter horizontal tail equation is 22.4% (fig. 19-6).

The vertical tail weight is:

tiltrotor or tiltwing Wvt = χvtSvt (0.00395S

0t

Vdive -0.4885)

helicopter or compound W 1 . 0460 S

0.9441A0.5332

vt = Xvt ftr vt vt

where ftr = 1 .6311 if the tail rotor is located on the vertical tail; 1 .0 otherwise. Based on 12 aircraft, the

average error of the helicopter vertical tail equation is 23.3% (fig. 19-7). Vdive is the design dive speed,

calculated or input; Vdive = 1 . 25Vmax , where Vmax is the maximum speed at design gross weight and SLS

conditions. The tail rotor weight is:

Wtr = χtr 1.3778R°0

897(PDSlimitR/Vtip )0.8951

Based on 19 aircraft, the average error of the helicopter tail rotor equation is 16.7% (fig. 19-8). The

auxiliary propulsion weight (as a propeller) is:

compound, propeller W 0. 0809484N T

1.04771(T /A )−0.07821

at = Xat at at at at
154 AFDD Weight Models

where Tat is at maximum speed, design gross weight, and SLS conditions, calculated or input. Parameters

are defined in table 19-5, including units as used in these equations.

Table 19-5. Parameters for tail weight.

parameter definition units

Sht horizontal tail planform area ft2

Svt vertical tail planform area ft2

Aht horizontal tail aspect ratio

Avt vertical tail aspect ratio

Vdive design dive speed at sea level kts

Rtr tail rotor radius ft

PDSlimit drive system rated power hp

R main rotor radius in hover ft

Vtip main rotor hover tip velocity ft/sec

Nat number of auxiliary thrusters

Tat thrust per propeller lb

Aat auxiliary thruster disk area ft2

19–4 Fuselage Group

The fuselage group consists of: basic structure; wing and rotor fold/retraction; tail fold/tilt; and

marinization, pressurization, and crashworthiness structure. The AFDD84 model is a universal bodyweight equation, used for tiltrotor and tiltwing as well as for helicopter configurations. The AFDD82

model is a helicopter body-weight equation, not used for tiltrotor or tiltwing configuration.

For the AFDD84 (UNIV) model, the basic structure weight is

(WMTO 

)0.4879 

nzWSD 10

.2075 0

.1676 0.1512 wbasic = 25.41fLGlocfLGretframp 1 1000 1000 ) Sbody

Wbasic = χbasicwbasic

where fLGloc = 1 . 1627 if the landing gear is located on the fuselage, and 1 .0 otherwise; fLGret = 1. 1437

if the landing gear is on the fuselage and retractable, and 1 . 0 otherwise; framp = 1 . 2749 if there is a cargo

ramp, and 1 . 0 otherwise. Based on 35 aircraft, the average error of the body equation is 6.5% (fig. 19-9).

The tail fold, wing and rotor fold, marinization, pressurization, and crashworthiness weights are:

wtfold = ftfold Wtail Wtfold = χtfoldwtfold

wwfold = fwfold (Wwing + Wtip) Wwfold = χwfoldwwfold

wmar = fmarWbasic Wmar = χmarwmar

wpress = fpressWbasic Wpress = χpresswpress

wcw = fcw (Wbasic + Wtfold + Wwfold + Wmar + Wpress) Wcw = χcwwcw

Typically ftfold = 0.30 for a folding tail, and fcw = 0.06. For wing folding the weight on the wing tip

(Wtip) is required (calculated as for the wing group). Parameters are defined in table 19-6, including
AFDD Weight Models 155

units as used in these equations.

Table 19-6. Parameters for fuselage weight (AFDD84 model).

parameter definition units

WMTO maximum takeoff weight lb

WSD structural design gross weight lb

Sbody wetted area of body ft2

R main rotor radius ft

nz design ultimate flight load factor at WSD g

B length of fuselage ft

ftfold tail fold weight (fraction tail weight)

fwfold wing and rotor fold weight (fraction wing/tip weight)

Wtail tail group weight

Wwing + Wtip wing group weight plus weight on wing tip

fmar marinization weight (fraction basic body weight)

fpress pressurization (fraction basic body weight)

fcw crashworthiness weight (fraction fuselage weight)

For the AFDD82 (HELO) model, the basic structure weight is

^

WMTO

^0.4908

wbasic = 5.896 framp 1000n

0.1323 Sb.2dy44 0 .6100

Wbasic = χbasicwbasic

where framp = 1. 3939 if there is a cargo ramp, and 1 . 0 otherwise. Based on 30 aircraft, the average error

of the body equation is 8.7% (fig. 19-10). The tail fold, wing and rotor fold, marinization, pressurization,

and crashworthiness weights are:

wtfold = ftfold Wbasic Wtfold = χtfoldwtfold

wwfold = fwfold (Wbasic + Wtfold) Wwfold = χwfoldwwfold

wmar = fmarWbasic Wmar = χmarwmar

wpress = fpress Wbasic Wpress = χpresswpress

wcw = fcw (Wbasic + Wtfold + Wwfold + Wmar + Wpress) Wcw = χcwwcw

Typically ftfold = 0. 05 for a folding tail, and fcw = 0. 06. Parameters are defined in table 19-7, including

units as used in these equations.

Table 19-7. Parameters for fuselage weight (AFDD82 model).

parameter definition units

WMTO maximum takeoff weight lb

Sbody wetted area of body ft2

R main rotor radius ft

nz design ultimate flight load factor at WSD g

B length of fuselage ft

ftfold tail fold weight (fraction basic structure)

fwfold wing and rotor fold weight (fraction basic structure and tail fold)

fmar marinization weight (fraction basic body weight)

fpress pressurization (fraction basic body weight)

fcw crashworthiness weight (fraction fuselage weight)
156 AFDD Weight Models

19–5 Alighting Gear Group

The alighting gear group consists of: basic structure, retraction, and crashworthiness structure.

There are two models, parametric (AFDD82) and fractional. The basic landing gear weight is:

parametric wLG = 0.4013WMTO

2NLG360 (W/S)0 .1525

fractional wLG = fLGWMTO

\\

and WLG = χLGwLG. Typically fLG = 0.0325 (fractional method). Based on 28 aircraft, the average

error of the parametric equation is 8.4% (fig. 19-11). The retraction and crashworthiness weights are:

wLGret = fLGretWLG WLGret = χLGretwLGret

wLGcw = fLGcw (WLG + WLGret) WLGcw = χLGcwwLGcw

Typically fLGret = 0. 08, and fLGcw = 0. 14. Parameters are defined in table 19-8, including units as used

in these equations.

Table 19-8. Parameters for landing-gear weight.

parameter definition units

WMTO maximum takeoff weight lb

fLG landing gear weight (fraction maximum takeoff weight)

W/S wing loading (1.0 for helicopter) lb/ft2

NLG number of landing gear assemblies

fLGret retraction weight (fraction basic weight)

fLGcw crashworthiness weight (fraction basic and retraction weight)

19–6 Engine Section or Nacelle Group and Air Induction Group

The engine section or nacelle group consists of: engine support structure, engine cowling, and

pylon support structure. The weights (AFDD82 model) are:

W p p 0.0412(1 — f ) (W g /N g) 1.1433N

e .3762

s. pt = Xsu t airind en en g

Wcowl = χcowl0.2315Sn

c476

Wpylon = χpylon fpylonWMTO

Based on 12 aircraft, the average error of the engine-support equation is 11.0% (fig. 19-12). Based on

12 aircraft, the average error of the engine-cowling equation is 17.9% (fig. 19-13). The air induction

group weight (AFDD82 model) is:

Wairind = χairind0.0412fairind (Weng/Neng )

1.1433N

e .3762

g

Typically fairind = 0. 3 (range 0. 1 to 0. 6). Based on 12 aircraft, the average error of the air induction

equation is 11.0% (fig. 19-14). Parameters are defined in table 19-9, including units as used in these

equations.
AFDD Weight Models 157

Table 19-9. Parameters for engine section, nacelle, and air induction weight.

parameter definition units

WMTO maximum takeoff weight lb

Weng weight all main engines lb

Neng number of main engines

Snac wetted area of nacelles and pylon (less spinner) ft2

fairind air induction weight (fraction nacelle plus air induction)

fpylon pylon support structure weight (fraction WMTO)

19–7 Propulsion Group

The propulsion group consists of the engine system, fuel system, and drive system.

19-7.1 Engine System

The engine system consists of the main engines, the engine-exhaust system, and the engine accessories. The engine-system weights are:

Weng = χengNengWone eng

Wexh = χexhNeng (K0exh + K1exhP)

W 0.5919 0.7858

acc = χacc2.0088 flub (Weng /Neng) Neng

where flub = 1 . 4799 if the accessory weight includes the lubrication system weight, 1 . 0 if the lubrication

system weight is in the engine weight. The exhaust-system weight is per engine, including any IR

suppressor. The accessory-weight equation is the AFDD82 model. Based on 16 aircraft, the average

error of the accessories equation is 11.5% (fig. 19-15). Parameters are defined in table 19-10, including

units as used in these equations.

Table 19-10. Parameters for engine-system weight.

parameter definition units

Neng number of main engines

P installed takeoff power (SLS static, specified rating) per engine hp

K0exh, K1exh engine exhaust weight vs. power, constants

19-7.2 Propeller/Fan Installation

The rotor group equations are used for propellers.

19-7.3 Fuel System

The fuel system consists of tanks and support structure (including fuel tanks, bladders, supporting

structure, filler caps, tank covers, and filler material for void and ullage), and fuel plumbing (including

fuel-system weight not covered by tank weight). The fuel-system weights (AFDD82 model) are:

W

0.7717 0.5897 1.9491

tank = χtank0.4341Cint Nint fcwf

bt

Wplumb = χplumb [K0plumb + K1plumb (0.01Nplumb + 0.06Neng ) (F/Neng)

0 .866]
158 AFDD Weight Models

where fcw = 1 . 3131 for ballistically survivable (UTTAS/AAH level) and 1 .0 otherwise. The ballistic

tolerance factor fbt = 1 . 0 to 2 . 5. The fuel flow rate F is calculated for the takeoff power rating at static

SLS conditions; typically K0plumb = 120 and K1plumb = 3. Based on 15 aircraft, the average error of the

fuel tank equation is 4.6% (fig. 19-16). Parameters are defined in table 19-11, including units as used in

these equations.

Table 19-11. Parameters for fuel-system weight.

parameter definition units

Nint number of internal fuel tanks

Cint internal fuel tank capacity gallons

fbt ballistic tolerance factor

Nplumb total number of fuel tanks (internal and auxiliary) for plumbing

Neng number of main engines

K0plumb, K1plumb plumbing weight, constants

F fuel flow rate lb/hr

19-7.4 Drive System

The drive system consists of gear boxes and rotor shafts, drive shafts, and rotor brake. This

distribution of drive-system weights is based on the following functional definitions. Gearboxes are

parts of the drive system that transmit power by gear trains, and the structure that encloses them. Rotor

shafts are the structure (typically a shaft) that transmits power to the rotor. Drive shafts are the structure

(typically a shaft) that transmits power in the propulsion system, but not directly to the rotor or by a gear

train. The rotor brake weight encompasses components that can prevent the rotor from freely turning.

The gear-box and rotor-shaft weights for the AFDD83 model are:

wgbrs = 57.72P

0.8,,m f0 .0680N

g .0663 /1000) 0.0369 /Ω0 .6379

DSlimit Q gb (Ωeng rotor

Wgb = χgb (1 — frs) wgbrs

Wrs = χrsfrswgbrs

Based on 30 aircraft, the average error of the gear-box and rotor-shaft equation is 7.7% (fig. 19-17). The

gear-box and rotor-shaft weights for the AFDD00 model are:

wgb,, = 95. 7634N

0.38553P0.78137 Ω0.09899 /Ω0.80686

gbrs rotor DSlimit eng rotor

Wgb = χgb (1 — frs) wgbrs

Wrs = χrsfrswgbrs

Based on 52 aircraft, the average error of the gear-box and rotor-shaft equation is 8.6% (fig. 19-18).

Typically frs = 0. 13 (range 0. 06 to 0. 20). Parameters are defined in table 19-12, including units as used

in these equations.

The drive-shaft (AFDD82 model) and rotor-brake weights are:

Wds = χds1.166^/_^/0 .3828

 x

1.0455N0.3909 (0.01fP)0.2693

"G DSlimit hub ds

Wrb = χrb0.000871Wblade (0.01Vtip)

2
AFDD Weight Models 159

where fP = fQΩother/Ωmain. Based on 28 aircraft, the average error of the drive-shaft equation is 16.0%

(fig. 19-19). Based on 23 aircraft, the average error of the rotor-brake equation is 25.1% (fig. 19-20).

The clutch weight in the weight statement is associated with an auxiliary power unit, and is a fixed input

value. The conventional rotor drive-system clutch and free-wheeling device weights are included in the

gear-box and rotor-shaft weight equations. Parameters are defined in table 19-13, including units as

used in these equations.

Typically fP = fQ = 60% for twin main rotors (tandem, coaxial, and tiltrotor); for a single main

rotor and tail rotor, fQ = 3% and fP = 15% (18% for 2-bladed rotors).

Table 19-12. Parameters for drive-system weight.

parameter definition units

PDSlimit drive system rated power hp

Nrotor number of main rotors

Ngb number of gear boxes

Ωrotor main rotor rotation speed rpm

Ωeng engine output speed rpm

fQ second (main or tail) rotor rated torque %

(fraction of total drive system rated torque)

frs rotor shaft weight (fraction gear box and rotor shaft)

Table 19-13. Parameters for drive shaft and rotor brake weight.

parameter definition units

QDSlimit PDSlimit/Ωrotor hp/rpm

Nds number of intermediate drive shafts

xhub length of drive shaft between rotors ft

fP second (main or tail) rotor rated power %

(fraction of total drive system rated power)

Vtip main rotor tip speed ft/sec

19–8 Flight Controls Group

The flight controls group consists of cockpit controls, automatic flight control system, and system

controls. Wcc and Wafcs weights are fixed (input). System controls consist of fixed-wing flight controls,

rotary-wing flight controls, and conversion (rotor tilt) flight controls. The weight equations model

separately non-boosted controls (which do not see aerodynamic surface or rotor loads), boost mechanisms

(actuators), and boosted controls (which are affected by aerodynamic surface or rotor loads). The load

path goes from pilot, to cockpit controls, to non-boosted controls, to boost mechanisms, to boosted

controls, and finally to the component.

Fixed-wing flight controls consist of non-boosted flight controls and flight-control boost mechanisms. The weights are:

full controls w = 0.91000W

0.6

MTO

only stabilizer controls w = 0.01735WMT

o45Sht0952
160 AFDD Weight Models

and then

WFWnb = χFWnbfFWnb w

WFWmb = χFWmb (1 — fFWnb) w

For a helicopter, the stabilizer-control equation is used. Parameters are defined in table 19-14, including

units as used in these equations.

Table 19-14. Parameters for fixed-wing flight-control weight.

parameter definition units

Sht horizontal tail planform area ft2

WMTO maximum takeoff weight lb

fFWnb fixed wing non-boosted weight

(fraction total fixed wing flight control weight)

Rotary-wing flight controls consist of non-boosted flight controls, flight-control boost mechanisms,

and boosted flight controls. The non-boosted flight-control weight (AFDD82 model) is:

fraction method WRWnb = χRWnbfRWnb (1 — fRWhyd)wfc

parametric method WRWnb = χRWnb2.1785fnbsvWMT09Nrotor

5

where fnbsv = 1 . 8984 for ballistically survivable (UTTAS/AAH level); 1 .0 otherwise. The parametric

method assumes the rotor flight controls are boosted and computes the weight of the non-boosted portion

up to the control actuators. Based on 20 aircraft, the average error of the non-boosted flight controls

equation is 10.4% (fig. 19-21). The flight-control boost-mechanism weight and boosted flight-control

weight (AFDD82 model) are:

wf —0. 2873f

(N N )

0.6257C1.3286(0. 01V ,p)2.1129

c — , b. rotor blade \ t^ fRWred

WRWmb = χRWmb (1 — fRWhyd )wfc

WRWb = χRWb0.02324fbsv (NrotorNblade )

1.0042N0

rotor \

.1155C2.2296(0.01Vtip )3.1877

where fmbsv = 1 .3029 and fbsv = 1. 1171 for ballistically survivable (UTTAS/AAH level) and 1 .0 otherwise; and fRWred = 1 . 0 to 3 . 0. Typically fRWnb = 0. 6 (range 0. 3 to 1 . 8) and fRWhyd = 0.4. Based on 21

aircraft, the average error of the boost-mechanisms equation is 6.5% (fig. 19-22). Based on 20 aircraft,

the average error of the boosted flight-controls equation is 9.7% (fig. 19-23). Parameters are defined in

table 19-15, including units as used in these equations.

Table 19-15. Parameters for rotary-wing flight-control weight.

parameter definition units

WMTO maximum takeoff weight lb

Nrotor number of main rotors

Nblade number of blades per rotor

c rotor mean blade chord ft

Vtip rotor hover tip velocity ft/sec

fRWnb rotary wing non-boosted weight (fraction boost mechanisms weight)

fRWhyd rotary wing hydraulics weight

(fraction hydraulics plus boost mechanisms weight)

fRWred flight control hydraulic system redundancy factor
AFDD Weight Models 161

The conversion controls consist of non-boosted tilt controls and tilt-control boost mechanisms; they

are used only for tilting-rotor configurations. The weights are:

wCVmb = fCVmbWMTO

wCVnb = fCVnbWCVmb

WCVmb = χCVmbwCVmb

WCVnb = χCVnbwCVnb

Parameters are defined in table 19-16, including units as used in these equations.

Table 19-16. Parameters for conversion-control weight.

parameter definition units

WMTO maximum takeoff weight lb

fCVnb conversion non-boosted weight (fraction boost mechanisms weight)

fCVmb conversion boost mechanisms weight (fraction maximum takeoff weight)

19–9 Hydraulic Group

The hydraulic group consists of hydraulics for fixed-wing flight controls, rotary-wing flight controls,

conversion (rotor tilt) flight controls, and equipment. The hydraulic weight for equipment, WEQhud , is

fixed (input). The weights (AFDD82 model) are

WFWhyd = χFWhydfFWhydWFWmb

WRWhyd = χRWhydfRWhydwfc

WCVhyd = χCFhyd fCVhydWCVmb

Typically fRWhyd = 0.4. Parameters are defined in table 19-17, including units as used in these equations.

Table 19-17. Parameters for hydraulic group weight.

parameter definition units

fFWhyd fixed wing hydraulics weight (fraction boost mechanisms weight)

fRWhyd rotary wing hydraulics weight

(fraction hydraulics plus boost mechanisms weight)

fCVhyd conversion hydraulics weight (fraction boost mechanisms weight)

19–10 Anti-Icing Group

The anti-icing group consists of the electrical system and the anti-ice system. The weights are

obtained from the sum over all rotors, all wings, and all engines:

WDIelect = χDIelect E kelecAblade

WDIsys = χDIsys 

(E 

krotorAblade + E kwingtwing + kair E Weng lI

Parameters are defined in table 19-18, including units as used in these equations.
162 AFDD Weight Models

Table 19-18. Parameters for anti-icing group weight.

parameter definition units

Ablade total blade area of rotor, from geometric solidity ft2 or m2

^wing wing length (wing span less fuselage width) ftor m

kelect electrical system weight factor

krotor rotor deice system weight factor

kwing wing deice system weight factor

kair engine air intake deice system weight factor

19–11 Other Systems and Equipment

The following weights are fixed (input) in this model: auxiliary power group; instruments group;

pneumatic group; electrical group; avionic group (mission equipment); armament group (armament

provisions and armor); furnishing and equipment group; environmental control group; and load and

handling group. Typical fixed weights are given in table 19-19, based on medium to heavy helicopters

and tiltrotors.

Table 19-19. Other systems and equipment weight.

group typical weight (lb)

SYSTEMS AND EQUIPMENT

flight controls group

cockpit controls 100–125

automatic flight control system 35–200

flight control electronics, mechanical 35–100

flight control electronics, fly-by-wire 250

auxiliary power group 130–300

instruments group 150-250

hydraulic group

equipment 50–300

electrical group 400–1000

avionics group (mission equipment) 400–1500

furnishings & equipment group 600–1000

crew only 100–200

environmental control group 50–250

anti-icing group 50–300

load & handling group

internal 200–400

external 150–300

FIXED USEFUL LOAD

crew 500–800

fluids (oil, unusable fuel) 50–150
AFDD Weight Models 163

19–12 Folding Weight

Folding weights are calculated in a number of groups: wing Wfold (including extensions), rotor

Wfold , fuselage Wtfold and Wwfold . These are the total weights for folding and the impact of folding on

the group. A fraction f foldkit of these weights can be in a kit, hence optionally removable. Thus of the

total folding weight, the fraction ffoldkit is a kit weight in the fixed useful load of the weight statement;

while the remainder is kept in the wing, rotor, or fuselage group weight.

19–13 Parametric Weight Correlation

Figure 19-24 shows the error of the calculated weight for the sum of all parametric weight, accounting on average for 42% of the empty weight. This sum is composed of the structural group (based on

the AFDD00 equation for rotor blade and hub weights, and the AFDD84 equation for body weight), the

propulsion group (based on the AFDD00 equation for drive system weight), and the flight controls group.

Based on 42 aircraft, the average error of the sum of all parametric weight is 5.3%. The corresponding

average error is 6.1% for the structural group (8.6% for the rotor group alone), 10.9% for the propulsion

group, and 8.7% for the flight controls group.

19–14 References

1) Chappell, D.; and Peyran, R.: Methodology for Estimating Wing Weights for Conceptual Tilt-Rotor

and Tilt-Wing Aircraft. SAWE Paper No. 2107, Category No. 23, May 1992.

2) Chappell, D.P.: Tilt-rotor Aircraft Wing Design. ASRO-PDT-83-1, 1983.

3) Weight Trend Estimation for the Rotor Blade Group, Rotor Hub Group, and Upper Rotor Shaft of the

ABC Aircraft. ASRO-PDT-83-2, 1983.
164 AFDD Weight Models

15.

O

10. q

00 O

5. 0 E

-10. 0

-15.

0. 10000. 20000. 30000. 40000. 50000. 60000.

actual weight

Figure 19-1. Wing group (AFDD93).

25.

20.

15.

10.

5.

s

a

0.

to

-5.

-10.

-15.

-20.

-25.

0. 500. 1000. 1500. 2000. 2500. 3000. 3500.

actual weight

Figure 19-2. Rotor group, blade weight (AFDD82).
1500. 3000. 4500. 6000. 7500.

actual weight

-25.

0.

25.

20.

15.

10.

5.

0.

-5.

-10.

-15.

-20.

AFDD Weight Models 165

25.

20.

15.

10.

5.

s

a

0.

to

-5.

-10.

-15.

-20.

-25.

0 1500. 3000. 4500. 6000. 7500.

actual weight

Figure 19-3. Rotor group, hub weight (AFDD82).

Figure 19-4. Rotor group, blade weight (AFDD00).
166 AFDD Weight Models

25.

q^ x

20. Cr -o - - - --

C^

15. q

10.

5.

s

0 0. O

+ +

-5. 0

-10. OILJ0

O using actual blade weight

+ using calculated blade weight

^F q

-15. + x O O

-20. - -------- - - - - - - - -

+

-25.

0. 1500. 3000. 4500. 6000. 7500.

actual weight

Figure 19-5. Rotor group, hub weight (AFDD00).

50.

40.

O

30. 00 O

20. ----0---------------

O

10.

s

0

0.

^O-10.

-20. - C -- - - - - - - - - - - - - - -

O

-30. O 0

-40.

-50.

0. 10. 20. 30. 40. 50. 60. 70. 80. 90. 100. 110. 120.

actual weight

Figure 19-6. Empennage group, horizontal tail weight (AFDD82).
AFDD Weight Models 167

50.

40.

30.

20.

10.

s

a

0.

to

-10.

-20.

-30.

-40.

-50.

0. 10. 20. 30. 40. 50. 60. 70. 80. 90. 100. 110. 120.

actual weight

Figure 19-7. Empennage group, vertical tail weight (AFDD82).

50.

40.

30.

20.

10.

s

a

0.

to

-10.

-20.

-30.

-40.

-50.

0. 50. 100. 150. 200. 250. 300. 350. 400. 450. 500.

actual weight

Figure 19-8. Empennage group, tail rotor weight (AFDD82).
168 AFDD Weight Models

30. O

A O rotorcraft

20. -- - - - - - - - - - - - q fixed wing

0

p other RC

10.

O other FW

s q 0 O 0 O O

a 0. q

O 0 q O 0 0

O q 

-10.

pO

O

-20. O- - - - - - - - - - - - - - - - - -

0

-30.

0. 2500. 5000. 7500. 10000. 12500. 15000.

actual weight

Figure 19-9. Fuselage group, fuselage weight (AFDD84).

50.

40.

30.

20.

10. s

a 0.

to

-10.

-20.

-30.

-40.

-50.

0. 5000. 10000. 15000. 20000. 25000. 30000.

actual weight

Figure 19-10. Fuselage group, fuselage weight (AFDD82).
AFDD Weight Models 169

50. O rotorcraft

40. 0

q fixed wing

other RC 30.

20. - - - - -

10.

o q

0.

-10.

-20.

-30.

O

-40.

D

-50.

0. 2000. 4000. 6000. 8000. 10000. 12000.

actual weight

Figure 19-11. Alighting gear group, landing gear weight (AFDD82).

25.

20.

15.

10.

5.

s

a

0.

to

-5.

-10.

-15.

-20.

-25.

0. 50. 100. 150. 200. 250. 300.

actual weight

Figure 19-12. Engine section or nacelle group, engine support weight (AFDD82).
170 AFDD Weight Models

30.

25.

20.

15.

10.

5.

0.

-5.

-10.

-15.

-20.

-25.

-30.

0. 50. 100. 150. 200. 250. 300. 350. 400. 450.

actual weight

Figure 19-13. Engine section or nacelle group, cowling weight (AFDD82).

25.

20.

15.

10.

5.

0.

-5.

-10.

-15.

-20.

-25.

0. 50. 100. 150. 200. 250. 300.

actual weight

Figure 19-14. Air induction group, air induction weight (AFDD82).
AFDD Weight Models 171

25 .

20.

15.

10.

5.

0.

-5.

-10.

-15.

-20.

-25.

0. 50. 100. 150. 200. 250. 300. 350. 400. 450.

actual weight

Figure 19-15. Propulsion group, accessories weight (AFDD82).

25.

20.

15.

10.

5.

0.

-5.

-10.

-15.

-20.

-25.

0. 200. 400. 600. 800. 1000. 1200. 1400. 1600.

actual weight

Figure 19-16. Propulsion group, fuel tank weight (AFDD82).
172 AFDD Weight Models

25.

20.

15.

10.

5.

0.

-5.

-10.

-15.

-20.

-25.

0. 1000. 2000. 3000. 4000. 5000. 6000. 7000. 8000. 9000. 10000.

actual weight

Figure 19-17. Propulsion group, gear box and rotor shaft weight (AFDD83).

25.

O 0

20. 0 -e- - - - - - - - - - - - - - - - -O

15. q

10. q

s 

5. 0® 5] El

ig

.. 0.

O ®

o

-5.

o og o q

-10.

-15. 00

0

110 0

-20. 0 - - - - - - - - - - - - - - - - - -

-25.

0. 1000. 2000. 3000. 4000. 5000. 6000. 7000. 8000. 9000. 10000.

actual weight

Figure 19-18. Propulsion group, gear box and rotor shaft weight (AFDD00).
AFDD Weight Models 173

50.

40.

30.

20.

10.

s

a

0.

to

-10.

-20.

-30.

-40.

-50.

0. 100. 200. 300. 400. 500. 600. 700.

actual weight

Figure 19-19. Propulsion group, drive shaft weight (AFDD82).

75.

60.

45.

30.

15.

s

a

0.

to

-15.

-30.

-45.

-60.

-75.

0. 20. 40. 60. 80. 100. 120. 140. 160. 180. 200.

actual weight

Figure 19-20. Propulsion group, rotor brake weight.
174 AFDD Weight Models

25 .

20.

15.

10.

5.

0.

-5.

-10.

-15.

-20.

-25.

0. 50. 100. 150. 200. 250. 300. 350. 400. 450. 500.

actual weight

Figure 19-21. Flight controls group, rotor non-boosted control weight (AFDD82).

50.

40.

30.

20.

10.

0

0.

to

-10.

-20.

-30.

-40.

-50.

0. 100. 200. 300. 400. 500. 600. 700.

actual weight

Figure 19-22. Flight controls group, rotor boost mechanisms weight (AFDD82).
AFDD Weight Models 175

50.

40.

30.

20.

10.

s

a

0.

to

-10.

-20.

-30.

-40.

-50.

0 .200. 400. 600. 800. 1000. 1200. 1400. 1600. 1800. 2000. 2200.

actual weight

Figure 19-23. Flight controls group, rotor boosted control weight (AFDD82).

• all parametric weight

° structure group

v propulsion group

o flight control group

o °

•

v

0

25.

v

20. v

15. v

1 0. v

o • 4 v

5. • ^° v v v

0. °

'b °v A

-5. 7 • ov

-10. ^ g -

-15.

-20.

-25.

0. 5000. 10000. 15000. 20000. 25000. 30000. 35000. 40000.

weight

Figure 19-24. Sum of all parametric weight.
176 AFDD Weight Models