NDARC — NASA Design and Analysis of RotorcraftTheoretical Basis and Architecture

Wayne JohnsonAeromechanics Branch

National Aeronautics and Space AdministrationAmes Research Center, Moffett Field, California

wayne.johnson@nasa.gov

ABSTRACT

The theoretical basis and architecture of the conceptual design tool NDARC (NASA Design andAnalysis of Rotorcraft) are described. The principal tasks of NDARC are to design (or size) arotorcraft to satisfy specified design conditions and missions, and then analyze the performance of theaircraft for a set of off-design missions and point operating conditions. The aircraft consists of a set ofcomponents, including fuselage, rotors, wings, tails, and propulsion. For each component, attributessuch as performance, drag, and weight can be calculated. The aircraft attributes are obtained from thesum of the component attributes. NDARC provides a capability to model general rotorcraftconfigurations, and estimate the performance and attributes of advanced rotor concepts. The softwarehas been implemented with low-fidelity models, typical of the conceptual design environment.Incorporation of higher-fidelity models will be possible, as the architecture of the code accommodatesconfiguration flexibility, a hierarchy of models, and ultimately multidisciplinary design, analysis andoptimization.

INTRODUCTION..

The objectives of rotorcraft design work in a governmentlaboratory are to support research and to support rotorcraftacquisition. Research activities require a robust designcapability to aid in technology impact assessments and toprovide system level context for research. At the appliedresearch level, it is necessary to show how technology willimpact future systems, and justify the levels of investmentrequired to mature that technology to an engineeringdevelopment stage. Design provides one avenue toaccomplishing these objectives. The Department ofDefense (DoD) acquisition phases requiring rotorcraftdesign work include concept exploration, conceptdecision, concept refinement, and technologydevelopment. During these acquisition phases, performingquantitative evaluation and independent synthesis of awide array of aircraft designs is typically necessary, in

.Presented at the American Helicopter SocietyAeromechanics Specialists’ Conference, San Francisco,CA, January 20-22, 2010. This is a work of the U.S.Government and is not subject to copyright protection.

order to provide the foundation for specification andrequirement development.

Rotorcraft conceptual design consists of analysis,synthesis, and optimization to find the best aircraftmeeting the required capabilities and performance. Aconceptual design tool is used for synthesis and analysis ofrotorcraft. These tools historically have been low fidelityfor rapid application. Such sizing codes are built aroundthe use of momentum theory for rotors, classical finitewing theory, a referred parameter engine model, and semi-empirical weight estimation techniques. The successfuluse of a low-fidelity tool requires careful consideration ofmodel input parameters and judicious comparison withexisting aircraft to avoid unjustified extrapolation ofresults.

The helicopter industry has proprietary conceptual designtools, including PRESTO (Bell Helicopter), RDM(Sikorsky Aircraft), and HESCOMP and VASCOMP(Boeing). Until now the tools available to the U.S.government have been characterized by out-of-datesoftware and limited capabilities. Examples areHESCOMP and VASCOMP (the versions developed by

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Boeing in the 1970s), and RC (developed by the U.S.Army AFDD in the 1990s).

NASA, with support from the U.S. Army, conducted in2005 the NASA Heavy Lift Rotorcraft SystemsInvestigation (ref. 1), focused on the design and in-depthanalysis of rotorcraft configurations that could satisfy theVehicle Systems Program (VSP) technology goals. TheVSP technology goals and mission were intended toidentify enabling technology for civil application of heavylift rotorcraft. The goals emphasized efficient cruise andhover, efficient structure, and low noise. The requirementsincluded carrying 120 passengers over a 1200 nm range,350 knots at 30,000 ft altitude. The configurationsconsidered included the Large Civil Tiltrotor (LCTR),Large Civil Tandem Compound (LCTC), and LargeAdvancing Blade Concept (LABC). This project is anexample of the role of a rotorcraft sizing code within agovernment laboratory. The design tool used was theAFDD RC code. The project illustrated the difficultiesadapting or modifying a legacy code for configurationsother than conventional helicopters and tiltrotors.

Since 2005, there have been numerous other jointNASA/U.S. Army investigations of advanced rotorcraftconcepts, covering conventional tiltrotors and helicopters,slowed-rotor compound helicopters (ref. 2), a tilting-tandem concept, heavy-lift slowed-rotor tiltrotors (ref. 3),lift-offset rotor concepts (ref. 4), and a second generationlarge civil tiltrotor (LCTR2, ref. 5). These design projectshave gone well beyond the conventional boundaries of theconceptual design process, combining high-fidelityanalyses (including rotorcraft comprehensive analysis,computational fluid dynamics, and structural analysis)with the conceptual design tool. This approach has beenrequired because of the increasing sophistication of therequirements and the technology, and the increased levelof certainty needed to differentiate between systemconcepts.

Based on this experience, a new conceptual design toolhas been developed to support future needs of the NASASubsonic Rotary Wing project and the U.S. Army AFDDAdvanced Design Office: NASA Design and Analysis ofRotorcraft (NDARC). The software development startedin January 2007, and the initial code release occurred inMay 2009. This paper summarizes the NDARC theoreticalbasis and architecture; the complete description is inreference 6. A companion paper (ref. 7) presentsvalidation and demonstration results from the NDARCdevelopment.

REQUIREMENTS

Based on the recent experience of NASA and AFDD atAmes Research Center, the following requirements weredefined for the new conceptual design tool.

The principal tasks of the tool are to design rotorcraft tomeet specified requirements, and then analyze theperformance of the aircraft for a set of flight conditionsand missions. Multiple and flexible design requirements,from specific flight conditions and various missions, mustbe accommodated in the sizing task. The aircraftperformance analysis must cover the entire spectrum ofthe aircraft capabilities, and component performance andengine models must cover all operating conditions. Ageneral and flexible capability to define conditions andmissions is required.

For government applications and to support research, it isimportant to have the capability to rapidly model generalrotorcraft configurations, including estimates of theperformance and attributes of advanced rotor concepts andthe capability to model the impact of new technology atthe system and component level. In such an environment,software extensions and modifications will be routinelyrequired to meet the unique requirements of individualprojects. Complete and thorough documentation of thetheory and its software implementation is essential, tosupport development and maintenance and to enableeffective use and modification.

The code architecture must accommodate configurationflexibility and alternate models, including a hierarchy ofmodel fidelity. Although initially implemented with low-fidelity models, typical of the conceptual designenvironment, ultimately the architecture must allowmultidisciplinary design, analysis, and optimization.

Definition and development of the NDARC requirementsbenefited substantially from the experiences and computercodes of the preliminary design team of the U.S. ArmyAeroflightdynamics Directorate (AFDD) at AmesResearch Center. This background is described inreference 6. In the early 1990s, the RC code (forRotorCraft) emerged from the AFDD efforts, with RC97 amajor version that unified tiltrotor and helicopter analyses.NDARC is entirely new software, built on a newarchitecture for the design and analysis of rotorcraft. Fromthe RC theoretical basis, the equations of the parametricweight equations and the Referred Parameter TurboshaftEngine Model were used with only minor changes. Usewas also made of some RC component aerodynamicmodels and rotor performance models. The current users

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of RC, informed by past and recent applications,contributed significantly to the requirements definition.

NDARC TASKS

The NDARC code performs design and analysis tasks.The design task sizes the rotorcraft to satisfy specifieddesign conditions and missions. The analysis tasks caninclude off-design mission performance analysis, flightperformance calculation for point operating conditions,and generation of subsystem or component performancemaps. Figure 1 illustrates the tasks. The principal tasks(sizing, mission analysis, flight performance analysis) areshown in the figure as boxes with heavy borders. Blackarrows show control of subordinate tasks.

Figure 1. Outline of NDARC tasks.

The aircraft description (figure 1) consists of all theinformation, input and derived, that defines the aircraft.This information can be the result of the sizing task; cancome entirely from input, for a fixed model; or can comefrom the sizing task in a previous case or previous job.The aircraft description information is available to alltasks and all solutions (indicated by green arrows).

The sizing task determines the dimensions, power, andweight of a rotorcraft that can perform a specified set ofdesign conditions and missions. The aircraft size ischaracterized 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 designflight conditions and missions, the task can determine thetotal engine power or the rotor radius (or both power andradius can be fixed), as well as the design gross weight,maximum takeoff weight, drive system torque limit, andfuel tank capacity. For each propulsion group, the enginepower or the rotor radius can be sized.

Missions are defined for the sizing task and for themission performance analysis. A mission consists of aspecified number of mission segments, for which time,distance, and fuel burn are evaluated. For the sizing task,certain missions are designated for design gross weightcalculations, for transmission sizing, and for fuel tanksizing. The engines are sized to meet the powerrequirement of each segment. The mission parametersinclude mission takeoff gross weight and useful load. Forspecified takeoff fuel weight with adjustable segments, themission time or distance is adjusted so the fuel requiredfor the mission (burned plus reserve) equals the takeofffuel weight. The mission iteration is on time or distance (ifadjustable), or on fuel weight.

Flight conditions are specified for the sizing task and forthe flight performance analysis. For the sizing task, certainflight conditions are designated for design gross weightcalculations, for transmission sizing, for maximum takeoffweight calculations, and for antitorque or auxiliary-thrustrotor sizing. The engines are sized to meet the powerrequirement of each condition. The flight conditionparameters include gross weight and useful load.

For flight conditions and mission takeoff, the gross weightcan be maximized such that the power required equals thepower available.

A flight state is defined for each mission segment and eachflight condition. The aircraft performance can be analyzedfor the specified state, or a maximum-effort performancecan be identified. The maximum effort is specified byidentifying a target (such as best endurance, best range, orpower required equal power available) to be achieved byadjusting a variable (such as speed, rate of climb, oraltitude).

The aircraft must be trimmed by solving for the controlsand motion that produce equilibrium in the specified flightstate. Different trim solution definitions are required forvarious flight states, hence for each mission segment andflight condition.

Evaluating the rotor hub forces and blade pitch angles mayrequire solution of the blade flap equations of motion.

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The following sections describe the NDARC tasks in moredetail.

f) Wings: The aircraft can have one or more wings, or nowings.

AIRCRAFT DESCRIPTION

Decomposition of the aircraft system into fundamentalcomponents is critical to achieving the ability to rapidlymodel a wide array of rotorcraft concepts. Thus theaircraft consists of a set of components, includingfuselage, rotors, wings, tails, and propulsion. For eachcomponent, attributes such as performance, drag, andweight can be calculated. The aircraft attributes areobtained from the sum of the component attributes.Description and analysis of conventional rotorcraftconfigurations is facilitated, while retaining the capabilityto model novel and advanced concepts. Specific rotorcraftconfigurations considered are single main-rotor and tail-rotor helicopter; tandem helicopter; coaxial helicopter; andtiltrotors. Novel and advanced concepts typically aremodeled by starting with one of these conventionalconfigurations. For example, compound rotorcraft can beconstructed by adding wings and propellers.

The following components form the aircraft.

a) Systems: The systems component contains weightinformation (fixed useful load, vibration, contingency,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 norotors. In addition to main rotors, the component canmodel tail rotors, propellers, proprotors, and ducted fans.A rotor is designated a main rotor, tail rotor, or propeller;and can be tilting, ducted, and/or antitorque. The rotorpower required is evaluated using the energy method, as asum of induced, profile, and parasite power. The powercomponents are calculated in terms of an induced powerfactor and a mean drag coefficient. The power modelsaccount for the influence of speed, thrust, compressibility,stall, lift offset, and the induced interference between twinrotors. The calibration of these induced and profile powermodels reflects the level of technology being considered.Blade element theory is used to calculate rotor hub forcesand moments and to solve for blade pitch control orflapping.

e) Forces: The force component is a simple model for alift, propulsion, or control subsystem, including a weightand fuel flow description.

g) Tails: The aircraft can have one or more horizontal orvertical tail surfaces, or no tails.

h) Fuel Tank: There is one fuel tank component for theaircraft. There are one or more sizes of auxiliary fueltanks.

i) Propulsion Groups: There are one or more propulsiongroups. Each propulsion group is a set of rotors and enginegroups, connected by a drive system. The componentsdefine the power required, and the engine groups definethe power available. There are one or more drive states,with a set of gear ratios for each state. The power requiredequals the sum of component power, transmission losses,and accessory losses. There are drive system torque limits,and rotor and engine shaft ratings.

j) Engine Groups: Each engine group has one or moreengines of the same type. For each engine type an enginemodel is defined.

The engine model describes a particular engine, used inone or more engine groups. The engine performanceinformation includes mass flow, fuel flow, jet thrust, andmomentum drag at the required power. A ReferredParameter Turboshaft Engine Model (RPTEM) enables theaircraft performance analysis to cover the entire spectrumof operation. This model uses curve fits of referredperformance from an engine deck, including the effect ofturbine speed. The effects of size (a scaling model, basedon mass flow) and technology (specific power and specificfuel consumption) are included in the engine model.

The ability to define the aircraft control structure throughinput is a key feature for configuration generality. Aircraftcontrols are connected to component controls. Aircraftcontrols consist of pilot’s controls (for trim), configurationvariables (e.g. tilt of nacelle/pylon, engine, rotor shaft),and direct connections to component controls. There canbe one or more control states, with different connections tocomponents (for example, to model the controls of atiltrotor in helicopter mode and airplane mode flight).There are default control connections for eachconfiguration.

Weights are calculated or input for all components andsubsystems. Calculated weights are obtained fromparametric equations based on weights of existing turbinepowered helicopters and tiltrotors (and some fixed wingaircraft component weights). Multiplicative technologyfactors can be used for the weights of all elements of the

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aircraft, to either match measured weights or account foradvanced technology.

Engine Power. The engine size is described by the powerPeng , which is the sea-level static power available perengine at a specified takeoff rating.

SIZING TASK

The sizing task determines the dimensions, power, andweight of a rotorcraft that can perform a specified set ofdesign conditions and missions. The aircraft size ischaracterized by parameters such as design gross weightWD or weight empty WE , rotor radius R, and enginepower available Peng . The relationships betweendimensions, power, and weight generally require aniterative solution. From the design flight conditions andmissions, the task can determine the total engine power orthe rotor radius (or both power and radius can be fixed), aswell as the design gross weight, maximum takeoff weight,drive system torque limit, and fuel tank capacity. For eachpropulsion group, the engine power or the rotor radius canbe sized. Alternatively, Peng and R can be input ratherthan sized. Several aircraft parameters can be determinedby a subset of the design conditions and missions:

a) Design gross weight WD .

b) Maximum takeoff gross weight WMTO.

c) Drive system torque limit PDS limit .

d) Fuel tank capacity Wfuel_cap.

e) Antitorque or auxiliary-thrust rotor design thrustTdesign .

Alternatively, these parameters can be fixed at inputvalues. The design gross weight WD can be fixed. Theweight empty can be fixed (achieved by adjusting thecontingency weight).

For each flight condition and for each mission, the grossweight and useful load are specified. The gross weight canbe input, or maximized, or fallout. For flight conditions,the payload or fuel weight can be specified, and the othercalculated; or both payload and fuel weight specified, withgross weight fallout. For missions, the payload or fuelweight can be specified, the other fallout, and then time ordistance of mission segments adjusted; or fuel weightcalculated from mission, and payload fallout; or bothpayload and fuel weight specified (or payload specifiedand fuel weight calculated from mission), with grossweight fallout.

Component Sizing

Numerous choices are implemented for identification ofindependent (fixed) and dependent (fallout) designparameters, with parameter variation facilitated byautomating dependencies.

Main Rotor. The main rotor size is defined by the radiusR or disk loading W /A, thrust-weighted solidity a, hovertip speed Vtip, and blade loading CW /a = W / AVtip

2 ^ .

With more than one main rotor, the disk loading and bladeloading are obtained from an input fraction of design grossweight, W = fWWD .

If the rotor radius is fixed for the sizing task, three of R

(or W /A), CW/a, Vtip, and a are input; and the otherparameters are derived. Optionally the radius can becalculated from a specified ratio to the radius of anotherrotor.

If the sizing task determines the rotor radius ( R andW /A), then two of CW/a , Vtip, and a are input; and theother parameter is derived. The radius can be sized for justa subset of the rotors, with fixed radius for the others.

Antitorque or Auxiliary-thrust Rotor. For antitorque andauxiliary-thrust rotors, three of R (or W /A), CW/a , Vtip,

and a are input; and the other parameters are derived.Optionally the radius can be calculated from a specifiedratio to the radius of another rotor. The disk loading andblade loading are based on Tdesign , where Tdesign is themaximum thrust from designated design conditions.Optionally the tail rotor radius can be scaled with the mainrotor radius and the disk loading.

Wing. The wing size is defined by the wing area S orwing loading W / S; span (perhaps calculated from othergeometry), chord, and aspect ratio. With more than onewing, the wing loading is obtained from an input fractionof design gross weight, W = fWWD .

Two of the following parameters are input: area (or wingloading), span, chord, and aspect ratio; the otherparameters are derived. The span can be calculated fromthe rotor radius, fuselage width, and clearance (typicallyused for tiltrotors). The span can be calculated from aspecified ratio to the span of another wing.

Tail. The tail size is determined by the area or tail volume,span, chord, and aspect ratio.

Fuel Tank. The fuel tank capacity Wfuel_cap (maximumusable fuel weight) is determined from designated sizingmissions.

Weights. The structural design gross weight WSD andmaximum takeoff weight WMTO can be input, or specifiedas d + fW , for input increment d and fraction f . Thisconvention allows the weights to be input directly ( f = 0),

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or scaled with WD . For WSD , W is the design grossweight WD , or WD adjusted for a specified fuel state(input fraction of fuel capacity). For WMTO , W is thedesign gross weight WD , or WD adjusted for maximumfuel capacity. Alternatively, WMTO can be calculated asthe maximum gross weight possible at a designated sizingflight condition.

Drive System Rating. The drive system rating is defined asa power limit, PDS limit . The rating is properly a torquelimit, QDS limit = PDS limit /Q, but is expressed as a powerlimit for clarity. The drive system rating can be input;calculated from the engine takeoff power rating;calculated from the power available at the transmissionsizing flight conditions; or calculated from the powerrequired at the transmission sizing flight conditions. Thedrive system rating is a limit on the entire propulsionsystem. To account for differences in the distribution ofpower through the drive system, limits are also used forthe torque of each rotor shaft and of each engine group.

MISSION DEFINITION

Missions are defined for the sizing task and for themission performance analysis. A mission consists of aspecified number of mission segments. The takeoff grossweight is evaluated at the start of the mission, perhapsmaximized for zero power margin at a specified missionsegment. Then the aircraft is flown for all segments. Forcalculated mission fuel weight, the fuel weight at takeoffis adjusted to equal the fuel required for the mission(burned plus reserve). For specified takeoff fuel weightwith adjustable segments, the mission time or distance isadjusted so the fuel required for the mission (burned plusreserve) equals the takeoff fuel weight. The missioniteration is thus on mission fuel weight. A successivesubstitution method is used if an iteration is required.

Each mission consists of a specified number of missionsegments. The following segment types can be specified.

a) Taxi or warm-up (fuel burned but no distance added torange).b) Distance: fly segment for specified distance (calculatetime).c) Time: fly segment for specified time (calculatedistance).d) Hold: fly segment for specified time (loiter, so fuelburned but no distance added to range).e) Climb: climb or descend from present altitude to nextsegment altitude (calculate time and distance).f) Spiral: climb or descend from present altitude to next

segment altitude, fuel burned but no distance added torange.

The number of auxiliary fuel tanks can change with eachmission segment. The aircraft can refuel (either on theground or in the air) at the start of a mission segment, byeither filling all tanks to capacity or adding a specified fuelweight. Fuel can be dropped at the start of a missionsegment. For calculation of the time or distance in amission segment, a headwind or tailwind can be specified.

Mission fuel reserves can be specified in several ways foreach mission. Fuel reserves can be defined in terms ofspecific mission segments. Fuel reserves can be an inputfraction of the fuel burned by all (except reserve) missionsegments, or an input fraction of the fuel capacity.

The takeoff distance can be calculated, either as groundrun plus climb to clear an obstacle, or accelerate-stopdistance in case of engine failure. Landing and VTOLtakeoff calculations are not implemented, as these are bestsolved as an optimal control problem.

FLIGHT STATE

A flight state is defined for each flight condition and foreach mission segment. The flight state definition consistsof the speed, aircraft motion, altitude, atmosphere, heightabove ground level and landing gear state, aircraft controlstate, aircraft control values, and center-of-gravityposition. Parameters defined for each propulsion groupinclude drive system state, and rotor tip speed for primaryrotor. Specified for each engine group are the number ofinoperative engines, the infrared suppressor state, theengine rating, and the fraction of rated engine poweravailable. Aircraft and rotor performance parameters foreach flight state include payload drag, rotor performance,and aircraft trim state and trim targets.

The aircraft performance can be analyzed for the specifiedstate, or a maximum-effort performance can be identified.The maximum effort is specified by identifying a target(such as best endurance, best range, or power requiredequal power available) to be achieved by adjusting avariable (such as speed, rate of climb, or altitude). Twomaximum effort quantity/variable pairs can be specified,solved in nested iterations.

Environment and Atmosphere

The aerodynamic environment is defined by the speed ofsound cs , and density p, and kinematic viscosityv = μ / ^ of the air (or other fluid). These quantities can beobtained from the standard day (International StandardAtmosphere), or input directly. The International Standard

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Atmosphere (ISA) is a model for the variation withaltitude of pressure, temperature, density, and viscosity,published as International Standard ISO 2533 by theInternational Organization for Standardization (ISO) (ref.8). The ISA consists of a series of altitude ranges withconstant lapse rate (linear temperature change withaltitude).

Figure 2. Design and analysis tasks, with nested loops andsolution methods.

SOLUTION PROCEDURE

The NDARC code performs design and analysis tasks.Figure 1 illustrates the tasks. The principal tasks (sizing,mission analysis, flight performance analysis) are shownin the figure as boxes with dark borders. Black arrowsshow control of subordinate tasks. The aircraft descriptionconsists of all the information, input and derived, thatdefines the aircraft. This information can be the result ofthe sizing task; can come entirely from input, for a fixedmodel; or can come from the sizing task in a previous caseor previous job. The aircraft description information is

available to all tasks and all solutions (indicated by greenarrows in figure 1).

The nested iteration loops involved in the solution processare indicated by the subtitles in the boxes of figure 1, andillustrated in more detail in figure 2. The flight statesolution involves up to three loops. The innermost loop isthe solution of the blade flap equations of motion, neededfor an accurate evaluation of the rotor hub forces. The nextloop is the trim solution, which is required for most flightstates. The flight state optionally has one or two maximumeffort iterations. The flight state solution is executed foreach flight condition and for each mission segment. Aflight condition solution or any mission segment solutioncan optionally maximize the aircraft gross weight. Themission usually requires an iterative solution, for fuelweight or for adjustable segment time or distance. Thuseach flight condition solution involves up to four nestediterations: maximum gross weight (outer), maximumeffort, trim, and blade motion (inner). Each missionsolution involves up to five nested iterations: mission(outer), then for each segment maximum gross weight,maximum effort, trim, and blade motion (inner). Finally,the design task introduces a sizing iteration, which is theoutermost loop of the process. The solution method foreach iteration is indicated in figure 2. Details of thesolution methods and their implementation are given inreference 6.

COST

Costs are estimated using statistical models based onhistorical aircraft price and maintenance cost data, withappropriate factors to account for technology impact andinflation. The aircraft flyaway cost consists of airframe,mission equipment package, and flight control electronicscosts. The direct operating cost plus interest (DOC+I, incents per available seat mile) is the sum of maintenancecost, flight crew salary and expenses, fuel and oil cost,depreciation, insurance cost, and finance cost. Inflationfactors can be input, or internal factors used (either DoDdeflators or CPI factors).

The CTM rotorcraft cost model (refs. 9 to 11) gives anestimate of aircraft flyaway cost and direct operating costplus interest. The statistical equation for airframe costpredicts the price of 123 out of 128 rotorcraft within 20%(figure 3). The fuel burn, block time, and block range areobtained from a designated mission.

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1000.

800.o,o,

600.aUM.0 400.2U

200.R+

0.

no error± 10%

0. 200. 400. 600. 800. 1000.Figure 4. Aircraft and component controls

actual base price (1994$/lb)

Figure 3. Statistical estimation of rotorcraft flyaway cost.

AIRCRAFT MODEL

The aircraft disk loading is the ratio of the design grossweight and a reference rotor area: DL = WD / Aref. Thereference area is a sum of specified fractions of the rotorareas, Aref = E fA A. Typically Aref is the projected areaof the lifting rotors, where the lifting rotors are all thosenot designated antitorque or auxiliary-thrust. The aircraftwing loading is the ratio of the design gross weight and areference wing area: WL = WD / Sref . The reference area isa sum of the wing areas, Sref =I S .

Aircraft Controls

A set of aircraft controls cAC are defined, and connectedto the component controls. The connection to thecomponent control c is typically of the formc = STcAC + c0 , where T is an input matrix and c0 thecomponent control for zero aircraft control. The factor Sis available for internal scaling of the matrix. Figure 4illustrates the control relationships.

The connection (matrix T) is defined for several controlsystem states, allowing change of control configurationwith flight state. An example is control of a tiltrotor inhelicopter mode and airplane mode flight. The controlstate and initial control values are specified for each flightstate. Default control matrices are defined based on theconfiguration: helicopter, tandem, coaxial, or tiltrotor.

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'scontrols are contained in the matrix T.

These aircraft controls are available for trim of the aircraft.Any aircraft controls not selected for trim will remainfixed at the values specified for the flight state. Thus bydefining additional aircraft controls, component controlscan be specified as required for a flight state.

The tilt control variable ^tilt is intended for nacelle tiltangle or conversion control, particularly for tiltrotors.Typically this variable will be connected to nacelle/pylon,engine, rotor shaft, and even wings. The convention is

^tilt = 0 for cruise, and ^tilt = 90 deg for helicopter mode.

Trim

The aircraft trim operation solves for the controls andmotion that produce equilibrium in the specified flightstate. In steady flight (including hover, level flight, climband descent, and turns), equilibrium implies zero net forceand moment on the aircraft. There can be additionalquantities that at equilibrium must equal target values. Inpractice, the trim solution can deal with a subset of thesequantities. Usually it is at least necessary to achieveequilibrium in the aircraft lift and drag forces, as well as inyaw moment for torque balance. The basic purpose of thetrim solution is to determine the component states,including aircraft drag and rotor thrust, sufficient toevaluate the aircraft performance.

Different trim solution definitions are required for variousflight states. Therefore one or more trim states are defined

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for the analysis, and the appropriate trim state selected foreach flight state of a performance condition or missionsegment. For each trim state, the trim quantities, trimvariables, and targets are specified. The available trimquantities include aircraft total force and moment, loadfactor, power or power margin, rotor force or torque, rotorflapping, wing force or lift margin, and tail force. Theavailable trim variables include aircraft controls,orientation, speed, pullup rate, and turn rate.

Geometry

Layout of the geometry is typically in terms of station line(SL, positive aft), buttline (BL, positive right), andwaterline (WL, positive up), measured relative to somearbitrary origin. The x, y, and z axes are parallel to theSL, BL, and WL directions. One or more locations aredefined for each component of the aircraft. Eachcomponent will at least have a location that is the pointwhere component forces and moments act on the aircraft.Each location is input in fixed or scaled form. The fixedform input is SL/BL/WL (dimensional). The scaled forminput is x / L (positive aft), y / L (positive right), and z / L

(positive up), from a reference point. The reference lengthL is the rotor radius or wing span of a designatedcomponent, or the fuselage length. Fixed input can be usedfor the entire aircraft, or just for certain components.

Aircraft Motion

The aircraft velocity and orientation are defined by theflight speed, turn rate, orientation of the body framerelative to inertial axes (Euler angles), and orientation ofthe velocity frame relative to inertial axes (flight pathangles).

Aircraft conventions are followed for the direction andorientation of axes: the z-axis is down, the x-axisforward, and the y -axis to the right; and a yaw-pitch-rollsequence is used for the Euler angles.

The orientation of the body frame F relative to inertialaxes is defined by yaw ?4 F , pitch BF , and roll OF Eulerangles. The flight path is specified by the velocity V, inthe positive x-axis direction of the velocity axes. Theorientation of the velocity axes relative to inertial axes isdefined by yaw ?4 V (sideslip) and pitch V (climb) angles.In straight flight, all these angles are constant. In turningflight at a constant yaw rate, the yaw angle is VF = ?4 F t .

The aircraft angular velocity ^ACF

is obtained from theEuler angle rates. For steady state flight, ˙^ F =' F = 0 ; ?4

F

is nonzero in a turn. Accelerated flight is also considered,in terms of linear acceleration and pitch rate.

Loads and Performance

For each component, the power required and the net forcesand moments acting on the aircraft can be calculated. Theaerodynamic forces F and moments M are typicallycalculated in wind axes and then resolved into body axes,relative to the origin of the body axes (the aircraft center-of-gravity). The power and loads of all components aresummed to obtain the aircraft power and loads. Typicallythe trim solution drives the net forces and moments on theaircraft to zero.

The aircraft equations of motion, in body axes F withorigin at the aircraft center-of-gravity, are the equations offorce and moment equilibrium. The forces and momentsare the sum of loads from all components of the aircraft:fuselage, rotor, force, wing, tail, engine, and tank. Aparticular component can have more than one source ofloads; for example, the rotor component produces hubforces and moments, but also includes hub and pylon drag.

The component power required Pcomp is evaluated for allcomponents (rotors) of the propulsion group. The totalpower required for the propulsion group PreqPG isobtained by adding the transmission losses and accessorypower. The power required for the propulsion group mustbe distributed to the engine groups. From the powerrequired the fuel flow is calculated. The fuel flow of thepropulsion group is obtained from the sum over the enginegroups. The total fuel flow is the sum from all componentsof the aircraft: w = I w

reqEG + I w force

Aerodynamics

Each component has a position zF in aircraft axes F,relative to the reference point, and orientation ofcomponent axes B relative to aircraft axes given by arotation matrix. Acting at the component are interferencevelocities vint (velocity of air, in F axes) from all othercomponents. Then the total component velocity relative tothe air is the sum of velocity, angular velocity, andinterference terms. The aerodynamic environment isdefined in the component axes: velocity magnitudev = | v B |, dynamic pressure q = 12 Pv 2 , angle-of-attack a,and sideslip angle 6. From these the componentaerodynamic model calculates the loads in wind axes:drag, side force, and lift ( D, Y, and L) and the roll, pitch,and yaw moments; and from wind axis loads theaerodynamic loads in aircraft axes acting at the center-of-gravity are calculated.

The component aerodynamic models are based onequations intended to cover the principal operating

9

conditions, including small angle-of-attack and sideslipwith a representation of stall and maximum lift, verticalflight, sideward flight, and rearward flight.

Drag

Each component can contribute drag to the aircraft. Afixed drag can be specified, as a drag area D /q; or thedrag can be scaled, specified as a drag coefficient CD

based on an appropriate area S. There may also be otherways to define a scaled drag value. For fixed drag, thecoefficient is CD = (D / q) / S (the aerodynamic model isformulated 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 orcruise; typically this is the minimum drag value. For somecomponents, the vertical drag or sideward drag is defined.For some components, the aerodynamic model includesdrag due to lift, angle-of-attack, or stall.

Table 1 summarizes the component contributions to drag,and the corresponding reference areas. If no reference areais indicated, then the input is only drag area D /q. Anappropriate drag reference area is defined for eachcomponent; the reference area is either input or calculated.

Table 1. Component contributions to drag.

component drag contribution reference area

fuselage fuselage fuselage wetted area

Optionally the aircraft drag can be fixed. The quantityspecified is the sum (over all components) of the drag area

D /q (minimum drag, excluding drag due to lift andangle-of-attack), without accounting for interferenceeffects on dynamic pressure. The input parameter can beD /q; or the drag can be scaled, specified as a dragcoefficient based on the rotor disk area, so D / q = ArefCD

( Aref is the reference rotor disk area); or the drag can beestimated from the aircraft maximum take-off weight,D / q = k (WMTO /1000) 2/3 . Based on historical data, thedrag coefficient CD = 0.02 for old helicopters andCD = 0.008 for current low drag helicopters. Based onhistorical data, k = 9 for old helicopters, k = 2.5 forcurrent low drag helicopters, k = 1.6 for current tiltrotors,and k = 1.4 for turboprop aircraft ( k in English units). Ifthe aircraft drag is input, then the fuselage contingencydrag is adjusted so the total aircraft D /q equals the inputvalue.

The nominal drag areas of the components and the aircraftare part of the aircraft description and are used when theaircraft drag is fixed. While vertical drag parameters arepart of the aerodynamic model for the hub, pylon, andnacelle, aerodynamic interference at the rotor and at theengine group is not considered, so these terms do notcontribute to download. In the context of download, onlythe fuselage, wing, tail, and contingency contribute to thenominal vertical drag.

FUSELAGE

There is one fuselage component for the aircraft. There areseveral options for calculating the fuselage length, wettedarea (reference area for drag coefficients), and projectedarea (reference area for vertical drag).

The aerodynamic velocity of the fuselage relative to theair, including interference, is calculated in componentaxes. The drag area or drag coefficient is defined forforward flight, vertical flight, and sideward flight. Forsmall angle-of-attack, the drag increases proportional to(1 + K | ^ |X ) , using input factor K and exponent X. Inaddition, the forward flight drag area or drag coefficient isdefined for fixtures and fittings, and for rotor-bodyinterference. The drag force is

D = qSwet (CD + CD fit + ^ CD rb) + q((D / q) pay + (D / q) cont )

including the drag area of the payload (specified for theflight state) and the contingency drag area.

The fuselage lift, pitch moment, side force, and yawmoment are defined in fixed form (e.g. L /q) or scaledform (e.g. CL). Maximum lift coefficient and maximumside force coefficient are defined.

fuselage vertical fuselage projected area

fixtures and fuselage wetted areafittingsrotor-body interf fuselage wetted area

contingency —

payload —increment

gear landing gear —

rotor hub, hub vertical rotor disk area

pylon, pylon vert pylon wetted area

spinner spinner wetted area

wing wing, wing vert wing planform area

wing-body interf wing planform area

tail tail, tail vertical tail planform area

engine nacelle, nac. vert nacelle wetted area

momentum drag —

fuel tank auxiliary tank —

10

ROTORThe aircraft can have one or more rotors, or no rotors. Inaddition to main rotors, the rotor component can modeltail rotors, propellers, proprotors, ducted fans, thrustvectoring rotors, and auxiliary-thrust rotors. The principalconfiguration designation (main rotor, tail rotor, orpropeller) determines where the weights are put in theweight statement (rotor group, empennage group, orpropulsion group), and each configuration can possiblyhave a separate performance or weight model. Antitorquerotors and auxiliary-thrust rotors can be identified, forspecial sizing options. Other configuration features arevariable diameter and ducted fan.

Multi-rotor systems (such as coaxial or tandemconfiguration) are modeled as a set of separate rotors, inorder to accommodate the description of the position,orientation, controls, and loads. The performancecalculation for twin rotor systems can include the mutualinfluence of the induced velocity on the power.

Drive SystemThe drive system defines gear ratios for all thecomponents that it connects (rotors and engine groups).There is one primary rotor per propulsion group (forwhich the reference tip speed is specified); othercomponents are dependent (for which a gear ratio isspecified). For the primary rotor, a reference tip speedVtip-ref is defined for each drive system state.

GeometryThe rotor rotation direction is described by the parameterr: r = 1 for counter-clockwise rotation, r = — 1 forclockwise rotation (as viewed from the positive thrust sideof the rotor). The rotor solidity and blade mean chord arerelated by o = Nc /;rR . Generally thrust-weighted valuesare used, but geometric values are also required by theanalysis. A general blade chord distribution is specified asc (r) = crefc (r), where cref is the thrust-weighted chord.Linear taper is specified in terms of a taper ratiot = ctip /croot, or in terms of the ratio of thrust-weightedand geometric chords, f = 6t / 6g = c.75R / c.50R.

The rotor hub is at position zhubF. A component of theposition can be calculated, superseding the location input.The calculated geometry depends on the configuration.For a coaxial rotor, the hub locations can be calculatedfrom the input separation, and the input location midwaybetween the hubs. For a tandem rotor, the hub locationscan be calculated from the input overlap, and the inputlocation midway between the hubs. For a tail rotor, the

longitudinal position can be calculated from the main rotorradius, tail rotor radius, and tail-rotor/main-rotorclearance. For a tiltrotor, the lateral position can becalculated from the rotor radius (cruise value for variablediameter rotor), fuselage/rotor clearance, and fuselagewidth (with the pivot, pylon, and nacelle center-of-gravitylateral positions adjusted to keep the same relative positionto the hub). Alternatively for a tiltrotor, the lateral positioncan be calculated from the wing span so the rotors are atthe wing tips; or from a designated wing panel edge.

Control and LoadsThe rotor controls consist of collective, lateral cyclic,longitudinal cyclic, and perhaps shaft incidence (tilt) andcant angles. Rotor cyclic control can be defined in termsof tip-path plane or no-feathering plane command. Thecollective control variable is the rotor thrust amplitude T

or CT/6 (in shaft axes), from which the collective pitchangle can be calculated. This approach eliminates aniteration between thrust and inflow.

The relationship between tip-path plane tilt and hubmoment is M = (N /2)Ib Q 2 (v2 - 1)0 = Khubg, where Nis the number of blades, Q the rotor speed, and v thedimensionless fundamental flap frequency. The flapmoment of inertia Ib is obtained from the input Locknumber.

Optionally the rotor can have a variable diameter. Therotor diameter is treated as a component control. As acontrol it can be connected to an aircraft control and thusset for each flight state.

Tip-path plane command is characterized by direct controlof the rotor thrust magnitude and the tip-path plane tilt.Cyclic tilt of the tip-path plane, hence tilt of the thrustvector, consists of longitudinal tilt fic (positive forward)and lateral tilt Ps (positive toward retreating side).Alternatively, the cyclic control can be specified in termsof hub moment or lift offset, if the blade flap frequency isgreater than 1/rev. This control mode requires calculationof rotor collective and cyclic pitch angles from the thrustmagnitude and flapping.

No-feathering plane command is characterized by controlof rotor cyclic pitch angles, and direct control of the rotorthrust magnitude. Cyclic tilt of the no-feathering plane,usually producing tilt of the thrust vector, consists oflongitudinal cyclic pitch angle ^s (positive aft) and lateralcyclic pitch angle Oc (positive toward retreating side).This control mode requires calculation of rotor collectivepitch angle and tip-path plane tilt from the thrust

11

magnitude and cyclic control, including the influence ofinflow.

drag of the fuselage or a lifting surface in the aircraft trim.For example, fB can model the tail rotor blockage causedby operation near the vertical tail.

Rotor Axes and Shaft Tilt

The rotor hub is at position zub, where the rotor forces

and moments act; the orientation of the rotor shaft axesrelative to the aircraft axes is given by the rotation matrixC SF . The pivot is at position zpivotF. The hub or shaft axesS have origin at the hub center; the z -axis is the shaft,positive in the positive thrust direction; and the x -axisusually downstream. The rotor orientation is specified byselecting a nominal direction in body axes (positive ornegative x, y, or z -axis) for the positive thrust direction;the other two axes are then the axes of shaft control. For amain rotor the nominal direction would be the negative z -axis; for a tail rotor the nominal direction would be the±y -axis (depending on the direction of rotation of themain rotor); for a propeller the nominal direction would bethe positive x-axis. The hub and pivot axes have a fixedorientation relative to the body axes: hub incidence andcant, pivot dihedral, pitch, and sweep. The shaft anglecontrol consists of incidence and cant about the pivot axes.

For a tiltrotor aircraft, one of the aircraft controls is thenacelle angle, with the convention ^tilt = 0 for cruise, and^tilt = 90 deg for helicopter mode. The rotor shaftincidence angle is then connected to ^tilt by defining thecontrol matrix appropriately.

Hub Loads

The rotor controls give the thrust magnitude and the tip-path plane tilt angles fic and fis , either directly or from thecollective and cyclic pitch. The forces acting on the hubare the thrust T, drag H, and side force Y (positive in z,x, and y -axis directions respectively). The hub pitch androll moment are proportional to the flap angles. The hubtorque is obtained from the shaft power Pshaft and rotorspeed Q. The force and moment acting on the hub, inshaft axes, are then:

^ H ^ ^ 0 ^^ ^FS = ^ Y + 0

^^ T ^

^ ^^ —fBT ^

^^ Mx Khub (rIS )

M S = ^My = Khub (—Ic )

^^ —rQ ^^ —rPshaft / Q ^

^

The force includes a term equal to the rotor thrust times aninput blockage factor fB =AT /T . This term accounts forblockage or download, as an alternative to including the

The rotor loads in aircraft axes acting at the center-of-gravity ( FF and M F ) are then calculated from the shaftaxis loads ( F S and M S ) and Az = z hub

F

— z cgF.

The wind axes lift L and drag X are calculated from thenet rotor hub force FF and the rotor velocity vF . Thevelocity relative to the air gives the propulsive forcedirection eP = vF /| vF | (no interference), and thevelocity magnitude V = | v F |. The lift and dragcomponents of the force are XV = —(v F)T FF andL2 = | F F |2 —X 2.

Aerodynamics

The rotor velocity relative to the air isvF = vACF +

0)ACF X Az F in aircraft axes. The velocities in

shaft axes are

— μ x ^

^v S = CSF vF =QR ^ r μ y ^

^ ^μ z ^

where QR is the rotor tip speed. The advance ratio, inflowratio, and shaft angle-of-attack are defined asμ = (μ x

2 + μ y2)1/2 , A = Ai + μ z , and a = tan—1 ( μ z / μ ). The

rotor thrust coefficient is defined as CT = T /pA (QR) 2 .The dimensionless ideal induced velocity Ai is calculatedfrom μ, μz, and CT ; the dimensional velocity isvi = QR i . The ideal induced power is then Pideal = Tv i .Note that for these inflow velocities, the subscript “ i ”denotes “ideal.”

Ideal Inflow

The ideal wake-induced velocity is obtained from themomentum theory result of Glauert:

CTsAh

2

Ai = =2 A2 + μ 2A2 + μ 2

where A = Ai + μ z , Ah2 = | CT |/2 ( Ah is always positive),

and s = sign(CT ). This expression is generalized toAi = Ah s F ( μ / Ah , s μ z / Ah ) . If μ is zero, the equation forAi can be solved analytically. Otherwise, for non-axialflow, an iterative Newton-Raphson solution for Ai isneeded.

An approximate equation is used in the turbulent-wakeand vortex-ring states to eliminate the singularity of themomentum theory result at ideal autorotation. The

12

momentum theory result is also extended to the case of aducted fan.

The wake-induced velocity is reduced when the rotor diskis in the proximity of the ground plane,(Ai) IGE = fg (Ai) OGE. The factor fg is a function of thescaled rotor height above the ground, zg /D . The effectsof ground plane tilt and rotor velocity are modeled.Several empirical ground effect models are implemented.

As a simple approximation to nonuniform inducedvelocity distribution, a linear variation over the disk isused: AA, = ),x r cosip + ),yr sinip . There are contributionsto AA from forward flight and from hub moments, whichinfluence the relationship between flapping and cyclic.The models implemented for the forward flight gradientsare based on references 12 to 15. Differential momentumtheory is used to calculate the gradients caused by hubmoments.

significant, as for a rotor with large flap stiffness. Figure 5shows the tip-path plane tilt and thrust vector tilt withcyclic pitch control (no-feathering plane tilt) as a functionof flap stiffness (per-rev frequency) for several rotor thrustvalues. The difference between tip-path plane tilt andthrust vector tilt is caused by tilt of the thrust vectorrelative to the tip-path plane.

The profile inplane forces can be obtained from simplifiedequations, or calculated by blade element theory. Thesimplified method uses:

^ CHo\CYo ) 8 cd mean FH

\_μ y / μ /

where the mean drag coefficient cd mean is from the profilepower calculation. The function FH accounts for theincrease of the blade section velocity with rotor edgewiseand axial speed.

Rotor Forces

Direct control of the rotor thrust magnitude is used, so therotor collective pitch angle 00 must be calculated from thethrust CT /Q. If the commanded variable were thecollective pitch angle, then calculating the rotor thrustwould be necessary, resulting in a more complicatedsolution procedure; in particular, an iteration betweenthrust and inflow would be needed. There may be flightstates where the commanded thrust can not be producedby the rotor, even with stall neglected in the sectionaerodynamics. This situation will manifest as an inabilityto solve for the collective pitch given the thrust. In thiscircumstance the trim method should be changed so therequired or specified thrust is an achievable value.

The inplane hub forces are produced by tilt of the thrustvector with the tip-path plane, plus forces in the tip-pathplane, and profile terms (produced by the blade dragcoefficient). The orientation of the tip-path axes relative tothe shaft axes is C PS , which is determined by Pc and Ps .Then

^ CH 0 CxmP

CHO

CY = CsP 0 + rCY tpp + rCYO

CTCT 1 C33 0 0 ^

The inplane forces relative to the tip-path plane can beneglected or calculated by blade element theory. Note thatwith tip-path plane command and CH tpp and CYtpp

neglected, solving for the rotor collective and cyclic pitchangles is not necessary. In general the inplane forcesrelative to the tip-path plane are not zero, and may be

1.2 - - - - - - - CT/6=0.14b — — — CT/6 = 0.10

1.0 — CT/6 = 0.06

0.8 CT/6 = 0.02

0.6

Y 0.4

as 0.2F

0.01.0 1.2 1.4 1.6 1.8 2.0

flap frequency v (per-rev)

1.0 1.2 1.4 1.6 1.8 2.0

flap frequency v (per-rev)

Figure 5. Tip-path plane tilt and thrust vector tilt withcyclic pitch.

a, 1.2b1.0

E! 0.8U

0.6

0.4

0.2

0.0

13

Blade Element Theory

Blade element theory is the basis for the solution for thecollective and cyclic pitch angles (or flap angles) andevaluation of the rotor inplane hub forces. The sectionaerodynamics are described by lift varying linearly withangle-of-attack, cg = c laa (no stall), and a constant meandrag coefficient cd mean (from the profile powercalculation). The blade section aerodynamic environmentis described by the three components of velocity, fromwhich the yaw and inflow angles are obtained, and thenthe angle-of-attack a. The blade pitch consists ofcollective, cyclic, twist, and pitch-flap coupling terms. Theflap motion is rigid rotation about a hinge with no offset,and only coning and once-per-revolution terms areconsidered. The inflow includes gradients caused byedgewise flight and hub moments. The effect of the inflowproduced by hub moments is to introduce a lift-deficiencyfunction in the flap response.

Integrating the section lift, drag, and radial forces over theblade radial coordinate and azimuth gives the total rotorthrust, drag, and side forces. Integrating terms producedby the section drag coefficient gives the profile inplaneforces, CHo and CYo; using blade element theory toevaluate these accounts for the planform and root cutout,while using FH implies a rectangular blade and no rootcutout (plus at most a 1% error approximating the exactintegration). The remaining terms in the section forcesproduce the inplane loads relative to the tip-path plane,CH tpp and CYtpp.

Evaluating these inplane forces requires the collective andcyclic pitch angles, and the flapping motion. The thrustequation must be solved for the rotor collective pitch. Therelationship between cyclic pitch and flapping is definedby the rotor flap dynamics. The flap motion is rigidrotation about a central hinge, with a flap frequency v > 1for articulated or hingeless rotors. The flapping equationof motion, including the effects of precone and the inertialloads of shaft angular motion, is harmonically analyzed.The result is equations for the mean (coning) and 1/rev(tip-path plane tilt) flap motion. The solution for theconing is largely decoupled by introducing the rotor thrustcoefficient.

The thrust and flapping equations of motion must besolved for the unknown angles. For tip-path planecommand, the thrust and flapping are known, so theequations are solved for 00.75 , 0c , and 0s . For no-feathering plane command, the thrust and cyclic pitch areknown, so the equations are solved for 00.75, c , and s.

Power

The rotor power consists of induced, profile, and parasiteterms: P = Pi + Po + Pp. The rotor parasite power(including climb/descent power for the aircraft) isobtained from the wind axis drag force:Pp =—XV = (vF )T FF.

The induced power is calculated from the ideal power:Pi = KPideal = KfDTvideal. The empirical factor K accountsfor the effects of nonuniform inflow, non-ideal spanloading, tip losses, swirl, blockage, and other phenomenonthat increase the induced power losses ( K > 1). For aducted fan, fD = fW /2 is introduced, where fW is theratio of the far wake induced velocity to the inducedvelocity at the disk.

The profile power is calculated from a mean blade dragcoefficient: Po = pA (^R) 3 CPo, CPo = (6 /8)cd meanFP .

The function FP ( μ , μ z ) accounts for the increase of theblade section velocity with rotor edgewise and axial speed.

Two performance methods are implemented: the energymethod and the table method. The induced power factorand mean blade drag coefficient are obtained fromequations with the energy method, or from tables with thetable method. Optionally K and cd mean can be specifiedfor each flight state, superseding the performance methodvalues.

Energy Performance Method: Induced Power

The induced power is calculated from the ideal power:Pi = KPideal = KfDTvideal. Reference values of K arespecified for hover, axial cruise (propeller), and edgewisecruise (helicopter): Khover, Kprop, K edge. Two models areimplemented: constant model and standard model. Theconstant model uses K = K hover if μ = μ z = 0; or K = K prop

if | μ |< 0.1| μ z |; or K = Kedge otherwise.

The standard model calculates an axial flow factor K axial

from the values Khover , K climb, and Kprop. LetA = CT /6 — (CT /6) ind. For hover and low speed axialclimb, including a variation with thrust, the inflow factoris

2Kh = Khover + kh1Ah + kh 2A h

+ (K climb — K hover) 2 tan—1[((| μ z |/4 ) / M axial )X axial ]

where | μ z | /),h = Maxial is the midpoint of the transitionbetween hover and climb. Figure 6 illustrates K in hover(with a minimum value). A polynomial describes thevariation with axial velocity, scaled so K = Kh at μ z = 0

14

and K = Kp at μ z = μ z prop , including a variation withthrust:

x p = K prop + kp 1Ap + kp 2A2p

2 Xx axial = xh + ka1 μ z + Sa (ka 2 μ z + ka 3 μ z a )

where Sa accomplishes the scaling. A polynomialdescribes the variation with edgewise advance ratio, scaledso K = K axial at μ = 0 and K = foffK edge at μ = μ edge. Thusthe induced power factor is

K = Kaxial + ke1 μ + Se (ke 2 μ 2 + ke 3 μ X e)

where Se accomplishes the scaling. The functionf o ff = 1– ko1 (1– e– ko2 o

x ) accounts for the influence of liftoffset, ox = rMx /TR = (Khub /TR)Ps . Figure 7 illustratesK in edgewise flight. Minimum and maximum values ofthe induced power factor are also specified.

1.30

1.20k

1.10

1.000.00 0.03 0.06 0.09 0.12 0.15 0.18

CT/Cr

Figure 6. Induced power factor in hover (with minimumK = 1.12).

5.00 CT/cr = 0.08

F— — — CT/Cr = 0.14

4.00 ^ O ( μ edge, Kedge )

k 3.00

2.00

1.00 '0.00 0.10 0.20 0.30 0.40 0.50

μ

Figure 7. Induced power factor in edgewise flight.

Energy Performance Method: Profile Power

The profile power is calculated from a mean blade dragcoefficient: CPo = (6 /8)cd mean FP . Since the blade meanlift coefficient is c$ sg 6CT /6 , the drag coefficient is

estimated as a function of blade loading CT /6 (usingthrust-weighted solidity). With separate estimates of thebasic, stall, and compressibility drag, the mean dragcoefficient is

cd mean = XS (cd basic + cd stall + cd comp)

where X is a technology factor. The factorS = (Reref / Re)0.2 accounts for Reynolds number effectson the drag coefficient; Re is based on the thrust-weighted chord, 0.75Vtip, and the kinematic velocity ofthe flight state; and Reref corresponds to the input dragcoefficient information. Array and equation models areimplemented for the basic drag. In the array model thebasic drag cd basic is input as a function of CT /6 (linearlyinterpolated array).

In the equation model the basic drag cd basic is a quadraticfunction of CT /6 , plus an additional term allowing fastergrowth at high (sub-stall) angles of attack. LetA =| CT /6 — (CT /6) D min |, where (CT /6)D min

corresponds to the minimum drag; andA sep = | CT /6 | —(CT /6)sep. Values of the basic dragequation are specified for helicopter (hover and edgewise)and propeller (axial climb and cruise) operation:

cdh = d0 hel + d1 helA + d2 helA2 + ds Aep seP

cdp = d0 prop + d1propA+ d2propA2 + d sepAxP

The separation term is present only if A sep > 0. Thehelicopter and propeller values are interpolated as afunction of μ z :

cd basic = cdh + (cdp — cdh) 2^tan—1(| μ z | /)'h )

so | μ z | / a,h = 1 is the midpoint of the transition.

The stall drag increment represents the rise of profilepower caused by the occurrence of significant stall on therotor disk. Let A s = | CT /6 | —(fs / foff)(CT /6) s , where f s

is an input factor. The function f o ff = 1– do1 (1– e– d o2 ox )

accounts for the influence of lift offset,ox = rMx /TR = (Khub /TR)Ps . Then

cd stall = ds1A sX s 1 + ds2A sX s 2

(zero if As s 0). The blade loading at which the stallaffects the entire rotor power, (CT /6) s , is an inputfunction of the velocity ratio V = ( μ 2 + μ z2 )1/ 2 .

Figure 8 shows typical stall functions (CT /6) s for tworotors with different airfoils. For reference, typical rotorsteady and transient load limits are also shown.

The compressibility drag increment depends on theadvancing tip Mach number Mat . Drag divergence and

15

similarity models are implemented. For the dragdivergence model, let AM = Mat - Mdd , where Mdd isthe drag divergence Mach number of the tip section. Thenthe compressibility increment in the mean drag coefficientis

cd comp = dm1 AM + dm20MXm

(ref. 16). Mdd is a function of the advancing tip liftcoefficient (available from blade element theory) and thetip airfoil thickness-to-chord ratio.

Figure 9 illustrates the mean drag coefficient in hover,showing cdh without and with the separation term, and thetotal for the high-stall and low-stall cases. Figure 10illustrates the mean drag coefficient in forward flight,showing the compressibility term cd comp, and the growthin profile power with CT /o and μ as the stall dragincrement increases.

- - - transient limit- - steady limit

high stall

0.20 _ low stall

0.16 ^' \

p 0.12

U 0.08*'

0.04

0.00 1 1 1 1 11'

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70

μ

Figure 8. Profile power stall function.

cd mean (low stall)

• - ! cd mean (high stall)

0.0400 - - - - - cdh (with separation)- - - - - - - - - cdh (quadratic)

0.0300

0.0200 /i

ob

0.0100 - -^•r^

0.0000 ' 1 1 11'

0.00 0.03 0.06 0.09 0.12 0.15 0.18

CT/o

Figure 9. Mean drag coefficient in hover.

CT/o = 0.14

CT/o = 0.12

- CT/o = 0.10

0.0400 • - • - CT/o = 0.08

-----0.0300

9 0.0200

-----Zc:p

0.0100

(a) I J_ - I . I I0.0000 _ _ __ -

0.00 0.10 0.20 0.30 0.40 0.50

μ

0.0400 /

0.0300 /0.0200

U

0.0100 _

(b) .0.0000

0.00 0.10 0.20 0.30 0.40 0.50

μ

Figure 10. Mean drag coefficient in forward flight; (a)high stall; (b) low stall.

Twin Rotors

For twin rotors, the induced power is determined by theinduced velocity of the rotor system, not the individualrotors. The induced power is still obtained usingPi = Pideal = fDTvideal for each rotor, but the idealinduced velocity is calculated for an equivalent thrust CTe

based on the thrust and geometry of both rotors (see refs. 4and 6). The profile power calculation is not changed fortwin rotors.

Interference

The rotor can produce aerodynamic interference velocitiesat the other components (fuselage, wings, tails). Theinduced velocity at the rotor disk is ^v i, acting oppositethe thrust ( z-axis of tip-path plane axes). Sovind

P =-kP" - and v,nd = CFP v,pd . The total velocity of1 ,the rotor disk relative to the air consists of the aircraftvelocity and the induced velocity from this rotor:vt l = v F - v i

Fnd. The direction of the wake axis is thus

ewP = -CPF vt tal / | vtotalF |. The angle of the wake axis from

the thrust axis is X = cos-1 |(kP ) T ewP |.

16

The interference velocity vi.t at each component isproportional to the induced velocity v indF (and is in thesame direction), with factors accounting for the stage ofwake development and the position of the componentrelative to the rotor wake:

vintF

= KintfWfzfrftv indF

The factors fW fz account for axial development of thewake velocity, the factor fr accounts for immersion in thewake, and Kint is an input empirical factor. An additionalfactor ft for twin rotors is included.

Optionally the development along the wake axis can be astep function ( fW fz = 0, 1, or fW for above the rotor, onthe rotor disk, or below the rotor disk, respectively); or thewake develops with the nominal or input rate of change.Optionally the wake immersion can use the contractedradius or the uncontracted radius; be a step function( fr = 1 and 0 inside and outside the wake boundary); bealways immersed ( fr = 1 always); or use an inputtransition distance.

The interference factor Kint can be reduced from an inputvalue at low speed to zero at high speed, with linearvariation over a specified speed range. To account for theextent of the wing or tail area immersed in the rotor wake,the interference velocity can be calculated at several pointsalong the span and averaged.

Drag

The rotor component includes drag forces acting on thehub and spinner and on the pylon.

The hub drag can be fixed, specified as a drag area D /q;

scaled, specified as a drag coefficient CD based on therotor disk area; or estimated based on the gross weight,using either a squared-cubed relationship or a square-rootrelationship. Based on historical data, the drag coefficientCD = 0.004 for typical hubs, CD = 0.0024 for current lowdrag hubs, and CD = 0.0015 for faired hubs. For thesquared-cubed relationship: (D /q) hub = k (WMTO /1000) 2/3

where WMTO is the maximum take-off gross weight.Based on historical data, k = 1.4 for typical hubs, k = 0.8for current low drag hubs, and k = 0.5 for faired hubs ( kin English units). For the square-root relationship:

(D / q) hu b = k (WMTO / N rotor )1/2 where WMTO /Nrotor is the

maximum take-off gross weight per lifting rotor. Based onhistorical data (ref. 17), k = 0.074 for single rotorhelicopters, k = 0.049 for tandem rotor helicopters(probably a blade number effect), k = 0.038 for hingelessrotors, and k = 0.027 for faired hubs ( k in English units).

The pylon forward flight drag and vertical drag arespecified as drag area or drag coefficient, based on thepylon wetted area. The spinner drag is specified as dragarea or drag coefficient, based on the spinner wetted area.

FORCE

The force component is an object that can generate a forceacting on the aircraft, possibly used for lift, or propulsion,or control. The amplitude of the force can be a fixed value,or connected to an aircraft control for trim. The directionof the force can be fixed, or connected to aircraft control.

The force generation requires a fuel flow that is calculatedfrom an input thrust specific fuel consumption:w = A (sfc) , where A is the force amplitude, and the unitsof sfc are pound/hour/pound or kilogram/hour/Newton.

The force component weight is identified as either enginesystem or propeller/fan installation weight, both of thepropulsion group. The force component weight iscalculated from a specific weight and the design maximumforce Fmax, plus a fixed increment: W = SFmax + AW .

WING

The aircraft can have one or more wings, or no wings. Thewing is described by planform area S, span b, mean chordc = S / b , and aspect ratio AR = b2 /S . These parametersare for the entire wing. The geometry is specified in termsof two of the following parameters: S or wing loadingW /S, b (perhaps calculated from other geometry), c,

AR. With more than one wing, the wing loading isobtained from an input fraction of design gross weight,W = fWWD . Optionally the span can be calculated from aspecified ratio to the span of another wing.

For the tiltrotor configuration, the wing span can becalculated from the fuselage and rotor geometry (rotorradius, rotor-fuselage clearance, and fuselage width). Thewing span can be calculated from the rotor hub position(regardless of how the rotor position is determined).Optionally the wing span can be calculated from anappropriate specification of all wing panel widths.

Panels

The wing planform is defined in terms of one or morewing panels (figure 11). Symmetry of the wing isassumed. Each panel has a straight aerodynamic centerand linear taper. The aerodynamic center locus (in wingaxes) is defined by sweep, dihedral, and offsets at theinboard edge relative to the aerodynamic center of theprevious panel. The wing position is the meanaerodynamic center.

17

A panel is characterized by span bp (each side), meanchord cp , and area Sp = 2bp cp (both sides). The taper isdefined by inboard and outboard chord ratios. The span foreach panel (if there are more than two panels) can be afixed input, a fixed ratio of the wing span, or free. Thepanel outboard edge (except for the wing tip) can be at afixed input position, at a fixed station, calculated from thefuselage and rotor geometry, calculated from the hubposition, or adjusted. The specification of panel spans andpanel edges must be consistent, and sufficient to determinethe wing geometry. To complete the definition of thegeometry, one of the following quantities is specified foreach panel: panel area Sp ; ratio of panel area to wingarea, Sp /S; panel mean chord cp ; ratio of panel meanchord to wing mean chord, cp /c; inboard and outboardchord ratios; or free. The total wing area equals the sum ofall panel areas.

Figure 11. Wing geometry (symmetric, only right half-wing shown).

Controls

The wing control variables are flap, flaperon, aileron , andincidence. The flaperon and aileron are the same surface,generating symmetric and antisymmetric loadsrespectively, hence with different connections to pilotcontrols. With more than one wing panel, each panel canhave control variables.

Aerodynamics

The wing vertical drag can be fixed, specified as a dragarea (D / q) V ; or scaled, specified as a drag coefficientCDV based on the wing area; or calculated from an airfoilsection drag coefficient (for —90 deg angle-of-attack) andthe wing area immersed in the rotor wake (includingchanges in wing area due to flap and flaperon deflection).

The wing lift, pitch moment, and roll moment are definedin scaled form (coefficients). From the control surfacedeflection and geometry, the lift coefficient, maximum liftangle, moment coefficient, and drag coefficient incrementsare evaluated. The wing lift is defined in terms of liftcurve slope and maximum lift coefficient. The three-dimensional lift curve slope is input directly or calculatedfrom the two-dimensional lift curve slope and the wingaspect ratio.

The drag area or drag coefficient is defined for forwardflight and vertical flight. For small angle-of-attack, thedrag increases proportional to (1 + K | ^ |

X), using input

factor K and exponent X, plus an additional termrepresenting separation drag. The induced drag is obtainedfrom the lift coefficient, aspect ratio, and Oswaldefficiency e:

(CL — CL 0 ) 2

CDiace AR

Conventionally, the Oswald efficiency e can represent thewing parasite drag variation with lift, as well as theinduced drag (hence the use of CL0). The wing-bodyinterference is specified as a drag area, or a dragcoefficient based on the wing area. Then

D = qSCD = qS (CDp + CDi + CDwb )

is the drag force.

Interference

With more than one wing, the interference velocity atother wings is proportional to the induced velocity of thewing producing the interference, with an input factor Kint.

The induced velocity is obtained from the induced drag.For tandem wings, typically Kint = 2 for the interferenceof the front wing on the aft wing, and Kint = 0 for theinterference of the aft wing on the front wing. For biplanewings, the mutual interference is typically Kint = 0.7(upper on lower, and lower on upper). The induced drag isthen

CDi = (CL

CL 0 ) 2 + CL Kip.

ice AR

18

where the sum is over all other wings.

The wing interference at the tail produces an angle-of-attack change E = E(CL / CLa) , where E = de / da is aninput factor determined by the aircraft geometry. Thechange in orientation of the wing velocity produces theinterference velocity vF at the tail.int

Wing Extensions

The wing can have extensions, defined as wing portions ofspan bX at each wing tip. For the tiltrotor configuration inparticular, the wing weight depends on the distribution ofwing area outboard (the extension) and inboard of therotor and nacelle location. Wing extensions are defined asa set of wing panels at the tip. The extension span and areaare the sum of the panel quantities.

EMPENNAGE

The aircraft can have one or more tail surfaces, or no tailsurface. Each tail is designated as horizontal or vertical.The tail is described by planform area S, span b, chordc = S /b , and aspect ratio AR = b2 / S . The tail volume Vcan be referenced to rotor radius and disk area; to wingarea and chord for horizontal tails; or to wing area andspan for vertical tails. The geometry is specified in termsof S or V, and b or AR or c.

The horizontal tail can have a cant angle 0 (positive tilt toleft). The control variables are elevator and incidence.

The vertical tail can have a cant angle 0 (positive tilt toright). The control variables are rudder and incidence.

FUEL TANK

The fuel tank capacity Wfuel-cap (maximum usable fuelweight) is determined from designated sizing missions.The maximum mission fuel required, Wfuel-miss (excludingreserves and any fuel in auxiliary tanks), gives

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

where f fuel-cap z 1 is an input factor. Alternatively, the fueltank capacity Wfuel-cap can be input.

Auxiliary fuel tanks are defined in one or more sizes. Thecapacity of each auxiliary fuel tank, Waux-cap, is an inputparameter. The number of auxiliary fuel tanks on theaircraft, N auxtank for each size, can be specified for theflight condition or mission segment. Alternatively (if themission is not used to size the fuel tank), the number ofauxiliary fuel tanks at the start of the mission can bedetermined from the mission fuel. The weight and drag ofN auxtank tanks are included in the performance calculation.

The weight of the auxiliary fuel tanks is an input fractionof the tank capacity. The drag area for each auxiliary tankis specified, (D / q) auxtank .

PROPULSION

The propulsion group is a set of components and enginegroups connected by a drive system. The engine modeldescribes a particular engine, which is used in one or moreengine groups. The components (rotors) define the powerrequired. The engine groups define the power available.Figure 12 illustrates the power flow.

Figure 12. Power flow.

Drive System

The drive system defines gear ratios for all thecomponents that it connects. The gear ratio is the ratio ofthe component rotational speed to that of the primaryrotor. There is one primary rotor per propulsion group (forwhich the reference tip speed is specified); othercomponents are dependent (for which a gear ratio isspecified). There can be more than one drive system state,in order to model a multiple-speed or variable-speedtransmission. Each drive system state corresponds to a setof gear ratios.

For the primary rotor, a reference tip speed Vtip-ref isdefined for each drive system state. By convention, the

19

“hover tip speed” refers to the reference tip speed for drivestate #1. If the sizing task changes the hover tip speed,then the ratios of the reference tip speeds at differentengine states are kept constant.

For dependent rotors, either the gear ratio is specified (foreach drive system state), or a tip speed is specified and thegear ratio calculated ( r = Q dep /Q prim , Q = Vtip-ref /R).

For the engine group, either the gear ratio is specified (foreach drive system state), or the gear ratio calculated fromthe specification engine turbine speed^ spec = (2 ;r /60)N spec and the reference tip speed of theprimary rotor ( r = Q spec / Q prim , Qprim = Vtip -ref /R). Thelatter option means the specification engine turbine speedN spec corresponds to Vtip-ref for all drive system states.

The flight state specifies the tip speed of the primary rotorand the drive system state, for each propulsion group. Thedrive system state defines the gear ratio for dependentrotors and the engine groups.

Power Required

The component power required Pcomp is evaluated for aspecified flight condition, as the sum of the powerrequired by all the components of the propulsion group.The total power required for the propulsion group isobtained by adding the transmission losses and accessorypower: PreqPG = Pcomp + Pxmsn + Pacc. The power requiredfor the propulsion group must be distributed to the enginegroups. With only one engine group, PreqEG = PreqPG.

ENGINE GROUP

The engine group consists of one or more engines of aspecific type. For each engine type an engine model isdefined.

The engine size is described by the power Peng , which isthe sea-level static power available per engine at aspecified takeoff rating. The number of engines Neng isspecified for each engine group.

Power Available

Given the flight condition and engine rating, the poweravailable Pa is calculated (from the specific power SPa

and mass flow m a ). The flight condition informationincludes the altitude, temperature, flight speed, andprimary rotor speed; a power fraction fP ; and the states ofthe engine, drive system, and IR suppressor.

In the engine model, installation losses Ploss are subtractedfrom Pa ( Pav = Pa — Ploss), and then the mechanical limitapplied: Pav = min(Pav ,PmechR ).

The engine model gives the performance of a singleengine. The power available of the engine group isobtained by multiplying the single engine power by thenumber of engines operational (total number of enginesless inoperable engines): PavEG = (Neng — N inop )Pav . Thepropulsion group power available is obtained from thesum over the engine groups: PavPG = 1 fP PavEG , includinga specified power fraction fP .

The drive system rating at the flight condition is

(Q prim / Q ref )PDS limit. Optionally this limit is applied tothe propulsion group power: PavPG = min(PavPG , PDS limit) .

Similarly the engine shaft limit at the flight condition isoptionally applied to the engine group power.

Performance at Power Required

The engine performance (mass flow, fuel flow, and grossjet thrust) is calculated for a specified power required Pq ,

flight condition, and engine rating. The flight conditionincludes the altitude, temperature, flight speed, andprimary rotor speed; and engine, drive system, and IRsuppressor states. The engine turbine speed is

Neng = rengQprim.

The engine model deals with a single engine. The powerrequired of a single engine is obtained by dividing theengine 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 performance of the engine group is obtained bymultiplying the single engine characteristics by thenumber of engines operational (total number of enginesless inoperable engines):

m reqEG = (Neng — Ninop) m

req

w reqEG = (Neng — N inop )w

reqKffd

FNEG = (Neng — Ninop )FN

DauxEG = (Neng — N inop )Daux

The fuel flow has also been multiplied by a factor Kffd

accounting for deterioration of the engine efficiency.

Installation

The difference between installed and uninstalled power isthe inlet and exhaust losses Ploss : Pav = Pa — Ploss and

Pq = Preq + Ploss. The inlet ram recovery efficiency 27d isincluded in the engine model calculations. The inlet andexhaust losses are modeled as fractions of power available

20

or power required. Installation effects on the jet thrust areincluded in the engine model.

The momentum drag of the auxiliary air flow is a functionof the mass flow m

aux = f aux m req :

Daux = (1— rlaux )m aux V = (1 — rlaux )f aux m req V

where i7aux is the ram recovery efficiency.

Control and Loads

The engine orientation is specified by selecting a nominaldirection in body axes (usually thrust forward), thenapplying a yaw angle, then an incidence or tilt angle. Theyaw and incidence angles can be connected to the aircraftcontrols. Hence the incidence and yaw angles can be fixedorientation or can be control variables.

The engine group produces a jet thrust FN , acting in thedirection of the engine; a momentum drag Daux, acting inthe wind direction; and a nacelle drag Dnac, acting in thewind direction.

REFERRED PARAMETER TURBOSHAFTENGINE MODEL (RPTEM)

Aircraft gas turbine engine performance capabilities areformally specified by computer programs known as enginedecks, which are created by engine manufacturers in anindustry-standard format. Engine decks are typically basedon thermodynamic cycle analysis using real enginecomponent performance maps. The most importantperformance maps for turboshaft engines are compressor,gas generator turbine, and power turbine. Thesecomponent performance maps are critical to obtainingrealistic off-design engine performance. Design andanalysis codes calculate aircraft performance for a verywide range of operating conditions. This means thatengine performance must be realistic even far from theengine design point. A simple thermodynamic cycleanalysis that assumes design point component efficiencieseverywhere is not realistic for such an application. Ratherthan developing models for component performance, amodel for the total engine performance is used. The engineis not being designed.

The Referred Parameter Turboshaft Engine Model(RPTEM) is based on curve-fits of engine performancedata for existing or projected engines, over a range ofoperating conditions. The curve-fits are typically obtainedby exercising an engine deck (a computer program). Theuse of referred parameters tends to collapse the data, andprovides a basis for scaling the engine. The operatingcondition is described by pressure altitude, ambient air

temperature, flight Mach number, power turbine speed,exhaust nozzle area, and either engine rating or enginepower required. These curve-fits, typically based on realengines, are scaled to the required size and adjusted to theappropriate technology level to represent a notionalengine. Engine size is represented by mass flow. Enginetechnology is represented by specific power available andspecific fuel consumption at maximum continuous power(MCP), sea level/standard day (SLS), static (zero airspeed)conditions. Engine installation effects (inlet and exhaustlosses) are also modeled.

The use of referred parameters to curve-fit engineperformance data was suggested by David Woodley fromBoeing during the JVX program (1983). The RPTEM wasdeveloped and documented by Michael P. Scully andHenry Lee of ASRAO, U.S. Army AeroflightdynamicsDirectorate (AFDD), with a subsequent implementationwritten by Sam Ferguson (1995).

Operating Environment

The operating condition and atmosphere give the standardconditions (temperature and pressure) for a specifiedpressure altitude; the sea-level standard conditions(temperature T0 and pressure p 0 ); and the operatingtemperature T and pressure p = pstd . Here thetemperatures are in deg R or deg K. The enginecharacteristics depend on the temperature ratio 0 = T /T0

and pressure ratio 6 = p / p 0 . The flight Mach number Mis obtained from the aircraft speed.

The inlet ram air temperature ratio and pressure ratio areobtained from M and the inlet ram recovery efficiencyi7

d : ^M = (1 + 0.2M 2 ) and M =(1 + i7d 0.2M 2 ) 3.5, usingthe ratio of specific heats y = 1.4 .

Engine Ratings

The power available from a turboshaft engine depends onthe engine rating. Each engine rating has specificoperating limitations, most importantly an operating timelimit intended to avoid damage to the engine. Typicalengine ratings are MCP (maximum continuous power, notime limit), IRP (intermediate rated power, 30 min), MRP(maximum rated power, 10 min), and CRP (contingencyrated power).

Performance Characteristics

The engine performance is described by the poweravailable Pa , at each engine rating and the specificationengine turbine speed N spec; the mass flow m and fuel floww required to produce power required Pq at engine

21

turbine speed N; and the gross jet thrust Fg at a givenpower 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 staticconditions (subscript 0), and MCP (subscript C).

Referred or corrected engine parameters are used in themodel: power P /(8& ) , mass flow m /(8 / &) , specificpower (P / m ) / 0 , fuel flow w /(8&), thrust F /8, andturbine speed N / 0 .

For each rating R, the performance is characterized by thefollowing quantities for sea-level-standard staticconditions: power P0R , specific power SP0R , andmechanical power limit Pmech R . The mass flow is thenm

0R = P0R / SP0R . The gross jet thrust Fg 0C is given atMCP. These characteristics are at the specification turbinespeed N spec.

The installed power required Preq and power availablePav > Preq are measured at the engine output shaft. Inaddition to shaft power, the engine exhaust produces a netjet thrust FN , from mass flow that goes through the enginecore. The fuel flow and mass flow are the total required toproduce the shaft power and jet thrust. The forcesproduced by mass flow that does not go through theengine core (such IR suppressor or cooling air) are treatedas momentum drag Daux. The relationship between netand gross jet thrust is Fn = Fg — m

req V = m req (Vj — V) ,

where Vj is the engine jet exhaust velocity.

The uninstalled power required is Pq , the power availablePa , the gross jet thrust Fg , and net jet thrust Fn . Theengine model calculates Pa as a function of flightcondition and engine rating. Installation losses Ploss aresubtracted from Pa ( Pav = Pa — Ploss), and then themechanical limit applied: Pav = min(Pav ,PmechR ).

The engine performance (mass flow, fuel flow, and grossjet 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 toPreq ( Pq = Preq + Ploss).

Power Turbine Speed

The shaft power available is a function of the gas poweravailable PG and the power turbine efficiency 17t :Pa = 17tPG . Generally the power turbine speed N has asignificant effect on 17t , but almost no effect on PG . Themodel used for the efficiency variation is

r/t s 1— | (N / Nopt) —1 |X N

where Nopt is the speed for peak efficiency. Alternatively,the model parameters are defined at a set of engine speedratios N /N spec. Then the engine performance quantities atthe required engine speed N are obtained by linearinterpolation.

Scaling

The parameters of the engine model can be defined for aspecific engine, but as part of the aircraft sizing task theparameters must be scaled, in order to define an engine fora specified power. In addition, advanced technology mustbe represented in the model. Scaling and advancedtechnology are handled in terms of specific power andspecific fuel consumption (at SLS static conditions, MCP,and N spec).

The engine size is specified as takeoff power Pto = Peng :

power at rating R, for SLS static conditions andspecification turbine speed N spec. Hence the MCP isP0C = Pto / rp 0R , and the power at all other ratings follows.

The engine technology parameters SP0C and sfc0C areassumed to vary linearly with mass flow m

0C up to a limitm

lim, and constant thereafter at SPlim and sfc lim . Usuallythe effect of size is that specific power increases andspecific fuel consumption decreases with mass flow. Thespecific thrust available at MCP is assumed to be constant,and the specification power turbine speed decreases withthe square-root of the mass flow.

WEIGHTS

The design gross weight WD is a principal parameterdefining the aircraft, usually determined by the sizing taskfor the design conditions and missions. The aircraft weightstatement defines the empty weight, fixed useful load, andoperating weight for the design configuration. Thedefinition of the weight terms is as follows.

operating weight: WO = WE + WFUL

useful load: WUL = WFUL + Wpay + Wfuel

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

where WE is the weight empty; WFUL the fixed usefulload; Wpay the payload weight; and Wfuel the usable fuelweight. The weight empty consists of structure, propulsiongroup, systems and equipment, vibration, and contingencyweights. If the weight empty is input, then the contingencyweight is adjusted so WE equals the required value. If thedesign gross weight is input, then the payload or fuelweight must be fallout.

22

Table 2. Weight statement (* indicates extension ofRP8A).

WEIGHT EMPTYSTRUCTURE

useful load weight and operating weight. The grossweight, payload weight, and usable fuel weight (instandard and auxiliary tanks) completes the weightinformation for the flight state.

wing grouprotor groupempennage groupfuselage groupalighting gear groupengine section or nacelle groupair induction group

PROPULSION GROUPengine systempropeller/fan installationfuel systemdrive system

SYSTEMS AND EQUIPMENTflight controls groupauxiliary power groupinstruments grouphydraulic grouppneumatic groupelectrical groupavionics group (mission equipment)armament groupfurnishings & equipment groupenvironmental control groupanti-icing groupload & handling group

VIBRATION (*)CONTINGENCY

FIXED USEFUL LOADcrewfluids (oil, unusable fuel) (*)auxiliary fuel tanksother fixed useful load (*)kits (*)

PAYLOADUSABLE FUEL

standard tanks (*)auxiliary tanks (*)

The gross weight WG is specified for each flight conditionand mission, perhaps in terms of the design gross weightWD . For a each flight state, the fixed useful load may bedifferent than the design configuration, because of changesin auxiliary fuel tank weight or kit weights or incrementsin crew or furnishings weights. Thus the fixed useful loadweight is calculated for the flight state; and from it the

Weight Groups

Aircraft weight information is stored in a data structurethat follows SAWE RP8A Group Weight Statementformat (ref. 18), as outlined in table 2. The asterisksdesignate extensions of RP8A. There are 2 or 3 additionallevels in the data structure for some weight groups, basedon the weight breakdown for parametric estimation.

For each weight group, fixed (input) weights can bespecified or weight increments added to the results of theparametric weight model. The parametric weight modelincludes technology factors. Weights of individualelements in a group can be fixed by setting thecorresponding technology factor to zero.

The vibration control weight can be input, or specified as afraction of weight empty: Wvib = fvibWE . The contingencyweight can be input, or specified as a fraction of weightempty: Wcont = fcontWE . However, if the weight empty isinput, then the contingency weight is adjusted so WE

equals the required value.

AFDD Weight Models

For scaled weights of all components, the rotorcraftweight models developed by the U.S. ArmyAeroflightdynamics Directorate (AFDD) are implemented.The weights are estimated from parametric equationsbased on the weights of existing turbine-poweredhelicopters and tiltrotors (and some fixed wing aircraftcomponent weights). For some weight groups two modelsare available, designated AFDDnn. Table 3 summarizesthe statistics of the parametric weight estimationequations.

Figure 13 shows the error of the calculated weight for thesum of all parametric weight, accounting on average for42% of the empty weight. This sum is composed of thestructural group (based on the AFDD00 equation for rotorblade and hub weights, and the AFDD84 equation forbody weight), the propulsion group (based on theAFDD00 equation for drive system weight), and the flightcontrols group. Based on 42 aircraft, the average error ofthe sum of all parametric weight is 5.3%. Thecorresponding average error is 6.1% for the structuralgroup (8.6% for the rotor group alone), 10.9% for thepropulsion group, and 8.7% for the flight controls group.

23

Table 3. Statistics of parametric weight equations.

group number ofaircraft

averageerror %

wing 25 3.4rotor blade AFDD82 37 7.7rotor hub AFDD82 37 10.2rotor blade AFDD00 51 7.9rotor hub AFDD00 51 9.2horizontal tail 13 22.4vertical tail 12 23.3tail rotor 19 16.7fuselage AFDD82 30 8.7fuselage AFDD84 35 6.5alighting gear 28 8.4engine support 12 11.0engine cowling 12 17.9air induction 12 11.0accessory 16 11.5fuel tank 15 4.6gear box +rotor shaft

AFDD83 30 7.7

gear box +rotor shaft

AFDD00 52 8.6

drive shaft 28 16.0rotor brake 23 25.1rotary wingflight controls

non-boosted 20 10.4

rotary wingflight controls

boostmechanisms

21 6.5

rotary wingflight controls

boosted 20 9.7

• all parametric weight0 structure groupv propulsion group

25. v0 flight control group

v v_

V

"low

so • o $ o 1, -

0 0. fVo

V 0 -.-25.

0. 10000. 20000. 30000. 40000.

weight

Figure 13. Accuracy of sum of all parametric weight.

NDARC SOFTWARE

The NDARC program is entirely new software, built on anarchitecture that enables routine extensions andmodifications. The software has been implemented withlow-fidelity models, typical of the conceptual designenvironment. Incorporation of higher-fidelity models willbe possible, as the architecture of the code accommodatesconfiguration flexibility and a hierarchy of models.

The program is written in Fortran 95, using a special-purpose software tool to manage the data structures,construct the input manual, and automatically generatesome input and output subroutines. The program has beencompiled on several platforms and operating systems.

On typical computers, NDARC execution times rangefrom seconds for a job with just a few analysis tasks, tominutes for a job that sizes an aircraft based on multipleflight conditions and missions.

Input is in namelist-based text format. The program outputincludes text files formatted for printing and forspreadsheets, and special files to support functions such aspreparing layout drawings. Java graphical user interfacesare being developed by the user community to facilitatedealing with the input and output.

The program is supported by complete and thoroughdocumentation, including the theory manual (Ref. 6), inputmanual, and data structures manual. A NASA-hosted Wikihas been established to support user communication.

Distribution of the NDARC program is controlled by theSoftware Release Authority at NASA Ames ResearchCenter. Source code and documentation are available tousers, subject to a Software Usage Agreement.

CONCLUDING REMARKS

The theoretical basis and architecture of the conceptualdesign tool NDARC (NASA Design and Analysis ofRotorcraft) has been described. The principal tasks ofNDARC are to design a rotorcraft to satisfy specifieddesign conditions and missions, and then analyze theperformance of the aircraft for a set of off-design missionsand point operating conditions. NDARC provides acapability to model general rotorcraft configurations, andestimate the performance and attributes of advanced rotorconcepts.

24

REFERENCES1)Johnson, W.; Yamauchi, G.K.; and Watts, M.E. “NASAHeavy Lift Rotorcraft Systems Investigation.” NASA TP2005-213467, December 2005.2) Yeo, H., and Johnson, W. “Aeromechanics Analysis ofa Heavy Lift Slowed-Rotor Compound Helicopter.”Journal of Aircraft, Vol. 44, No. 2 (March-April 2007).

3) Yeo, H.; Sinsay, J.D.; and Acree, C.W., Jr. “BladeLoading Criteria for Heavy Lift Tiltrotor Design.”American Helicopter Society Southwest Region TechnicalSpecialists' Meeting on Next Generation Vertical LiftTechnologies, Dallas, TX, October 2008.4) Johnson, W. “Influence of Lift Offset on RotorcraftPerformance.” NASA TP 2009-215404, November 2009.

5) Acree, C.W., Jr.; Yeo, H.; and Sinsay, J.D.“Performance Optimization of the NASA Large CivilTiltrotor.” Royal Aeronautical Society InternationalPowered Lift Conference, London, UK, July 2008.6) Johnson, W. “NDARC, NASA Design and Analysis ofRotorcraft.” NASA TP 2009-215402, December 2009.7) Johnson, W. “NDARC — NASA Design and Analysisof Rotorcraft. Validation and Demonstration.” AmericanHelicopter Society Aeromechanics Specialists’Conference, San Francisco, CA, January 20-22, 2010.8) International Organization for Standardization.“Standard Atmosphere.” ISO 2533-1975(E), May 1975.

9) Harris, F.D., and Scully, M.P. “Rotorcraft Cost TooMuch.” Journal of the American Helicopter Society, Vol.43, No. 1, January 1998.10) Harris, F.D. “An Economic Model of U.S. AirlineOperating Expenses.” NASA CR 2005-213476, December2005.11) Coy, J.J. “Cost Analysis for Large Civil TransportRotorcraft.” American Helicopter Society Vertical LiftAircraft Design Conference, San Francisco, CA, January2006.12) Coleman, R.P.; Feingold, A.M.; and Stempin, C.W.“Evaluation of the Induced-Velocity Field of an IdealizedHelicopter Rotor.” NACA ARR L5E10, June 1945.13) Mangler, K.W., and Squire, H.B. “The InducedVelocity Field of a Rotor.” ARC R & M 2642, May 1950.14)Drees, J.M. “A Theory of Airflow Through Rotors andIts Application to Some Helicopter Problems.” Journal ofthe Helicopter Association of Great Britain, Volume 3,Number 2, July-September 1949.15) White, T., and Blake, B.B. “Improved Method ofPredicting Helicopter Control Response and Gust

Sensitivity.” Annual National Forum of the AmericanHelicopter Society, May 1979.16) Gessow, A., and Crim, A.D. “A Theoretical Estimateof the Effects of Compressibility on the Performance of aHelicopter Rotor in Various Flight Conditions.” NACATN 3798, October 1956.17)Keys, C.N., and Rosenstein, H.J. “Summary of RotorHub Drag Data.” NASA CR 152080, March 1978.18) “Weight and Balance Data Reporting Forms forAircraft (including Rotorcraft), Revision A.” Society ofAllied Weight Engineers, Recommended Practice Number8, June 1997.

NOMENCLATURE

AcronymsAFDD U.S. Army Aeroflightdynamics DirectorateIRP intermediate rated powerISA International Standard AtmosphereMCP maximum continuous powerMRP maximum rated powerRPTEM referred parameter turboshaft engine model

WeightsWD design gross weightWE empty weight

WMTO maximum takeoff weightWSD structural design gross weightWG gross weight, WE + WUL = WO + Wpay + Wfu el

WO operating weight, WE + WFUL

WUL useful load, WFUL + Wpay + Wfuel

Wpay payload

Wfuel fuel weight

WFUL fixed useful load

Wburn mission fuel burn

Wvib vibration control weight

Wcont contingency weight

Fuel Tanks

Wfuel–cap fuel capacity, maximum usable fuel weight

Nauxtank number of auxiliary fuel tanks

Waux-cap auxiliary fuel tank capacity

Power

PreqPG power required, propulsion group;

Pcomp + Pxmsn + Pacc

PreqEG power required, engine group

PavPG power available, propulsion group;min(l: fPPavEG, (^prim /^ ref )PDS limit)

PavEG power available, engine group;(N eng – Ninop)Pav

25

Pcomp component power required D /q drag area, SCD ( S = reference area)

Pxmsn transmission lossesPacc accessory power Aircraft

Ninop number of inoperative engines, engine group DL disk loading, WD / Aref

PDS limit drive system torque limit (specified as power Aref reference rotor area, E fA A; typically

limit at reference rotor speed) projected area of lifting rotors

PES limit engine shaft rating WL wing loading, WD / Sref

Sref reference wing area, E S; sum area all wingsEngine cAC aircraft controlPeng sea level static power available per engine at T control matrix

specified takeoff rating c component control, c = STcAC + c0

N eng number of engines in engine group ^tilt tilt control variablePav power available, installed;

min(P P Pa — loss, mechR )Rotor

Pa power available, uninstalled W/A disk loading, W = fWWD

Preq power required, installed; Pq — PlossCW/ a design blade loading, W / AVtip

2 cr ( Vtip =

Pq power required, uninstalled hover tip speed)

Ploss installation losses R blade radius

Pm echR mechanical power limit A disk area

SP specific power, P / m (conventional units) a solidity (ratio blade area to disk area)

sfc specific fuel consumption, w /P Tdesign design thrust of antitorque or auxiliary-thrust

(conventional units) rotor

m mass flow (conventional units) r direction of rotation ( r = 1 for counter-

w fuel flow (conventional units) clockwise, r = — 1 for clockwise)

FN net jet thrust μ advance ratio

Daux momentum drag A inflow ratio

N specification turbine speed Mat advancing tip Mach numberV blade flap frequency (per-rev)

Tip Speed and Rotation y blade Lock numberVtip-ref reference tip speed, propulsion group CT/ CF thrust coefficient divided by solidity,

primary rotor; each drive state T /pA(QR) 26r gear ratio; Q dep /Q prim for rotor, 16c , Ps

longitudinal, lateral flapping (tip-path planeQ spec /Q prim for engine tilt relative shaft)

Q prim primary rotor rotational speed, Vtip-ref /R 80.75 blade collective pitch angle (at 75% radius)

Qdep dependent rotor rotational speed ^c, s lateral, longitudinal blade pitch angleQ spec specification engine turbine speed H, Y , T drag, side, thrust force on hub (shaft axes)N spec specification engine turbine speed (rpm) Mx , My roll, pitch moment on hub

Q shaft torqueAxis Systems Pi , Po, Pp induced, profile, parasite powerI inertial x induced power factor, Pi = PidealF aircraft cd mean profile power mean drag coefficient,

Aerodynamics and Loads CPo = (a /8)cd meanFP

v velocity relative air (including interference) Wingq dynamic pressure, 12 v 2

W /S wing loading, W = fWWD

a angle-of-attack S area16 sideslip angle b spanF force c chord, S/bM moment AR aspect ratio, b2 /SD, Y, L drag, side, lift forces (aerodynamic axes)CD , CY , CL drag, side, lift force coefficients

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