Propeller Aeroelastic Modelling for Industry - Postgraduate Review

Harry Smith

September 1, 2011

Abstract

This document is presented to meet the criteria as laid out in the PGR guidelines by theSchool of Engineering at the University of Glasgow. This project is a CASE studentship for

Dowty Propellers (DP) and the Aircraft Research Association (ARA) and NDAs have been signedby the student and both industrial and academic supervisors. Consequently, a brief outline of theproject shall be provided but details will not be elaborated upon due to commercial sensitivity.

Some of this document is an extract from the provisional literature and scheme of work surveyprovided to DP and ARA prior to industrial placements in July 2011, with key detail removed.

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Contents

1 Summary 3

2 The Project 4

3 Propeller Aerodynamics 53.1 History of Propeller Aerodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

4 Unsteady Rotary Aerodynamics 7

5 Air Mass Dynamics/Non-Uniform Inflow Historical Perspective 9

6 Industrial Placements 12

7 Proposed Work 13

Bibliography 13

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1 Summary

The exploration of the various relevant literature is fully detailed in this report. A brief summary ofthe key findings is included here for clarity and brevity.

Propeller aerodynamic theory for design and analysis is found to be geared towards performancecalculations and little else - no major theoretical work has been completed for propeller calculationssince that of Theodorsen. Vortex theory has been used to model, to some degree, the propeller wake- the ‘ideal’ propeller can be designed by extension of wake theory to predict minimum-loss propellerloading. No major work has been done to calculate unsteady propeller loading or off-design cases - thisis the area upon which this project aims to expand the analytical tools available by taking advantageof developments in the rotorcraft field for unsteady aerodynamic analysis.

Unsteady rotary aerodynamics have been to date largely unexplored for propellers. This is per-haps due to the mindset that if a propeller works then detailed analysis of vibratory loads, secondaryflows etcare unnecessary as a propeller provides propulsion and not lift. Conversely, a helicopter mainrotor provides propulsion and lift and its continued safe performance is more mission-critical. Accord-ingly, the bulk of unsteady aerodynamic theory for this project has been taken from rotorcraft-focussedliterature.

Further work has been explored in unsteady aerodynamics and structural-dynamicmodels, but thedetail will be left out.

The scheme of work over the next eighteen months has been laid out in a report to the sponsorsand the timescales of project components are being managed in a suitable manner, with my progressbeing regularly reviewed both at DP and UoG. Following from the placements at ARA and DP, thescope of the project has been elucidated and this has enabled provisional planning of my thesis.

Essentially, propeller modelling to date has not changed much in the last century. Developmentshave been made in computing power and in theoretical work used in rotorcraft analysis, but littlehas been utilised for propellers - the latest theoretical work is nearly sixty years old. This projectwill help work towards a thesis that is likely to be entitled “Mathematical modelling of the propellerfor industry”, or similar and involve a ground-up approach to the propeller problem with justifica-tion for any assumptions or simplifications that can be made to assist with mathematical modelling.Preparatory research on techniques that may be used has been undertaken, and it is clear that thereare untested techniques for modelling that may be utilised, but the end form of such analysis is asyet unclear. What is clear, however, is that the end result of the work for the industrial sponsor willenable publication of literature pertinent to performing similar analyses in the general case.

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2 The Project

The project specifications will not be provided here - suffice to say that the work requires formulatinga mathematical model of a propeller suitable for steady freestream calculations as a design tool, whichplaces constraints on the mathematical techniques available.

No one single model will be capable of such analysis alone. Instead, the final model will be formedfrom separate models of propeller steady and unsteady aerodynamics and propeller structural dy-namics/aeroelasticity. As shall be discussed further, the use of a dynamic inflow model will likely beamenable to this task. Accordingly, Figure 2.1, below, is a re-working of similar diagrams presentedin rotorcraft dynamic inflow literature, adapted for the propeller problem. By looking at the differentareas required, the work may be split down into subtasks, research areas and subroutines within thefinal model.

PropellerAerodynamicEnvironment

Blade AerodynamicModel

Induced FlowMode/Unsteady

Aerodynamic Model

Structural DynamicModel

a/c FlightEnvironment &

Prop. Conditions

Vb(s)

αb(s)

CT , CM , CN

or τmcn , τms

n

dL, dD

BE Forces

L(s), D(s)

Inertial Loads

λ(s, ψ) λ(s, ψ)

h(s, ψ)

h(s, ψ)

θ(s, ψ)

θ(s, ψ)

θ,Ω, V∞

αac, βac

Figure 2.1: Basic Model Structure Required

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3 Propeller Aerodynamics

Before developing an unsteady aerodynamic model, one needs to have an appreciation of the steadyaerodynamics in play on a propeller. Propellers have been in use since the birth of powered aviation -the Wright Brothers, although experimenters and not theorists, took the step of changing to a pair ofthin, highly twisted blades turning at high speed as opposed to the cumbersome, windmill-like bladesof other designers of the time. With little power to play with from the engines of the day, this step waskey in securing their role as the founders of heaver-than-air flight. Although, as stated above, they wereexperimenters and not theorists, there is evidence that they may have been the first to combine Blade-Element and Momentum theory to determine the AoA distribution along the blade span (Wald, 2001).

Since this simple modification in blade shape, much work has been completed on propeller theoryand design to bring us to the present day where we see scimitar-type blades capable of producingexcellent propulsive efficiency, even at high-subsonic flight regimes, comparable to the performance ofa turbofan. The history of propeller theory and design for optimum efficiency is laid out chronologi-cally in great detail in a 2006 review of propeller aerodynamics (Wald, 2006). This forty-page workprovides an overview of not only the key developments, but goes into detail on the theory of propellersand shows the different means via which propeller calculations have been performed through differentflight regimes; propeller, vortex-ring and windmill states. This document will likely be of use for theduration of this project. It should be noted that some of the works and citations discussed in thefollowing section are taken directly from Wald (2006), as the works themselves are not written inEnglish and no translations readily available.

3.1 History of Propeller Aerodynamics

The earliest applicable theoretical work on propellers was performed by Froude and Rankine (1865),working on marine propellers. The basic theory of the propulsive fluid momentum equation was for-mulated, and thus created actuator-disc theory. Although this crude approximation allows designersto set performance and sizing requirements, it involves no real detail on actual propeller aerodynamics,instead proposing an infinitesimal imaginary disc through which there is a discontinuity in pressureand momentum, resulting in thrust.

In 1892 (and later in 1901), Drzewiecki modelled propeller blade elements moving through a fluidmedium on a helical path, but did not account for the propeller-induced velocity due to the shedwake. It was not until Prandtl’s lifting line theory with free and bound vorticity that the modern the-ory of propeller aerodynamics would be possible. Using this theory, the propeller may be consideredto be a lifting surface about which there is a circulation associated with bound vorticity and a vortexsheet shed continuously from the trailing edge - propeller wake theory gives a better appreciation ofthe actual physical flow field than the crude approximations of actuator-disc theory.

In 1919, Betz showed that the loading distribution for ‘lightly-loaded’1 propellers is such that theshed vorticity forms regular helicoidal sheets moving aft uniformly from the propeller. This work wasreprinted by Prandtl and Betz (1927) with an appendix showing a closed-form approximation to thisflow for small advance ratios.

Goldstein (1929) solved the problem of the potential field and circulation distribution based on ahelicoidal vortex system as outlined by Betz. This circulation distribution was presented in tabulatedform for two and four-bladed propellers as the ‘ideal’ distribution. Ideal design via Goldstein’s func-tion is thereby a design procedure to fit this ideal circulation distribution .

1A lightly loaded propeller is described as one where the induced velocities are small compared to the propellervelocity (Makinen, 2005) - this was never clearly defined by Betz in the original literature.

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Although the work by Betz, Prandtl and Goldstein marked the development of a better understand-ing of propeller aerodynamics via vortex theory, work continued on the refinement of Blade ElementMomentum Theory (BEMT), notably by Glauert. This development as in (Glauert, 1926b, 1943)is still applied in practical calculations today, despite the dependence on an “independence of bladeelements”, which was shown to be without physical justification2.

The long out-of-print ‘Theory of Propellers’ (Theodorsen, 1948) is essentially a collection of NACATNs 775-778 (Theodorsen, 1944a,b,c,d) - Theodorsen followed from Goldstein’s ideal circulation dis-tribution and studied the shed wake far downstream, rather than at the propeller itself and as suchwas able to negate the need for the lightly-loaded condition that had previously been invoked.

Tibery and Wrench Jr (1964), applied mathematicians, used modern computational and mathematicaltechniques to tabulate accurately functions related to the Goldstein function over a wider range ofpropeller parameters than had previously been attempted.

A design procedure developed by Larrabee (1979), capable of determining the geometry of mini-mum induced loss propellers based upon specific conditions (e.g., ; loading, λ, n etc.) has the benefitof convenience, quoted as “efficient...adaptable to pocket calculators” (Wald, 2006). However, thetheory behind the work is based on the original vortex assumptions by Prandtl and Betz (1927) anddoes not utilise the more rigorous and general concepts introduced by Theodosen.

The preceding section gives a comprehensive overview of the key developments in steady propelleraerodynamics. It should be noted that the most recent significant work dates to 1979, with the mostmajor theoretical developments being that of Theodorsen in 1948. Despite the main principles ofpropeller design being over sixty years old, they have enabled designers and blade manufacturers toproduce efficient and powerful propellers. Where there has been a lack of propeller-focussed research,and where there are gains to be made in terms of reduced fatigue etc., is in the propeller unsteadyaerodynamics. By analysing how the blade forces and moments change with respect to time-varyingor non-uniform inflow, correlation between 1P-loading and design features may be determined andblades designed/adapted for maximum life between service intervals.

Although there has been a lack of propeller-focussed unsteady aerodynamic theory, there is a wealthof information on rotorcraft-based investigation. This is down to a combination of the fact that a he-licopter rotor is generally subject to more unsteady flow (e.g.; more manoeuvring flight, edgewise flowand transition between propeller, vortex-ring and windmill-brake states) and the fact that a helicopterrotor provides the main lifting surface in addition to propulsion. Therefore unsteady aeroelastic phe-nomena such as flutter etcare clearly not only more likely without careful and safe design procedure,but also more catastrophic than they would be on a propeller (although one would still hope to avoidblade flutter on a propeller). It should be noted that work on nonuniform pressure distributions wascompleted by Glauert (1926a) during development of autogyro theory, but not applied to propelleraerodynamics - likely because of the reduced effect of inflow variation on the propeller problem, andthe aforementioned critical aspect of a rotor vs. a propeller as lifting surface vs. pure propulsion.

2Taken from Wald (2006)

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4 Unsteady Rotary Aerodynamics

As mentioned in the previous section, there is little information available on propeller unsteady theory,and the key sources found that deal with rotary unsteady aerodynamics have come from helicopter-focussed literature. An excellent starting point for rotorcraft unsteady aerodynamics is in “Principlesof Helicopter Dynamics” (Leishman, 2000). This book gives an overview of the different unsteadytheories and how they compare under different flight environments/analytical procedure. Before go-ing into detail on the model that is anticipated shall be most suitable for propeller analysis, thissection shall briefly cover the key developments in unsteady aerodynamics as laid out by Leishman(2000, Chapter 8 - Unsteady Aerodynamics), and why they are suited or unsuited to propeller analysis.

To explain the relevance of unsteady aerodynamics, Leishman presents a breakdown of the sources ofunsteady aerodynamic loading that may be found on helicopter rotor. Figure 4.2, below, is an adapta-tion of Leishman’s breakdown of the sources of unsteady aerodynamic loading, put into context hereon a propeller. The relative frequencies and amplitudes of different effects are given; the frequencydictates the level to which unsteady effects dominate the flow via reduced frequency k = ωb

V

Blade Motion

Pitch Flap

Torsion(high freq.,high amp.)

Control(low freq.,high amp.)

Flap*(low freq.,high amp.)

Bending(high freq.,low amp.)

(a) Unsteady Aerodynamics due to Blade Motion *(only applicable if including the proposed pin-jointedhinge)

Flowfield Structure

Periodic Aperiodic

Velocity(low freq.,low amp.)

Sweep(low freq.,low amp.)

Downwash(low freq.,low amp.)

Fuselageflowfield

(low amp.)

Discretevortices

(high amp.)

Wakedistortion(low amp.)

(b) Unsteady Aerodynamics due to Flowfield Structure

Figure 4.2: Unsteady Aerodynamic Sources on a Propeller; adapted from Leishman (2000, takenfrom Beddoes (1980))

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Before going into the detail of different unsteady theories, Leishman lays out the requisites ofany useful unsteady aerodynamic model, for use in practical rotary aerodynamics. The following arelargely common sense, but fundamental to choosing any aerodynamic model to be used as part of alarger calculation procedure and hence are included here.

1. The assumptions and limitations of any model need to be fully assessed, understood andjustified if invoked. e.g., incompressibility requires not only local M 1 but Mk 13.

2. The model must be written in a form that is easily coupled with structural dynamicmodel. e.g., the model may be in terms of ODEs at radial blade elements, or written in state-space form at the disc level.

3. If choosing an integral approach, i.e., a BE model, the concentric radial and azimuthal integrationloops place a strict limit on the computational cost of any unsteady model4.

Leishman discusses reduced frequency to be used as a measure of the degree to which unsteadyeffects dominate the flowfield, and the reduced time, s = 1

b

∫ t0 V dt, as a non-dimensional ordinate

representing distance that the blade travels through the flowfield in semi-chords during a given timeinterval, t. An preliminary analysis of the effect of unsteady flow in the propeller problem will bebeneficial before constructing the model, and by calculating k over the disc it is anticipated that abetter appreciation of when and where in the flowfield unsteady effects dominate. This shall assistwith preliminary qualitative analysis of initial results.

The book goes through the development of unsteady attached theory, quasi-steady thin-aerofoil the-ory, Theodorsen’s theory/function with returning-wake additions by Loewy and Jones and throughother frequency-domain theories. As shall be discussed below, such frequency domain based modelsare not foreseen to be used in the proposed model as it shall be preferable to have a time domainbased solution. However, it is important that they have been researched as they may prove useful ata later date. Indeed, as shall be discussed in section ??, the proposed extension of a Dynamic Inflowmodel to the propeller problem and validation thereof is not guaranteed and this work may have torely on a Theodorsen method or similar to model unsteady aerodynamics. This will be determinedat a decision gate after the theoretical work to extend dynamic inflow to the propeller case is complete.

Leishman (2000, 8.14.2: State-Space Solution) highlights the advantage of state-space form for anunsteady aerodynamic model that is to be used as part of a larger aeroelastic model - in such aform, the governing differential equations may be used in conjunction with the time domain basedstructural-dynamic model. Another fundamental benefit of using a system that is in state-space formis that it is possible to calculate the response of a propeller/rotor system at a given time (i.e., for aparticular state), rather than a model’s validity being limited to the stability boundary.

Efforts to formulate a state-space model of 2D unsteady aerofoil theory via Laplace transform methodsof the indicial response have been performed by many using approximations to Theodorsen and Wag-ner’s functions - however, the approximations essentially amount to curve-fitting and have no physicalbasis. For a problem with little previous theoretical work and without a wealth of experimental data,it is difficult to justify such a method and as such it will be preferable to use a more mathematicallyrigourous formulation.

3Incompressibility may not be assumed in a blade-level model as tip speeds will be high subsonic/transonic. Thevalidity of assuming the wake compressibility may be negated is discussed in Section ??

4As shall be discussed in Section ??, use of vectorisation in code construction negates the concentric loops for somecalculations, although a numerical integration scheme shall require them still.

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5 Air Mass Dynamics/Non-Uniform Inflow Historical Perspective

From the earliest rotorcraft design, the inflow over the the rotor was presumed to be uniform, enablingeasy calculations by momentum theory providing reasonable insight into hover performance, despitebeing a large simplification of the actual flow situation (Bramwell et al., 2000).

During forward flight (or, infact, any translational flight), there is a radial and azimuthal variation ininflow due to different factors:

• A one-dimensional variation in the axis of flight direction due to translational lift causing rotor-induced upwash at the leading edge, much like an aerofoil of equivalent size - this effect is much,much smaller on a propeller at an angle of attack than on a rotor blade.• A radial variation of local tangential velocity and therefore inflow angle along the blade axis.• A lateral (azumuthal) variation of inflow due to asymmetric summation of forward speed and

local rotational velocity on advancing and retreating blades at AoA 6= 0.

The autogyro is clearly not capable of axial flight, and as part of his work on autogyro theory, thevariation in inflow on the rotor was first accounted for by Glauert (Glauert, 1926b) by modelling thedisc as an elliptically-loaded circular wing with associated downwash distribution. He proposed amean and longitudinal variation in inflow of the form:

vi0 =T

2ρAV(5.1)

vi = vi0(1 +Kcx cosψ) (5.2)

Where the nondimensional radial ordinate is x = rR and Kc is a coefficient slightly larger than one,

chosen so that the equation 5.2 produces an upwash at the leading edge and a positive linear gradientof vi towards the trailing edge i.e., along the line (ψ = 0 or π).

Much work was done over the following years to refine the definition of and values for Kc, throughboth experimental and theoretical means. Most of the work for this shall not be included in thisdocument - for a full review of the development of Kc for autogyro and helicopter flight, the Ph.D.thesis by Murakami (2008) provides a good perspective. Arguably the first innovative addition toGlauert’s work was provided by Coleman and Feingold (1945) where they defined Kc to be a functionof χ, the wake skew angle.

Kinner (1937)5 introduced the use of elliptical co-ordinates to the problem and proposed the useof pressure distributions as a series of the associated Legendre functions - satisfying the Laplace equa-tion and providing a discontinuity in pressure (i.e., lift) across the disc. This step was key to gainingthe dynamic inflow models that are used today, but it should be noted that Kinner was not exploringunsteady induced flow, rather the distribution of the steady induced flow.

Mangler and Squire (1950) extended the work of Kinner by associating his lift distribution withthe induced velocity field - using Euler’s equations and a variant of actuator disc theory, they wereable to calculate the acceleration of the induced flow distribution so as to satisfy the requisite hubloading. This coupling, as Murakami (2008) notes, may be regarded as the theoretical forebearer ofmodern dynamic inflow theory - more recent dynamic inflow models couple pressure distribution toinflow, but the concept is the same of coupling the loads.

Joglekar and Loewy (1970) extended Mangler and Squire’s work to incorporate the effects of theshed wake geometry, but the technique is less sophisticated than in modern methods, as noted in

5German language paper, but theory described by Joglekar and Loewy (1970)

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(Murakami, 2008) - additionally various sources show that the models based on Mangler and Squire’swork, although innovative, fail to show good correlation with experimental data; (Chen, 1989).

Around the same time as Mangler and Squire’s work, Sissingh (1951) attempted to account for across-coupling in roll and pitch damping of a helicopter rotor, first reported by Amer (1950), by as-suming a nonuniform distribution of induced velocity based on the first harmonic variation of the liftcoefficient, or:

v = v0 + v1 cosψ (5.3)

His results for rotor damping cross-couping matched predictions and the results were well received,finding use in commercial simulation programmes (Gaonkar and Peters, 1988). However, the modelabove utilises a Fourier distribution based only on the azimuthal variation, neglecting the radial vari-ation. Additionally, Sissingh’s work presumes that the reaction of inflow to the change in rotor loadsis instantaneous, and does not account for the aerodynamic hysteresis due to dynamic flow effects andis termed the quasi-steady model. Nonetheless, the step of explaining the cross-coupling by linkingwith nonuniform inflow lays the foundations for dynamic inflow theory.

The relationship between rotor load and induced flow had been noted much earlier than the above anal-yses in Wheatley’s 1935 analysis of autogyro rotor blade vibration, concluding that Glauert and Lock’sprevious work on autogyro theory was “quantitatively usable except for the blade motion...[which] can-not be calculated rigourously without the accurate determination of the induced flow”. Wheatley’sreport proved the link between induced flow and rotor loads, but it was not until the advent of thehingeless rotor nearly thirty years later that inclusion of this effect gained momentum as a necessarystep in a rotor modelling. This is due to the ability of the hingeless rotor to apply moments directlyto the hub noted in by Ormiston and Peters (1972)“The nonuniform inflow is a direct result of thelarge hub moments developed by hingeless rotors...neither the induced inflow nor elastic blade bendingare of comparable importance for articulated rotor response”. This is a key point for the proposedpropeller analysis as current propeller design is essentially a hingeless design, rather than articulated6

and as such, the fundamental principles of work based on the correlation between rotor loads andinflow may be carried forward.

Carpenter and Fridovich (1953) explored time delay in control response of rotorcraft - observing alag in the build up of rotor thrust in response to a change of collective pitch whilst exploring the“jump take-off ” of heavily-loaded helicopters. For this, a rapid increase in collective is applied toa pre-rotated rotor, effecting a vertical take-off. The observed time delay or ‘lift hysteresis’ was at-tributed to Newton’s second law and the force required to effect a momentum change in the fluidsurrounding the disc. Following the theory of NACA TN 197 (Munk, 1924), they added an “appar-ent addition mass” to the 1D momentum equation of 63.7% of the mass of a sphere of fluid with thesame diameter as the rotor disc, and their method is now referred to as “unsteady momentum theory”7.

In their work on nonuniform rotor loads and the relationship with pitch and roll moments Curtissand Shupe (1971) developed a model for axial flight in a quasi-steady formulation. Rather than incor-porating dynamic flow effects using a rigorous physical formulation, they developed the reduced Locknumber, effectively the same as modelling a heavier blade. The two methodologies for modelling thelift hysteresis have been explored by Banerjee and Crews (1979) and found that the dynamic inflowmodel works better for low advance ratio regimes (Murakami, 2008).

6Aside from the proposed pin-jointed hinge proposed as conceptual design, the effect of which on 1p-loading is to bedetermined.

7In (Carpenter and Fridovich, 1953) they do not follow the method of TN 197 fully, rather they adapt a two-dimensonalmethod to match with experimental data. This is not documented in the paper, but the reliability of their method isnoted in other sources and by prominent researchers e.g., Peters and colleagues.

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The models developed up until this point had been proved valid only in axial flight, and showedpoor correlation with forward flight. Although our propeller case is closer in flowfield situation to theaxial case, the aircraft will be cruising at an angle of attack and it is this skewed inflow/wake thatwe want to determine the effects of. Work from the 1960’s developed techniques for experimentalmeasurement of the induced velocity distribution, and the results showed that unsteady momentumtheory of Carpenter and Fridovich matched the result poorly (Gaonkar and Peters, 1986). Dynamicinflow needed developing in order to cover forward flight conditions, and much work was done toamend the theory.

Methods for developing new methodologies to cover the forward flight regime included local momen-tum theory and a variety of methods combining blade-element and momentum theory, but few showedsuccess. Development of these methods is explained in (Murakami, 2008) and shall be disregarded here.

Ormiston and Peters (1972) developed a quasi-steady model in which nonuniform inflow distribu-tion was included in a hingeless rotor model. Their model was based on Kutta-Joukowski circulationtheory and developed in a matrix formulation relating induced flow and rotor thrust and moments.Peters (1974) extended this model using unsteady momentum theory to develop the basic formulationof dynamic inflow in its whole form. The model was limited to hovering flight (off-diagonal gain matrixelements are not included). Murakami (2008) also notes that the elements of the matrices “were notmathematically rigorously determined”.

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6 Industrial Placements

During July 2011, consecutive fortnight-long placements were undertaken at ARA and DP. The majorachievements were:

• Elucidation of the project scope• Data received for validation• Familiarity gained in existing codes in use (DP)• Methods used to formulate databanks explored (ARA)• Positive feedback of literature review and proposed model structure

No more needs to be stated other than that the industrial and didactic goals were met as laid out.Advisors at ARA and DP are happy with my progress and proposed scheme of work, and my supervisor,Dr. Gillies, is up-to-date with my work and has given appropriate feedback in the attached form.

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7 Proposed Work

A Gantt chart of the proposed work was submitted along with the initial review, outlining my pro-posed scheme of work for the next eighteen months. It will not be reproduced here.

The proposed scheme of work was acceptable to all advisors and has been kept to since being backat UoG. Detail on the content does not need to be included here, suffice to say that the initialpost-placement stages are theoretical work, moving on to mathematical derivation and concurrent for-mulation of model components that may be used interchangeably with existing codes, moving towardsvalidation and comparison of different techniques and finally the full model.

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References

Amer, K. . (1950). Theory of Helicopter Damping in Pitch or Roll and Comparison with FlightMeasurements. NACA.

Banerjee, D. and Crews, S. (1979). Parameter identification applied to analytic hingeless rotor mod-eling. Journal of the American . . . .

Beddoes, T. S. (1980). Unsteady Flows Associated With Helicopter Rotors. Agard Report 679.

Bramwell, A. R. S., Done, G., and Balmford, D. (2000). Helicopter Dynamics. Elsevier Ltd., secondedition edition.

Carpenter, P. and Fridovich, B. (1953). Effect of a rapid blade-pitch increase on the thrust andinduced-velocity response of a full-scale helicopter rotor.

Chen, R. T. N. (1989). A survey of nonuniform inflow models for rotorcraft flight dynamics and controlapplications. Technical report, Ames Research Center, Moffett Field, California, Ames ResearchCenter, Moffett Field, California.

Coleman, R. P. and Feingold, A. M. (1945). Evaluation of the Induced-Velocity Field of an Ideal-ized Helicoptor Rotor. Technical report, National Advisory Committee for Aeronautics, NationalAdvisory Committee for Aeronautics.

Curtiss, H. K. and Shupe, N. K. (1971). A Stability and Control Theory for Hingless Rotors. InAnnual Forum of the American Helicopter Society, Washington, D.C.

Gaonkar, G. H. and Peters, D. A. (1986). Effectiveness of Current Dynamic Inflow Models in Hoverand Forward Flight. Journal of the American Helicopter Society.

Gaonkar, G. H. and Peters, D. A. (1988). Review of Dynamic Inflow Modeling for Rotorcraft FlightDynamics. Vertica, 12(3):213–242.

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Goldstein, S. (1929). On the vortex theory of screw propellers. In Proceedings of the Royal Society ofLondon., pages 440–465.

Joglekar, M. and Loewy, R. (1970). An Actuator-Disc Analysis of Helicopter Wake Geometry and theCorresponding Blade Response. Technical report, US Army Air Mobility Research and DevelopmentLab (USAAVLABS), Fort Eustis No. 69-66, US Army Air Mobility Research and Development Lab(USAAVLABS), Fort Eustis No. 69-66.

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Makinen, S. (2005). Applying dynamic wake models to large swirl velocities for optimal propellers. PhDthesis, Washington University Sever Institute of Technology Department of Mechanical Engineering.

Mangler, K. W. and Squire, H. B. (1950). The induced velocity field of a rotor. Technical report.

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