HN

LECTURE NOTES ON HELICOPTER AEROMECHANICS

Pre-requisite: Aerodynamics, Flight Mechanics and Engineering Mechanics

Course objectives:       

Perpetuate, cultivate and advance the understanding of a still unusual and very capable aircraft: the helicopter Learning the first principles of helicopter flight Using the knowledge framework from the previous years to understand the multidisciplinary aspects of helicopter (aerodynamics, structural dynamics, performance, Aeroelasticity, and optimisation) To understand basic and advanced concepts related to aerodynamic loads, vehicle performance, basic rotor dynamics, and control of helicopters and tilt-rotor aircraft (i.e. VTOL aircraft). To develop the students' understanding of helicopter aerodynamics. To develop the students' understanding of momentum theory and blade element theory in estimating helicopter performance. To develop the students' understanding of helicopter performance problems and how to estimate the performance of an example helicopter.

Program outcome:         

Understand the characteristics of helicopter rotor flow fields for all phases of helicopter flight, hover, vertical flight, forward flight, autorotation, etc. Understand theoretical and empirical techniques used to analyse and predict the aerodynamic characteristics of helicopters in hover, vertical flight and forward flight. Understand helicopter main rotor and tail rotor aerodynamic design considerations including airfoil selection and rotor configuration trade-offs. Understand momentum theory uses and limitations and apply it to the estimation of rotor performance of a selected helicopter. Understand and apply rotor blade element theory to estimate rotor performance and trim conditions for a selected helicopter. Understand and apply combined momentum and blade element methods to analyse helicopter rotors. Understand and apply empirical corrections to both momentum and blade element techniques. Understand and apply component drag build-up techniques to the estimation of the total drag of a selected helicopter. Understand and apply performance estimation techniques to hover, vertical climb, autorotation, range and endurance.

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2

References: 1. J. Gordon Leishman, “Principles of Helicopter Aerodynamics”, Cambridge University Press, 2000 2. Prouty, R.W., “Helicopter Performance, Stability and Control”, R.E. Krieger Pub. Co. Florida, 1990 3. Seddon, J., “Basic Helicopter Aerodynamics”, B.S.P. Professional Books, 1990 4. Johnson, W., “Helicopter Theory”, Princeton University Press, New Jersey, 1980 5. Mil, M. et al, “Helicopters – Calculation and Design: Vol. I Aerodynamics”, “NASA TTF-494, 1967. 6. Mil, M. et al, “Helicopters – Calculation and Design: Vol. II Vibrations and Dynamics”, “NASA TTF-519, 1968. 7. Gessow, A. and Publication

Meyers, G.C., “Aerodynamics of the Helicopter”, Dover

8. Bramwell, A.R.S., “Helicopter Dynamics”, Edward Arnold Pub., London, 1976 9. Stepniewski, W.Z. and and Keys, C.N, “Rotary Wing Aerodynamics, Vol. I and II”, Dover Publications, 1984. 10. Padfield, G.D., “Helicopter Flight Dynamics: The Theory and Application of Flying Qualities and Simulation Modeling”, AIAA series, 1996

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Pitch Angle ( θ ): The blade pitch angle (θ) is the angle between the plane perpendicular to the rotor shaft and the chord line of a reference station on the blade. For a hovering helicopter, angle of incidence (i) is different from θ. As the rotor blade rotates, a downward velocity ( v i ) is

induced. The resultant velocity VR is a combination of this induced velocity vi  and the linear velocity (Ωr) in the plane of rotation at a distance r from the hub. The angle between induced velocity vi  and the linear velocity (Ωr) is defined as inflow angle φ and the angle of incidence (i) is reduced from θ by the inflow angle φ.

Azimuth Angle (  ): The helicopter rotor blade moves through 360 degree azimuth. The azimuth position is measured positively in the direction of rotation from its downstream position.

Collective change of Pitch Within limits of stall, lift coefficient increase with increase in . If the pitch of all the blades is increased simultaneously, the overall lift and hence thrust increases. Therefore, changing the thrust to values more than or less than weight will cause the helicopter to climb or descend. The means of achieving this change of pitch of all blades simultaneously is called “collective” pitch change.

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4

Cyclic change of pitch With cyclic pitch lever, the pilot can increase the blade pitch at one azimuth position (A) and decreases it at a diagonally opposite position (B). As a result all the blades coming to position A steadily have increasing pitch values those receding from A and going to B have steadily decreasing values. This causes increased angle of attack at position A and decreased angle of attack at position B. this cyclic variation of pitch along azimuth position is called “cyclic” pitch change.

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5

Rotor Hinges The development of the autogyro and, later, the helicopter owes much to the introduction of hinges about which the blades are free to move. The use of hinges was first suggested by Renard in 1904 but the first successful practical application of hinges was due to Juan de la Cierva in the early 1920s. There are three hinges in the so-called fully articulated rotor: i. Flapping hinge ii. Drag or lag hinge iii. Feathering hinge

Flapping hinge The flapping hinge solves the problem of rolling moment when the helicopter is in forward flight. In hover, pitch is maintained the same throughout the azimuth position. However, when the rotor moves forward horizontally at a velocity V, the advancing blade (at = 90 Rotorcraft Aeromechanics

6

degree) is at a velocity V + r and the retreating blade (at =270 degree) is at V-r. Thus, if the pitch is same, the advancing blade gives higher lift than the retreating blade. This production of unequal lifts on either side of the helicopter would result in undesirable rolling moment and excessive alternating air blades on the blade. One way of correcting this is by setting the pitch on the advancing side lower and the retreating side higher by use of some sort of lateral control.

Flapping Hinge with offset „a‟ As the blade advances and develops more lift, it begins to flap upward. This then introduces a downward vertical component of velocity in relation to the blade which reduces its angle of incidence and hence the lift of the advancing blade. As it retreats, the opposite is true, for a downward flapping of the blade produces an increased lift. The changes in speed in advancing and retreating blades are compensated by opposing changes in angle of incidence (and lift) and net rolling moment about flapping hinge becomes zero. •

Advancing Side (flapping up reduces angle of incidence)

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7

Retreating Side (flapping down increases angle of incidence) It is to be noted that this flapping motion is caused automatically by unequal velocities only (i.e. without any control force by pilot) and it is referred to as aerodynamic flapping.

Drag (Lag) Hinge The next important hinge is the drag hinge. In addition to the flapping hinge, a hinge is essential to cater for the lead-lag motion of the blade; this is the drag hinge. The blade is hinged about a vertical axis near the center of rotation so that is free to oscillate or “lead and lag” in the plane of rotation. This flexibility makes the net moment about drag hinge zero. Both the flapping and drag hinge (in a so-called fully articulated rotor).

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8

Feathering Hinge Pitch of the blades can be increased or decreased by the pilot simultaneously or differentially (collective and cyclic pitch change) by the use of feathering hinge.

Types of rotors Three fundamental types of rotors have been developed so far:

Rigid rotor In these rotors, the blades are connected rigidly to the shaft. Such rotors do not have either flapping or drag hinge. Usually, such rotors are two-bladed.

See-saw (or teetering) rotor Rotors in which blades are rigidly interconnected to a hub but the hub is free to tilt with respect to shaft. These rotors are two bladed. The blades are mounted as a single unit on a “see-saw” or “teetering” hinge. No drag hinges are fitted and therefore lead-lag motion is not permitted. However, bending moments my still be reduced by under-stinging the rotor. The principle of see-saw rotor is similar to that fully articulated rotor (having both flapping and drag hinges) except that blades are rigidly connected to each other. The “see-saw” hinge is like the flapping hinge located on the axis of rotation and because of rigid interconnection between two blades, when the advancing blade, flaps up, the opposite (retreating) blade flaps down.

Fully articulated rotor Rotors in which blades are attached to the hub by hinges, free to flap up and down also swing back and forth (lead and lag) in the plane of rotation. Such rotors may have two, four or more blades, such rotors usually have drag dampers which present excessive motion about the lag hinge.

General Expression to determine the location of Kth blade Azimuth of Kth blade is given by (

)

N- Number of blades

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9

Rotor Provides   

Lift (thrust) Horizontal Propulsive Force Forces needed for control

Hover: Helicopters Designed to be operationally efficient there. For forward flight the rotor is tilted forward 

Blades Flap



Vortices create vibration and noise



Higher forward speed: Transonic flow o Increased drag and noise (swept blades help) o Retreating side, High AOA, Dynamic stall, Vibration



Forward flight speeds are limited

High Speed Forward Flight Limitations •

As the forward speed increases, advancing side experiences shock effects, retreating side stalls. This limits thrust available.

•

Vibrations go up, because of the increased dynamic pressure, and increased harmonic content.

•

Shock Noise goes up.

•

Fuselage drag increases, and parasite power consumption goes up as V3

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10

Power plant Limitations •

Helicopters use turbo shaft engines.

•

Power available is the principal factor.

• An adequate power plant is important for carrying out the missions. Momentum Theory of Rotors (Actuator Disk Theory) •

Developed for marine propellers by Rankine (1865), Froude (1885).

•

Extended to include swirl in the slipstream by Betz (1920)

•

This theory can predict performance in hover, and climb.

•

First approximation, no details of shape

Applications: Propellers, Rotor and Ducted Fans

Assumptions: •

Rotor is modeled as an actuator disk (infinitely thin disk of area A which offers no resistance to air passing through it) which adds momentum and energy to the flow.

•

Flow is incompressible (compressibility corrections can be made).

•

Far Upstream and Far Downstream the pressure is freestream static pressure

•

Flow is steady, inviscid (no drag and no momentum diffusion), and irrotational.

•

Flow is purely one-dimensional

•

Thrust loading and Inflow velocity are uniform over the rotor disk (equivalent to assuming infinite number of blades).

•

There is no swirl in the wake (no rotational effect)

•

Low Disk Loading

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11

Consider an actuator disk (thin circular disk with radius of rotor radius and has infinite number of blades. It is permeable to the air flow and supports the pressure difference on the top and bottom surface of the disk) of area A and total thrust T. It is assumed that the loading is distributed uniformly over the disk. Let be the induced velocity at the rotor disk and w be the wake-induced velocity infinitely far downstream. A well-defined smooth slipstream is assumed with and w uniform over the slip-stream cross section. The rotational energy in the wake due to the rotor torque is neglected. Mass flow through the rotor ̇

(1)

By conservation of mass the mass flux is constant all along the wake. Mass conservation (

)

(2) (3)

√ √ Actual Value

0.78

Thrust from Momentum equation is given by, considering stations (0) – (3) (

Rotorcraft Aeromechanics

)

(4)

12

Applying Bernoulli‟s equation from station (0) – (1) (5) Bernoulli‟s equation cannot apply between 1-2 stations Below the disk, between stations (2) & (3), the application of Bernoulli‟s equation gives (6) (

)

Because the jump in pressure is assumed to be uniform across the disk, this pressure jump must be equal to the disk loading, T/A, that is From equation (5) & (6) (

(7)

)

Therefore rotor disk loading is equal to dynamic slipstream pressure. . /

(8)

( )

. /

(9)

( )

Therefore, the static pressure is reduced by ( ⁄ )( ⁄ ) above the rotor disk and increased by ( ⁄ )( ⁄ ) below the disk. From equation (4) & (7) (

(10)

)

(11) From equation (4) & (11) ( ̇

√

)

(12) (13) √( )

Power required to hover: √

Rotorcraft Aeromechanics

√

(

)

(14)

13

̇

(

)

(15)

Power to hover reduces when induced velocity is small and mass flow rate through the disk is large (large rotor disk area) √

√

(16)

where

Pressure Variation Plot in Axial Direction:

According to momentum theory, the velocity deficit in the far wake is twice the velocity deficit at the rotor disk. Momentum theory gives an expression for velocity deficit at the rotor disk. It also gives an expression for maximum power produced by a rotor of specified dimensions. Actual power produced will be lower, because momentum theory neglected many sources of losses- viscous effects, tip losses, swirl, non-uniform flows, etc.

Preliminary Remarks Momentum theory gives rapid, back-of-the-envelope estimates of Power. This approach is sufficient to size a rotor (i.e. select the disk area) for a given power plant (engine), and a given gross weight. This approach is not adequate for designing the rotor.

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Limitations of the Momentum Theory: The analysis made by the simple momentum theory is idealised because it neglects profile drag losses, non-uniformity of induced flow (including the energy losses due to spilling of the air about blade tips, commonly known as tip losses) and slipstream rotation losses. Thus an actual rotor would require more power to hover with a given load than an “ideal” rotor (i.e., a rotor having zero profile drag and uniform inflow) and therefore would be less efficient. The order of magnitude of the rotor losses not considered by simple momentum theory, expressed as a percentage of the total power required is as follows: Profile drag losses: 30% Non uniform inflow: 6% Slipstream rotation: 0.2% Tip losses: 3% Lastly, it does not provide any information as to how the rotor blades should be designed for a given thrust.

Hover Power Losses Momentum theory gives the induced power loss of an ideal rotor in hover, √ .A real rotor has the power losses as well, in particular the profile power loss due to the drag of the blades in a viscous fluid. There is also an induced power loss due to the non-uniform inflow of a real, non-optimum rotor design. The distribution of the power losses of the rotor in hover is approximately as follows: Induced Power – 60% Profile Power – 30% Non-uniform Inflow – 5% to 7% Swirl in the wake – less than 1% Tip losses – 2% to 4% The main rotor absorbs most of the helicopter power, but there are other losses as well. The engine and transmission absorbs 4% to 5% of the total power with turbine engines. The tail rotor absorbs about 7% to 9% of the total helicopter power, and there is an additional loss of about 2% due to aerodynamic interference.

Disk Loading and Power Loading T/A or DL = Disk Loading T/P or P/L = Power Loading At hover T = W Induced (ideal) power:

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15

√

√

(

)

(17)

For a single rotor helicopter in hover, the rotor thrust, T, is equal to the weight of the helicopter, W; the disk loading is sometimes written as W/A. Disk loading is measured in pounds per square foot. The power loading is defined as T/P, which is denoted by PL, Power loading is measured in pounds per horsepower or newton per kilowatt. Power required to hover is given by . This means that the ideal power loading will be inversely proportional to the induced velocity at the disk. The ratio T/P decreases quickly with disk loading. Therefore, vertical lift aircraft that have a low effective disk loading will require relatively low power per unit of thrust produced (i.e. they will have high ideal power loading) and will tend to be more efficient.

Disk loading for helicopters are usually in the range of 100 – 500 N/m^2 and the corresponding inflow velocity is in the range of 6.4 – 14.3 m/s (at sea level density condition). Rotor Propeller Ducted fan Jet

⁄ N/m^2 100 – 500 2500 2500 - 5000 50000

The higher the disk loading, the higher the induced velocity, and the higher the power. For helicopters, disk loading is between 5 and 10 lb/ft2 (24 to 48 kg/m2).Tilt-rotor vehicles tend to have a disk loading of 20 to 40 lbf/ft2. They are less efficient in hover. VTOL aircraft have very small fans, and have very high disk loading (500 lb/ft2).

Power Loading For a given gross weight, that is, a high power loading with a large value of T/P is required. Power loading is the ratio of the thrust produced to the power required to hover, that is,

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16

(18)

(19) (

)

This quantity should be as close as possible to the ideal value for best hovering efficiency. Because ( ) ( ) Maximizing the power loading requires a low tip speed (

).

On the basis of simple momentum theory considerations, the ratio P/T is given by

√

√

(

√

)

(20)

To maximize the power loading (that is minimize the ratio P/T) the disk loading should be low (i.e. the disk area should be large for a given gross weight to give a low induced velocity and the tip speed should be low). When using the modified momentum theory, the ratio P/T is given by √ : √ ( √ √

√ √

; )

√

This means that the best rotor efficiency (maximum power loading) is obtained when the disk loading is minimum and the figure of merit is a maximum. Pure helicopters have a power loading between 6 to 10 lb/HP. Tilt-rotors have lower power loading – 2 to 6 lb/HP. VTOL vehicles have the lowest power loading – less than 2 lb/HP.

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17

Induced Inflow Ratio 

Induced Velocity (Hover)

The induced inflow velocity, , at the rotor disk can be written as

(Non-dimensional quantity) = Angular speed; R = Rotor radius;

Thrust and Power Co-efficient In helicopter analysis the rotor thrust co-efficient is formally defined as

Where the reference area is the rotor disk area A and the reference speed is the blade tip speed. All velocity components are non-dimensionalized by tip speed so the inflow ratio is related to the thrust co-efficient in hover by √

√

(

)

√

This is based on the 1-D flow assumption made in the preceding analysis, which means that this value of inflow is assumed to be distributed uniformly over the disk. If Thrust coefficient goes up, Inflow ratio goes up and if inflow goes up, power goes up. As the goes up Thrust coefficient decreases and tip speed increases beyond the critical value. The rotor power coefficient is defined as

Rotorcraft Aeromechanics

18

So that based on momentum theory the power coefficient for the hovering rotor is (

)

(

(

)

).

/

√

Again this is calculated on the basis of uniform inflow and no viscous losses, so is called the ideal power coefficient. The corresponding rotor shaft torque coefficient is defined as

In non-US countries an extra ½ is used in the denominator for

Comparison on Theory and Measured Rotor Performance The Comparison of measured and theoretical results is shown in the above figure. In terms of coefficients it is apparent that the ideal power according to the simple momentum theory can be written as

√ Notice that the momentum theory under predicts the actual power required, but the predicted trend that is essentially correct. These differences between the momentum theory and experiment occur because viscous effects (i.e. non-ideal effects) have been totally neglected so far.

Non-Ideal Effects on Rotor Performance Rotorcraft Aeromechanics

19

In hovering flight the induced power predicted by the simple momentum theory can be approximately described by an empirical modification to the momentum result in

√ Where k is called an induced power factor. This factor is derived from physical effects such as non-uniform inflow, tip losses, wake swirl, less than wake contraction, finite number of blades and so on. Induced power factor =

= Profile power;

= Induced Power ∫

Where is the number of blades and D is the drag force per unit span at a section on the blade at a distance y from the rotational axis. The drag force can be expressed conventionally as (

)

Where c is the blade chord. If the section profile drag coefficient, , is assumed to be constant (= ) and independent of Reynolds number and Mach number and the blade is not tapered in planform (i.e. a rectangular blade), then the profile power integrates out to be

(

) (

(

)

)

: Solidity ratio, which is the ratio of blade area to rotor disk area. Typical values of solidity ratio for a helicopter range between 0.05 and 0.12. Armed with these estimates of the induced and profile power losses, it is possible to recalculate the rotor power requirements by using the modified momentum theory result that

√

Figure of Merit The figure of merit is a measure of rotor hovering efficiency, defined as the ratio of the minimum possible power required to hover to the actual power required to hover. The figure of merit is equivalent to a static thrust efficiency and defined as the ratio of the ideal power Rotorcraft Aeromechanics

20

required to hover to the actual power required, that is,

Ideal power is the power required to lift the weight of the helicopter (i.e. minimum power without any loss). Ideal power is nothing but induced power. √ √ As thrust coefficient increases FM reaches a maximum or drops off slightly. This is because of the higher profile drag coefficients (> ) obtained at higher rotor thrust and higher blade section AoA. increases as increases; behaviour depends on airfoil stall characteristics. In practice, FM values between 0.7 and 0.8 represent a good hovering performance for a helicopter rotor.

Because a helicopter spends considerable portions of time in hover, designers attempt to optimize the rotor for hover (FM~0.8). A rotor with a lower figure of merit (FM~0.6) is not necessarily a bad rotor. It has simply been optimized for other conditions (e.g. high speed forward flight). Using the modified form of the momentum theory with the non-ideal approximation for the power, the figure of merit can be written as

⁄√ ⁄√

⁄

Notice that at low operating thrusts the figure of merit is small. This is because the profile drag term in the denominator is large compared to the numerator. As the value of increases, however, the importance of the profile power term decreases relative to the Rotorcraft Aeromechanics

21

induced term and FM increases. This continues until the induced power dominates the profile term and the figure of merit will begin to approach a value of 1/k. In practice, however, the profile drag contribution decreases this value. In practice, at higher values of rotor thrust the profile drag (and power) increases quickly as the blade begin to stall, which will again cause a reduction in FM. High solidity (lot of blades, wide-chord, large blade area) leads to higher Power consumption, and lower figure of merit. Figure of merit can be improved with the use of low drag airfoils. If the solidity ratio ( ) value is small FM value goes up. We know that √ If we need high figure of merit, value should be small. But in the equation (A), value goes down value goes up and reaches stall condition. So optimum value of should be decided to compromise both the equations.

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22

Induced Tip Loss The formation of trailed vortex at the tip of each blade produces a high local inflow over the tip region and effectively reduces the lifting capability. Tip vortex reduces lift

tip loss

B: tip loss factor BR =

=effective radius (< R) (

)

Adding inner root cut-out

Prandtl (vortex theory) showed that when accounting for the tip loss, the effective blade radius, , is given by (

)

(

√

)

√ For climb B still holds using climb velocity

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23

(

√

)

√

√

B is shown to decrease with decreasing (less blade-to-blade interference) and also with increasing (spacing of vortex sheet below the rotor). In practice, values of B for helicopter rotors are found to range from about 0.95 to 0.98, depending on the number of blades. Gessow & Myers (1952) suggest an empirical tip-loss factor based on blade geometry alone where

c: tip chord Sissingh (1939) has proposed the alternative geometric expression ( Where is the root chord of the main blade and to the root chord).

Rotorcraft Aeromechanics

) is the blade tip ratio (ratio of the tip chord

24

Blade Element Theory Two primary limitations of the momentum theory are that it provides no information as to how the rotor blades should be designed, so as to produce a given thrust. Also, profile drag losses are ignored. The blade element theory provides means for removing these limitations. The blade element theory, which was put in practical form by Drzewiecki, is based on the assumption that element of a propeller or rotor can be considered as an aerofoil segment that follows a helical path. Lift and drag are then calculated from the resultant velocity acting on aerofoil, each element being considered independent of the adjoining element. The thrust and torque of the propeller or rotor are then obtained by integrating the individual contribution of each element along radius. It is a “strip” theory. The blade is divided into a number of strips, of width dr. The aerodynamic force lift generated by that strip, and the power consumed by that strip, are computed using 2-D airfoil aerodynamics. The contributions from all the strips from all the blades are summed up to get total thrust, and total power. The resultant local flow velocity at any blade element at a radial distance y from the rotational axis has an out-of-plane component (perpendicular to the rotor) as a result of climb and induced inflow An in-plane component (parallel to the rotor) because of blade rotation The resultant velocity at the blade element is, therefore √

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25

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26

The relative inflow angle (or induced angle of attack) at the blade element will be (

)

Thus, if the pitch angle at the blade element is , then the aerodynamic or effective AoA is

( ) For low axial velocities of rotation

(

)

and a radial position sufficiently outboard from the axis

( )

(

)

Aerodynamic section of the rotor blades starts from 20 to 25% of the rotor. The elemental lift and drag, associated with a segment of width dr is

For a constant chord blade, and assuming a constant drag coefficient, the profile power coefficient can be evaluated as

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27

For an accurate calculation of the profile power loss, the variation of the drag coefficient with the angle of attack and Mach number should be included. Consider a profile drag polar of the form

By properly choosing the constants , and the variation of drag with lift for a given airfoil can be well represented for angle of attack below stall. This representation for drag coefficient was used by Bailey (1941) and his numerical example is frequently found in helicopter calculations. The lift dL and drag dD act perpendicular and parallel to the resultant flow velocity respectively. The quantity c is the local blade chord. When

is small,

is large and

is small

( ,(

) ,(

∫

) )

(

( ,(

∫

)

-

)

-

) (

(

)

)

These forces can be resolved perpendicular and parallel to the rotor disk plane giving

Therefore the contributions to the thrust, torque and power of the rotor are

: Number of blades comprising the rotor For hover and axial flight independent of azimuth angle ( ) (

Rotorcraft Aeromechanics

)

28

(

)

(

)

For helicopters the following simplifying assumptions can be made: √ This is valid approximation except near the blade root, but the aerodynamic forces are small here anyway. The induced angle

is small, so that

The drag is at least one order of magnitude less than lift, so that the contribution ( ) is negligible. Applying these simplifications to the preceding equations results in

(

)

(

)

Non-dimensionalization: ̅

(

)

(

)

(

)

The inflow ratio can be written as (

)

. /

. (

)

/

(

(

) (

)(

. / )

̅

)

̅

. / ̅

For a rectangular blade (c = constant)

Rotorcraft Aeromechanics

29

̅

̅

Similarly (

( (

)

) )(

)(

(

)

)̅

(

)̅

̅ ̅

For a rectangular blade the thrust coefficient is ∫

̅

̅

Torque / Power coefficient: ∫ ( ∫ ( ̅

)̅ ̅

̅ ) ̅

Since

Drawbacks of Blade Element Theory It does not handle tip losses. Solution: Numerically integrate thrust from the cutout to BR, where B is the tip loss factor. Integrate torque from cut-out all the way to the tip. It assumes that the induced velocity v is uniform. It does not account for swirl losses. The Predicted power is sometimes empirically corrected for these losses.

√

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30

Integrated Rotor Thrust and Power To evaluate and it is necessary to predict the span wise variation in the inflow as the sectional aerodynamic force coefficients .

as well

If 2-D aerodynamics are assumed (

)

(

)

Where Re and M are the local Reynolds number and Mach number

Thrust Approximations Based on steady linearized aerodynamics, the local blade lift coefficient can be written as (

)

(

) of the airfoil section

For an incompressible flow per radian (

)

For symmetric airfoil ∫

̅

∫ ( ̅

) ̅ ̅

⁄ ∫ ( ̅

̅) ̅

∫ ( ̅

̅) ̅

Untwisted Blades, Uniform Inflow Zero twist

Uniform Inflow

6

Rotorcraft Aeromechanics

̅

̅

7

31

[

]

√

√

< This can be solved iteratively to find

=

for a given √

√

is almost equal to 6 ⁄ gives the average angle that required during hovering condition or angle need to be provided during hover. ⁄ is also known as blade loading or mean pitch angle required for operation. When the weight increases, pitch angle also increases. After certain angle, helicopter cannot lift-off, because blade reaches stall angle.

Linearly Twisted Blades, Uniform Inflow Many helicopter rotor blades are designed with a linear twist, so that

Rotorcraft Aeromechanics

takes the form

32

( ) where

̅

is the blade twist rate per radius of the rotor.

Using this variation in ( ), we get ∫ ,(

)̅

̅

6

̅

̅

[ Using

at 0.75 radius (

̅- ̅

7

]

) ( )

(̅

∫ *,(

)

(̅

∫ *,( ̅

) ̅

[

)̅ -

̅+ ̅

̅ )-

̅+ ̅

] [

]

Ideal Twist This twist distribution while not physically realizable at the root is of interest because it will be found to give uniform inflow with the constant chord blades. ̅ The induced velocity „ ‟can be made constant across the disk by allowing the pitch angle

Rotorcraft Aeromechanics

to

33

vary inversely with non-dimensional radial distance ( ̅) So [

]

Rotor Torque Another important quantity associated with rotor behaviour is the torque, or the moment needed to overcome the drag and keep the rotor turning at a certain RPM in steady state conditions. One can express the elemental torque

Total torque ∫ ( ∫

) (

)

Elemental Torque due to all the blades [

(

)

]

In non-dimensional form (

∫ [ ∫ 0

( ̅ ∫

)

( ̅ ̅ )1 ∫ ∫ , ̅

Rotorcraft Aeromechanics

̅

̅)

̅ ]

∫ ̅

̅

̅ ̅

̅ ̅- ̅

34

∫

∫

∫

̅

̅

If the climb velocity is constant ∫ In non-dimensional form the climb power will be the product of velocity and the total weight of the helicopter. Induced velocity in climbing condition is different from induced velocity in hovering condition. For hovering flight

becomes ∫

∫

̅

̅

In this case at each section is assumed to be constant. If different aerofoils used in the rotor blade then will become the function of span and Mach number.

∫

[

(

∫ 8

) (

6

)

(

6 [(

)

)

] 7

9

7

]

(

)

Combining the above equations

Rotorcraft Aeromechanics

35

[(

) ]

[(

) ]

The first term is usually called the induced torque (because it is due to the induced drag) and the second term is called the profile torque.

The power coefficient is defined as (

)

Next it is interesting to connect the expressions which have been derived with momentum theory in hovering flight. Recall for this case power is given by

(

)

(

)

Since for hover

So

Ideal torque coefficient in absence of friction is identical from both blade element theory and momentum theory. In hovering flight √

√

<

√

=

From which

Rotorcraft Aeromechanics

36

(

√

)

√

(

) (

)

) (

This is a quadratic equation for

(

)

, when the value of collective pitch

is given.

Thus

√ Where

= Profile torque coefficient

Ideal torque coefficient,

√ Another useful quantity often used in helicopter engineering is the Rotor Figure of Merit (FM) which is defined as

√ (

)

So that the ideal rotor figure of merit is

√ (

)

For an actual rotor the Figure of Merit indicates the magnitude of the losses due to nonuniformity of flow, tip loss, and profile drag for a particular rotor. For a good rotor FM 0.75 where “good” implies a well-designed rotor. In the equations used above is given by

has appeared a number of times, a good approximation for in radians, Angle of Attack

Rotorcraft Aeromechanics

37

For a reasonable angle of attack this normally yields a value of

0.012

In hover case

√ k = Empirical factor which represents additional losses (+15%) In this case we are integrating the equations from zero to one directly (i.e. along the full span of the blade) but near the tip the above equation will not look like this. In real rotors, at the tip there will be lift drop. But the equation gives the lift per section.

Lifting-line theory is not strictly valid near wing tips. When the chord at the tip is finite, blade element theory gives a non-zero lift all the way out to the end of the blade. In fact, however, the blade loading drops to zero at the tip over a finite distance because of threedimensional flow effects. Since the dynamic pressure is proportional to . Beyond 0.5 the lift value will be large. In the actual rotor near ̅ , lift drops and then to zero value at the tip exactly. At the tip, lift becomes zero bu there will be always drag value and non-uniform inflow. Analysis at the tip region is very complicated and it cannot be done by Momentum Theory.

Tip-Loss Factor Effective Blade

B = Prandtl tip-loss factor; B = 0.95 – 0.98 ∫ For Untwisted blades

̅

̅

∫

̅) ̅

and uniform inflow [

For a twist

( ̅

]

̅ (ideal twist, = constant)

Rotorcraft Aeromechanics

38

∫ (

6(

̅

)

)̅ ̅

7

(

)

Because B is between 0.95 and 0.98, we find a 6-10% reduction in rotor thrust resulting from tip-loss effects for a given blade-pitch setting under the stated assumptions. The correct interpretation is to consider that for the same thrust the induced inflow will be increased to a value √

√

(

√

)

(or ) is increased by a factor compared to the case with no assumed tip losses. For untwisted blades and uniform inflow with tip losses alone, the thrust becomes [

]

For ideal twist and uniform inflow, the thrust now becomes [

]

Because of tip-loss effects, a real rotor will always have a higher overall average induced velocity compared to that given by momentum theory and so the induced power will also be increased relative to the simple momentum result. Tip loss constitutes an additional source of non-uniform inflow, and would normally be factored into the value of k. Using the BET, the induced power can be written as ∫

∫

̅

̅

Using untwisted rectangular blades and uniform inflow assumptions, then with a tip loss the total power can be approximated by [

]

Where the induced power factor k includes the effects of both tip loss and non-uniform inflow over the remainder of the blade. In general we can write that

Rotorcraft Aeromechanics

39

Combined Blade Element-Momentum Theory A combination of momentum theory and blade element theory has been developed later. Identical equations may be derived by means of vortex theory, but it is believed that the combination of momentum and blade element theory has greater physical significance and can be easily grasped. The combined theory can be applied for performance analysis of helicopter in hover, vertical climb and forward flight. In the previous sections we dealt with momentum theory which gave estimates for power and induced velocity in terms of the thrust. We also studied blade element theory, which allowed us to incorporate features such as number of blades, airfoil section drag and lift characteristics, taper and twist, etc. The latter, however, assumed that the flow through the rotor disk is uniform as in the momentum theory. Consider a small annulus segment of the rotor disk, shown.

Rotorcraft Aeromechanics

40

The annulus is at a distance y from the rotational axis, and has a width dy.

Equating differential thrust over annular area from BET and Momentum Theory. Assume that the annular area is not affected by either the flow inside or outside that. On the basis of simple one-dimensional momentum theory, the incremental thrust on the rotor annulus as the product of the mass flow rate through the annulus and twice the induced velocity at that section. In this case the mass flow rate over the annulus of the disk is ( ̇

)

(

)

(

)

so that the incremental thrust on the annulus is (

)

This has also been known as the Froude-Finsterwalder equation. ( ( (

)(

) ( ( )

)

)

(

)

.

) /. / . /

̅ ̅ Therefore, the incremental thrust coefficient on the annulus can be written as (

̅ ̅

)̅ ̅

This assumes no swirl in the wake. Consider first the hovering state where incremental thrust and power contributions of the annulus are given by ̅ ̅

. The

̅ ̅

The total thrust coefficient of the rotor is ∫

∫

̅ ̅

And the corresponding induced power coefficient is ∫

̅ ̅

From the BET the incremental thrust produced on an annulus of the disk is ( ̅

̅) ̅

Equating the incremental thrust coefficients from the momentum and blade element theories we find that

Rotorcraft Aeromechanics

41

( ̅

̅)

(

)̅

Which gives ̅ .

/

̅

This quadratic equation in has the solution (̅

)

√(

)

̅

(

)

In hovering flight condition, the above equation simplifies to :√

( ̅)

̅

;

The numerical implementation of combined BEM theory is identical to classical blade element theory. The only difference is the inflow is no longer uniform. It is computed using the formula given earlier, reproduced below: (̅

)

√(

)

̅

(

)

Average Lift Coefficient Let us assume that every section of the entire rotor is operating at an optimum lift coefficient and the rotor is untapered. ̅ ̅

) ̅

(

∫

̅ (

̅

) ̅

Rotor will stall if average lift coefficient exceeds 1.2, or so. Thus, in practice, CT/s is limited to 0.2 or so.

Rotorcraft Aeromechanics

42

With ideal twist the performance of the rotor can now be recalculated. If ̅

∫ (

)̅ ̅

(

( )

:√

(

)

)

;

√

A solution for the blade pitch angle can be found from

Rotorcraft Aeromechanics

43

√

Alternate Derivation The elemental thrust of the blade elements contained in the annular ring based on the blade element theory is given by ( )

[ ( )

]

For the same annular ring shown, the elemental thrust based on momentum theory is given by ,

( )-

( )

Equating this two expressions yields ( )

[ ( )

,

]

( )-

( )

Which is a quadratic equation for ( ) Rewrite the equation in non-dimensional form ( ̅)

( ̅)

̅ 6 ( ̅) ( ̅) ( ̅)

( ̅) ( ̅)

( ̅ ),

( ̅)

( ̅) 6

(

̅

( ̅)

̅ 6 ( ̅) ̅

̅ ( ̅)

̅ 6 ( ̅) ̅ ( ̅) ( ̅)

( ̅ )-

( ̅)

,

( ̅ )-

( ̅)

7

7

7

( ̅ )-

,

,

-

√ (

7

( ̅)

̅ 6 ( ̅)

)

̅ ,

7

( ̅ )- ( ̅ ) ̅ [ ( ̅)

̅

( ̅)

]

̅̅̅

[ ( ̅)

, ̅

]

̅̅̅

)

The above equation is an important and useful equation for determining the inflow in hover

Rotorcraft Aeromechanics

44

or axial flight. Once the induced velocity is known, the inflow angle at the blade element can be determined from ̅ It is a completely general expression which allows one to determine the inflow velocity for any blade planform and pitch distribution. A number of special forms of this equation are quite useful. CASE (A) when chord is constant and the blade twist is ideal (inversely proportional to( ̅ ), one obtains ( ̅ ) constant over the disk

( ̅)

(

,

)

-

√ (

)

CASE (B) Another useful relation is obtained for the case of hover (V=0) and constant chord. Assuming that the inflow velocity at 0.75 R is representative of a uniformly distributed inflow for one has

( ̅)

(

)

√

,

-

( ( ̅)

( ̅)

) ) :√

(

(

) :√

;

;

which is an approximate relation for uniform inflow frequently used in aero-elastic calculations.

Optimum Rotor for Hover Here we are interested in the optimum rotor for hover including real fluid effects. We are seeking for max (L/D)

Still want

= constant over the disk.

Rotorcraft Aeromechanics

45

( )

( )

6 ( ) ( )

7

( )

6 ( )

7

We want ( ) Let ( )

( ) Therefore

( )

( ) .

( ̅)

̅

/

√ ̅

Ideal Rotor vs Optimum Rotor ⁄̅

•

Ideal rotor has a non-linear twist:

•

This rotor will, according to the BEM theory, have a uniform inflow, and the lowest induced power possible.

•

The rotor blade will have very high local pitch angles q near the root, which may cause the rotor to stall.

•

Ideally twisted rotor is also hard to manufacture.

•

For these reasons, helicopter designers strive for optimum rotors that minimize total power, and maximize figure of merit.

•

This is done by a combination of twist, and taper, and the use of low drag airfoil sections.

Optimum Rotor •

We try to minimize total power (Induced power + Profile Power) for a given T.

•

In other words, an optimum rotor has the maximum figure of merit.

•

From earlier work, figure of merit is maximized if is maximized.

Rotorcraft Aeromechanics

46

•

All the sections of the rotor will operate at the angle of attack where this value of coefficient of lift and drag are produced.

•

We will call this lift coefficient the optimum lift coefficient.

All radial stations will operate at an optimum value at which is maximum Once angle of attack is selected, we find

from .

√

/

This determines how the blade must be twisted. Variation of chord for the optimum rotor is given by ( Note that

)

is constant (the optimum value). It follows that ( )

6

7

Such planforms and twist distributions are hard to manufacture, and are optimum only at one thrust setting. Manufacturers therefore use a combination of linear twist, and linear variation in chord (constant taper ratio) to achieve optimum performance.

Accounting for Tip Losses We have already accounted for two sources of performance loss-non-uniform inflow, and blade viscous drag. We can account for compressibility wave drag effects and associated losses, during the table look-up of drag coefficient. Two more sources of loss in performance are tip losses, and swirl. An elegant theory is available for tip losses from Prandtl. Prandtl suggest that we multiply the sectional inflow by a function F, which goes to zero at

Rotorcraft Aeromechanics

47

the tip, and unity in the interior. (

)

When there are infinite number of blades, F approaches unity, there is no tip loss. (

)

All we need to do is multiply the lift due to inflow by F (incorporation of Tip Loss model in BEM) Thrust generated by the annulus (

)

From BET ( (

) )

.

/

Resulting Inflow (Hover) will be ( )

0 means power is consumed; P < 0 means power is extracted. Consider now power-off descent in terms of the blade aerodynamic loading. The inflow ratio

is directed upward through the disk, so there is a forward tilt of the lift vector. For power equilibrium at the blade section, the inflow angle must be such that there is no net in-plane force and hence no contribution to the rotor torque.

Rotorcraft Aeromechanics

65

The rotor efficiency is about as high as possible; a low descent rate can be achieved only with very low disk loadings. In normal case power is supplied to the rotor to support the weight. Power supplied to the rotor to support the weight slowly decreases with increase in the descent velocity. At a particular condition, (V+ ) becomes zero and there is no power supplied to rotor to support the weight. After the zero velocity condition, as the velocity increases rotor generates power from the flow to the disk. In autorotation state loss of potential energy is converted into kinetic energy (i.e. spinning of the rotor). In autorotation state, rotors act like parachute. In condition (a) and (d) momentum theory is valid. In condition (b) and (c), there is no theory to calculate the inflow at this state.

Rotorcraft Aeromechanics

66

Blade Element Theory Treatment of Descent Problem

Again assume small angles and

then [

]

and we have to redefine the inflow for vertical descent as

∫

[

]

Thus only the difference between the vertical climb and descent is the sign in the square ( ) ( ) bracket. The positive sign convention for this case is [

]

From this figure it is clear that the in-plane component of lift opposes the profile drag term. Torque in descending flight is given by

Rotorcraft Aeromechanics

(

)

∫ ,

-

67

,

∫ { ∫

( )-}

[

]

[

]

[

]

Partial Powered Vertical Descent Solve torque expression for ( For a given throttle setting

)

(fixed by putting a particular throttle setting) is prescribed (

)

and recall √ From momentum theory and therefore (

)√

(

)

(

(

)

)

√

In the windmill brake range, where momentum theory is valid one can assume approximately (

)

On the other hand

Rotorcraft Aeromechanics

68

For minimum rate of descent, power off and (

)

Therefore √

For a parachute Therefore

√

√

√

Autorotation Engine shut down (power-off), here energy balance between friction losses and rotational energy is important

And

Rotorcraft Aeromechanics

69

( (

)

)

√( ) Knowing get

⁄

(

)

From ( )

)√

( Or (

) √

Using the above equation one can use the figure () to determine in which region one is. If one is in TW state, need to empirical part, if in wind mill brake state use momentum theory. Two equations with two unknowns, one has to pick ,

so as to get the condition one wants ( )-

( )

Rotorcraft Aeromechanics

70

At some station

In-plane component of profile drag balances in-plane component of lift. Further in the lift is greater and in the outboard direction the opposite is true. At equilibrium

( ) ,

( )-

Which yields a quadratic equation for

( )

( ) √( )

( ) ( )

( )

( )

This determines . / location of the equilibrium section. ( )

( )

Angle of attack increases as one goes inboard on an auto-rotating blade. In such a blade stall occurs inboard. If stall exceeds 40% of the blade the vehicle can be destroyed.

Forces acting on an airfoil in autorotation (Blade Element)

During the power flight condition, shaft is given by the power. When the engine fails, rotor shaft disengages from the engine.

Rotorcraft Aeromechanics

71

If is positive, velocity of the rotor blade decreases (since the rotor is rotating opposite to ). If is negative, it will accelerates the rotor blade. There is a particular angle ( ) at which will be zero. If the is zero then the rotor continues to rotate with same velocity.

Near the rotor root,

is nearly zero.

( ) at which autorotation is possible. Rotor is powered by air means the resultant flow rotates the rotor (Power from air to rotor). For power equilibrium at the blade section, the inflow angle must be such that there is no net ( ) in-plane force and hence no contribution to the rotor torque . Because autorotation involves induced and profile torques of the entire rotor, generally only one section will be in equilibrium itself, while the others are either producing or absorbing power. Inflow angle is large inboard and decreases toward the tip. Then on the inboard sections, which produce an accelerating torque ( ) on the rotor and absorb power from the air; and on the outboard sections, which produce a decelerating torque and deliver power to the airstream. Since there is no net power to the rotor, the accelerating and decelerating torques must balance. For a given descent rate, the rotor tip speed will adjust itself until this equilibrium is achieved. may be positive, negative or zero, depending on the inflow angle . Autorotation depends on the inflow angle (D and L also contribute to it). When

is positive, section of the rotor blade decelerates (

decreases).

When is zero, Autorotation (because no force in the x-direction, rotor continues to rotate with (constant)). When

is negative, section of the rotor blade accelerates (

Rotorcraft Aeromechanics

increases).

72

̅ In autorotation ̅

Condition for autorotation of a particular element is

When

becomes

When

is positive

When

is zero

When

is negative

Autorotation Diagram ⁄ , plotted against the blade The figure consist of the airfoil section characteristics, section angle of attack, , both quantities being drawn to the same scale.

Rotorcraft Aeromechanics

73

The diagram is applied to a particular section by marking off the pitch angle of the section from the origin of the plot along the AoA axis, and then by constructing a 45-degree line from the pitch angle. A perpendicular dropped from any point on this line to the horizontal axis will then define the blade-element inflow angle on the horizontal axis, in as much as the inflow angle may be obtained by subtracting the pitch angle from the blade section angle ). Also, because of the equality of the two legs of the right triangle, the of attack( vertical leg is also equal to . ⁄ . Consider a condition of operation as represented by point (a) is one in which For this condition, the resultant force on the airfoil is displaced from the axis of rotation in such a manner as to cause the blade element to accelerate, with a consequent increase in . As increases, decreases. The element continues to accelerate until its rotational speed has increased to the point where the element is operating at condition (b). Autorotative ⁄ at that condition. equilibrium is established at (b), because (1) For a given pitch angle, , the intersection of the 45-degree line with (a) Any point above the curve [such as point (a)] represents an accelerating condition wherein the resultant vector falls ahead of the rotor axis. (b) Any point of the curve [such as point (b)] represents autorotative equilibrium wherein the resultant vector falls along the rotor axis. (c) Any point below the curve [such as point (c)] represents a decelerating condition wherein the resultant vector falls behind the rotor axis. (2) The highest possible value of the pitch angle at which autorotation may exist is such that the 45-degree line is tangent to the curve, as at point (d). It is important to note that autorotation is a stable phenomenon as long as the pitch angle is less than the maximum as defined by point (d). Any disturbance which slows down the rotor increases and accelerates the rotor to autorotative equilibrium. Similarly, if a disturbance causes the rotor to speed up, is decreased, thereby tilting the resultant vector rearward and decelerating the rotor to equilibrium. If changes in the inflow velocity are neglected, then varies inversely with . On the diagram, then, the highest rotational speed corresponds to the lowest . The pitch for maximum rotor speed is therefore the pitch defined by the intersection of a 45-degree line ⁄ curve and the horizontal axis. As the pitch is increased, through the minimum of the the rotational speed will decrease more and more rapidly until the highest possible pitch for Rotorcraft Aeromechanics

74

autorotation is reached [point (d)].

Reverse Flow At higher rotor advance ratios, a considerable amount of reverse flow will exist on the retreating side of the rotor disk, that is, the blade sections operate with the trailing edge into the relative wind (at the root region .is small). (Azimuth angle) always measured from the rear X-axis. The locus of the region where

means that

Boundary of the reverse flow region circle is given by

Beyond this boundary is large and below this boundary is small. Consider the angle between the line that extended from the root of the blade and Y-axis is . In the retreating side

So (

)

This is the condition for reverse flow region boundary. When

,

In non-dimensional form ( ⁄ ( ⁄ Rotorcraft Aeromechanics

) )

75

̅ The reverse flow region is a circle of diameter on the retreating side of the rotor disk. For low advance ratio the influence of reverse flow is small, since it is confined to a small area where the dynamic pressure is low. Therefore, up to about reverse flow effects may be neglected. At higher advance ratios, the reverse flow region occupies a large portion of the disk and must be accounted for in calculating the aerodynamic forces on the blade.

The normal aerodynamic force neglecting stall was given for small angle as (

)

The positive directions of the various quantities are as follows: and L upward, nose up, downward, and from the leading edge to the trailing edge. In the reverse flow region the angle of attack is |

|

Just as in normal flow. However, in reverse flow a positive angle of attack gives a negative (downward) lift: |

|(

)

Thus an expression valid in both reverse and normal flow is |

Rotorcraft Aeromechanics

|

|

|(

)

76

Momentum Analysis in Forward Flight Consider a rotor in forward flight. In this case there is an edgewise component of velocity, as illustrated below, because the thrust vector has to be tilted so as to provide a force component in the direction of flight.

Rotorcraft Aeromechanics

77

The mass flow rate through the actuator disk is now ̇ Where U is the resultant velocity at the disk and is given by )

√(

(

)

,

(

)

-

,

(

)

-

,

-

Under these conditions the axisymmetric flow that was used in the derivation of the momentum theory in axial flight is lost. However by assumptions originally introduced by Glauert it is possible to construct a momentum theory that allows one to calculate rotor performance in forward flight. The application of the conservation of momentum in a direction normal to the disk gives ̇( Rotorcraft Aeromechanics

) ̇

̇

78

Using conservation of energy (

)

̇(

)

̇( ( ̇

̇

)

)

̇(

)

This is the same result shown previously for the axial flight cases. Therefore √ ̇ Notice that for hovering flight

, so that the above equation reduces to

This confirms that the forward flight result above reduces to the hover result as required. As the forward flight speed increases such that , then

This is called Glauert‟s “High speed” Approximation. In forward flight, the rotor thrust is given by )

√( ̇

(

)

√ (

)

(

)

For hovering flight

Substitute

Divide by

and rearrange to get (

)

Rotorcraft Aeromechanics

(

) (

)

(

) (

)

79

Power Required (=

) ̇ ,(

)

(

)

-

̇,

̇, ̇(

(

)-

)(

)

(

)

Ideal power required by the rotor is given by the product of the thrust and velocity normal to the disc. If the ideal (induced) power required to hover and produce the same thrust . Then

(

From given T,

)

calculate

(

)

(

solve (

To obtain

and then calculate

)

)

from (

)

Special cases of thrust constant equations

Consider the general expression for induced velocity (

)

(

) (

)

(

) (

)

The induced velocity equation is biquadratic (

(

Rotorcraft Aeromechanics

)

)

(

(

) (

)

)

√ (

)

80

⁄

(

)

<

(

)

(

(

)

(

)(

√ (

)

)(

)

=

)

√ (

)

Constant Power Equations Variation of and T with forward velocity can be developed at a given power

for constant power. In hover the thrust that

With the same power the rotor at an angle of attack in a freestream velocity produces a T and induced velocity given by the equation (for hovering rotor) (

)

(

)

[ ( Dividing by

]

(

)

)

we get ( ) 6(

Rotorcraft Aeromechanics

( )

) (

(

) (

) (

)

)

( (

) (

) (

)

) 7( )

81

6(

)

(

) (

)

(

) (

) 7[

]

Then the induced velocity in forward flight can be written as

)

√(

(

)

The idea of a tip speed ratio or advance ratio ( ), is now used. By using the velocity parallel to the plane of the rotor, then we define

The inflow ratio is

√ But it is also known from the hover case that √

√ Finally, the solution for the inflow ratio, √ Special Case, If the disk AoA is zero, an exact analytical solution for the inflow ratio can be determined. The induced velocity ratio in forward flight is

√

√

Squaring both sides of the above equation and rearranging gives

Dividing by

gives

Rotorcraft Aeromechanics

82

( )

( ) ( )

This quadratic has the solution