Steady, Level Forward Flight

Steady, Level Forward Flight
I . Introductory Remarks © Lakshmi Sankar 2002

The Problems are Many.. © Lakshmi Sankar 2002

The Dynamic Pressure varies Radially and Azimuthally
© Lakshmi Sankar 2002

Consequences of Forward Flight
The dynamic pressure, and hence the air loads have high harmonic content. Above some speed, vibrations can limit safe operations. On the advancing side, high dynamic pressure will cause shock waves, and too high a lift (unbalanced). To counter this, the blade may need to flap up (or pitch down) to reduce the angle of attack. Low dynamic pressure on the retreating side. The blade may need to flap down or pitch up to increase angle of attack on the retreating side. This can cause dynamic stall. Total lift decreases as the forward speed increases as a consequence of these effects, setting a upper limit on forward speed. © Lakshmi Sankar 2002

Forward Flight Analysis thus requires
Performance Analysis – How much power is needed? Blade Dynamics and Control – What is the flapping dynamics? How does the pilot input alters the blade behavior? Is the rotor and the vehicle trimmed? Airload prediction over the entire rotor disk using blade element theory, which feeds into vibration analysis, aeroelastic studies, and acoustic analyses. We will look at some of these elements. © Lakshmi Sankar 2002

Steady, Level Forward Flight
II. Performance © Lakshmi Sankar 2002

Inflow Model To start this effort, we will need a very simple inflow model. A model proposed by Glauert is used. This model is phenomenological, not mathematically well founded. It gives reasonable estimates of inflow velocity at the rotor disk, and is a good starting point. It also gives the correct results for an elliptically loaded wing. © Lakshmi Sankar 2002

Force Balance in Hover In hover, T= W That is all!
Drag Thrust Rotor Disk Drag Weight In hover, T= W That is all! No net drag, or side forces. The drag forces on the individual blades Cancel each other out, when summed up. © Lakshmi Sankar 2002

Force Balance in Forward Flight
Thrust, T Vehicle Drag, D Flight Direction Weight, W © Lakshmi Sankar 2002

Simplified Picture of Force Balance
Rotor Disk, referred to As Tip Path Plane (Defined later) aTPP Flight Direction © Lakshmi Sankar 2002

Recall the Momentum Model
V V+v V+2v © Lakshmi Sankar 2002

Glauert’s Conceptual model
Freestream, V∞ Freestream, V∞ Induced velocity, v Freestream, V∞ 2v © Lakshmi Sankar 2002

Total Velocity at the Rotor Disk
V∞cosαTPP V∞sinαTPP Freestream V∞ Total velocity Induced Velocity, v © Lakshmi Sankar 2002

Relationship between Thrust and Velocities
In the case of hover and climb, recall T = 2 r A (V+v) v Induced Velocity Total velocity Glauert used the same analogy in forward flight. © Lakshmi Sankar 2002

In forward flight.. This is a non-linear equation for induced velocity v, which must be iteratively solved for a given T, A, and tip path plane angle aTPP It is convenient to non-dimensionalize all quantities. © Lakshmi Sankar 2002

Non-Dimensional Forms

Approximate Form at High Speed Forward Flight
In practice, advance ratio m seldom exceeds 0.4, because of limitations associated with forward speed. © Lakshmi Sankar 2002

Variation of Non-Dimensional Inflow with Advance Ratio
Notice that inflow velocity rapidly decreases with advance ratio. © Lakshmi Sankar 2002

Power Consumption in Forward Flight
Parasite Power Induced Power © Lakshmi Sankar 2002

Power Consumption in Level Flight

Induced Power Consumption in Forward Flight
Induced Power, Tv V∞ © Lakshmi Sankar 2002

Parasite Power Consumption in Forward Flight
V∞ © Lakshmi Sankar 2002

Profile Power Consumption in Forward Flight
Blade Profile Power V∞ © Lakshmi Sankar 2002

Power Consumption in Forward Flight
Total Power Power Parasite Power Induced Power, Tv Blade Profile Power V∞ © Lakshmi Sankar 2002

Non-Dimensional Expressions for Contributions to Power

Empirical Corrections
The performance theory above does not account for Non-uniform inflow effects Swirl losses Tip Losses It also uses an average drag coefficient. To account for these, the power coefficient is empirically corrected. © Lakshmi Sankar 2002

Empirical Corrections
Profile Power Induced power Parasite Power © Lakshmi Sankar 2002 1.15

Excess Power Determines Ability to Climb
Rate of Climb= Excess Power/W Available Power Excess Power Total Power Power Parasite Power Induced Power, Tv Blade Profile Power V∞ © Lakshmi Sankar 2002

Forward Flight Blade Dynamics © Lakshmi Sankar 2002

Background Helicopter blades are attached to the rotor shaft with a series of hinges: Flapping hinges (or a soft flex-beam) , that allow blades to freely flap up or down. This ensures that lift is transferred to the shaft, but not the moments. Lead-Lag Hinges. When the blades rotate, and flap, Coriolis forces are created in the plane of the rotor. In order to avoid unwanted stresses at the blade root, lead-lag hinges are used. Pitch bearing/pitch-link/swash plate: Used to control the blade pitch. The blade loads are affected by the motion of the blades about these hinges. From an aerodynamic perspective, lead-lag motion can be neglected. Pitching and flapping motions must be included. © Lakshmi Sankar 2002

Articulated Rotor © Lakshmi Sankar 2002

Flap Hinge http://www.unicopter.com/0941.html#delta3
~w-p/bookaut/scbprts.htm © Lakshmi Sankar 2002

Pitch Horn http://www.unicopter.com/0941.html#delta3

Lead-Lag Motion © Lakshmi Sankar 2002

Coordinate Systems Before we start defining the blade motion, and the blade angular positions, it is necessary to define what is the coordinate system to use. Unfortunately, there are many possible coordinate systems. No unique choice. © Lakshmi Sankar 2002

Hub Plane z Y-axis is perpendicular to Fuselage symmetry plane
And perpendicular to z =180 deg Y =270 deg =90 deg Z axis is normal to shaft X-axis runs along fuselage symmetry plane And is normal to Z =0 deg X © Lakshmi Sankar 2002

Tip Path Plane This plane is defined by two straight lines.
The first connects the blade tips at azimuth angle =0 and =180 deg. The second connects the blade tips at azimuth angle =90 and =270 deg. Z is perpendicular to TPP. In TPP, the blade Does not appear to be Flapping.. X Z Blade is at =0 deg. © Lakshmi Sankar 2002

Tip Path Plane 2Rb1c b1c b0+b1c R(b0+b1c) R(b0-b1c) X b0-b1c

No-Feathering Plane (NFP) or Control Plane
Blade at =90 deg Blade at =270 deg Pitch link Z Swash Plate X Pilot Input The pilot controls the blade pitch by applying a collective control (all blades pitch up or down by the same amount), or by a cyclic control (which involves tilting the swash-plate). Some of the pitch links move up, while others move down. The airfoils connected pitch up or down). © Lakshmi Sankar 2002

Differences between Various Systems
For an observer sitting on the tip path plane, the blade tips appear to be touching the plane all the time. There is no flapping motion in this coordinate system. For an observer sitting on the swash plate, the pitch links will appear to be stationary. There is no vertical up or down motion of the pitch links, and no pitching motion of the blades either. In the shaft plane, the blades will appear to pitch and flap, both. One can use any one of these coordinate systems for blade element theory. Some coordinate systems are easier to work with. For example, in the TPP we can set the flapping motion to zero. © Lakshmi Sankar 2002

Blade Flapping Dynamics and Response
Forward Flight Blade Flapping Dynamics and Response © Lakshmi Sankar 2002

Background As seen earlier, blades are usually hinged near the root, to alleviate high bending moments at the root. This allows the blades to flap up and down. Aerodynamic forces cause the blades to flap up. Centrifugal forces causes the blades to flap down. Inertial forces will arise, which oppose the direction of acceleration. In forward flight, an equilibrium position is achieved, where the net moments at the hinge due to these three types of forces (aerodynamic, centrifugal, inertial) cancel out and go to zero. © Lakshmi Sankar 2002

Schematic of Forces and Moments
We assume that the rotor is hinged at the root, for simplicity. This assumption is adequate for most aerodynamic calculations. Effects of hinge offset are discussed in many classical texts. © Lakshmi Sankar 2002

Velocity encountered by the Blade
Velocity normal to The blade leading edge is Wr+V∞sin V∞ Wr  © Lakshmi Sankar 2002 X

Moment at the Hinge due to Aerodynamic Forces
From blade element theory, the lift force dL = Moment arm = r cosb ~ r Counterclockwise moment due to lift = Integrating over all such strips, Total counterclockwise moment = © Lakshmi Sankar 2002

Moment due to Centrifugal Forces
The centrifugal force acting on this strip = Where “dm” is the mass of this strip. This force acts horizontally. The moment arm = r sinb ~ r b Clockwise moment due to centrifugal forces = Integrating over all such strips, total clockwise moment = © Lakshmi Sankar 2002

Moment at the hinge due to Inertial forces
Small segment of mass dm With acceleration r b b is positive if blade is flapping up Flap Hinge Associated moment at the hinge = Integrate over all such segments: Resulting clockwise moment at the hinge= © Lakshmi Sankar 2002

At equilibrium.. Note that the left hand side of this ODE resembles a
spring-mass system, with a natural frequency of W. We will later see that the right hand side forcing term has first harmonic (terms containing Wt), second, and higher Harmonic content. The system is thus in resonance. Fortunately, there is Adequate aerodynamic damping. © Lakshmi Sankar 2002

How does the blade dynamics behave when there is a
forcing function component on the right hand side of the form AsinWt, and a damping term on the left hand side of form c db/dt ? To find out let us solve the equation: To solve this equation, we will assume a solution of form: In other words, the blade response will be proportional to the amplitude A of the resonance force, but will lag the force by 90 degrees. © Lakshmi Sankar 2002

What happens when the pilot tilts the swash plate back?
The blade, when it reaches 90 degrees azimuth, pitches up. Lift goes up instantly. The blade response occurs 90 degrees later (recall the phase lag). The front part of the rotor disk tilts up. The exact opposite happens with the blade at 270 degree azimuth. Blade at =90 deg Blade at =270 deg Z X © Lakshmi Sankar 2002

What happens when the pilot tilts the swash plate back?
TPP T Blade reaches its Highest position At =180 deg Blade reaches its lowest position At =0 deg NFP SwashPlate The tip path plane tilts back. The thrust points backwards. The vehicle will tend to decelerate. © Lakshmi Sankar 2002

What happens when the pilot tilts the swash plate to his/her right?
Blade at = 180 deg pitches up. Lift goes up. Blade responds by flapping up, and reaches its maximum response 90 degrees later, at = 270 deg. The opposite occurs with the blade at = 0 deg. TPP tilts towards the pilot’s right. The vehicle will sideslip. T TPP NFP Helicopter viewed from Aft of the pilot © Lakshmi Sankar 2002

Blade Flapping Motion =t First and higher harmonics Coning Angle

b1c determines the fore-aft tilt of TPP
X This angle is measured in one of the three coordinate systems (Shaft plane, TPP, or NFP) that we have chosen to work with. Some companies like to use TPP, others like NFP or shaft plane. If TPP is our reference coordinate system, what will b1c be? Ans: Zero. © Lakshmi Sankar 2002

b1s determines the lateral tilt of TPP
Y Helicopter viewed from the back of the pilot © Lakshmi Sankar 2002

Pilot Input Lateral control Collective Longitudinal control
The pitch is always, by convention Specified at 75%R. The pitch at all the other radial locations may be found If we know the linear twist distribution. The pilot applies a collective pitch by vertically raising the swash plate up or down. All the blades collectively, and equally pitch up or down. The pilot applies longitudinal control (i.e. tilts the TPP fore and aft) by Tilting the swash plate fore or aft as discussed earlier. Lateral control means tilt the swash plate (and the TPP) laterally. © Lakshmi Sankar 2002

Longitudinal control Line Parallel to NFP Blade at 90 deg azimuth b1c
q1s X q1s NFP Note that q1s+ b1c is independent of the coordinate system in which these angles are measured. © Lakshmi Sankar 2002

Lateral Control Line parallel to NFP Blade at =180 b1s q1c TPP NFP
Y Note that q1c-b1s is independent of the coordinate system in which q1c And b1s were measured. © Lakshmi Sankar 2002

As a result.. © Lakshmi Sankar 2002

In the future.. We will always see b1c+q1s appear in pair.
We will always see b1s-q1c appear in pair. As far as the blade sections are concerned, to them it does not matter if the aerodynamic loads on them are caused by one degree of pitch that the pilot inputs in the form of q1c or q1s, or by one degree of flapping (b1c or b1s). One degree of pitch= One degree of flap. © Lakshmi Sankar 2002

Angle of Attack of the Airfoil And Sectional Load Calculations
Forward Flight Angle of Attack of the Airfoil And Sectional Load Calculations © Lakshmi Sankar 2002

The angle of attack of an airfoil depends on
Pilot input: collective and cyclic pitch How the blade is twisted Inflow due to freestream component, and induced inflow Velocity of the air normal to blade chord, caused by the blade flapping Anhedral and dihedral effects due to coning of the blades. The first two bullets are self-evident. Let us look at the other contributors to the angle of attack. © Lakshmi Sankar 2002

Blade Flapping Effect r db/dt © Lakshmi Sankar 2002

Angle of Attack caused by the Inflow and Freestream
Edgewise component V∞sina+v Normal Component Rotor Disk a V∞ Induced inflow, v Wr+V∞cosa © Lakshmi Sankar 2002

Anhedral/Dihedral Effect
V∞ V∞sinb V∞sinb Blade at =0 will See an upwash equal to V∞sinb V∞cosaV∞ Blade at =180 will See an upwash equal to V∞sinb Blade at any  will see an upwash equal to - V∞sinb cos  © Lakshmi Sankar 2002

Summing them all up.. Zero lift line a q UP UT= Wr+V∞cosa

Small Angle of Attack Assumptions
The angle of attack a (which is the angle between the freestream and the rotor disk) is small. The cyclic and collective pitch angles are all small. The coning and flapping angles are all small. Cos(a) = Cos(b)= Cos(q) ~ 1 sin(a) ~ a, sin(b) ~ b, sin(q) ~ q © Lakshmi Sankar 2002

Angle of Attack Subscript s: All angles are in the shaft plane

Angle of attack (continued)

Angle of Attack (Continued)
After some minor algebra, Notice b1c+q1s appears in pair, as pointed out earlier. Also q1c-b1s appears in pairs. One degree of pitching is equivalent to one degree of Flapping. © Lakshmi Sankar 2002

Relationship between aTPP and as
V∞ as b1c TPP aTPP=as+b1c © Lakshmi Sankar 2002

Angle of Attack (Concluded)

Calculation of Sectional Loads
Once the angle of attack at a blade section is computed as shown in the previous slide, one can compute lift, drag, and pitching moment coefficients. This can be done in a number of ways. Modern rotorcraft performance codes (e.g. CAMRAD) give the user numerous choices on the way the force coefficients are computed. © Lakshmi Sankar 2002

Some choices for Computing Sectional Loads as a function of a
In analytical work, it is customary to use Cl=aa, Cd=Cd0 = constant, and Cm= Cmo, a constant. Here “a” is the lift curve slope, close to 2p. In simple computer based simulations using Excel or a program, these loads are corrected for compressibility using Prandtl-Glauert Rule. More sophisticated calculations will use C-81 tables, with corrections for the local sweep angle. © Lakshmi Sankar 2002

Prandtl-Glauert Rule Zero lift line V∞ a q UP UT= Wr+V∞cosa ψ
Compute Mach number=M= UT/a∞ X © Lakshmi Sankar 2002

More accurate ways of computing loads
In more sophisticated computer programs (e.g. CAMRAD-II, RCAS), a table look up is often used to compute lift, drag, and pitching moments from stored airfoil database. These tables only contain steady airfoil loads. Thus, the analysis is quasi-steady. In some cases, corrections are made for unsteady aerodynamic effects, and dynamic stall. It is important to correct for dynamic stall effects in high speed forward flight to get the vibration levels of the vehicle correct. © Lakshmi Sankar 2002

Calculation of Sectional Forces
L’=Lift per foot of span After Cl, Cd, Cm are found, one can find the lift, drag, and pitching moments per unit span. These loads are normal to, and along the total velocity, and must be rotated appropriately. Zero lift line a D’ q UT= Wr+V∞cosa © Lakshmi Sankar 2002

Integration of Sectional Loads To get Total loads
Forward Flight Integration of Sectional Loads To get Total loads © Lakshmi Sankar 2002

Background In the previous sections, we discussed how to compute the angle of attack of a typical blade element. We also discussed how to compute lift, drag, and pitching moment coefficients. We also discussed how to compute sectional lift and drag forces per unit span. We mentioned that these loads must be rotated to get components normal to, and along reference plane. In this section, we discuss how to integrate these loads. In computer codes, these integrations are done numerically. Analytical integration under simplifying assumptions will be given here to illustrate the process. © Lakshmi Sankar 2002

Assumptions for Analytical Integration
c= constant (untapered rotor) v = constant (uniform inflow) Cd = constant Linearly twisted rotor No cut out, no tip losses. © Lakshmi Sankar 2002

Blade Section Zero lift line a q UP UT= Wr+V∞cosa

Effective Angle of Attack
As discussed earlier, © Lakshmi Sankar 2002

Some algebra first.. Notice that we have first, second, and third harmonics present! These fluctuations will be felt by the passengers/pilots as vibratory loads. © Lakshmi Sankar 2002

Thrust Thrust is computed by integrating the lift radially to get
instantaneous thrust force at the hub, then averaging the thrust force over the entire rotor disk, and multiplying the force per blade by the number of blades. Computer codes will do the integrations numerically, without any of the assumptions we had to make. © Lakshmi Sankar 2002

Anlaytical Integration of Thrust
We can interchange the order of integration. Integrate with respect to ψ first. Use the formulas such as © Lakshmi Sankar 2002

Result of Azimuthal Integration

Next perform radial integration and Normalize
Note that we will get the hover expressions back if advance ratio m is set to zero. © Lakshmi Sankar 2002

Torque and Power We next look at how to compute the instantaneous torque and power on a blade. These are azimuthally-averaged to get total torque and total power. It is simpler to look at profile and induced components of torque are power separately. © Lakshmi Sankar 2002

Induced Drag Zero lift line a q UP UT= Wr+V∞cosa L’ Di

Induced Torque and Power
Performing the analytical integration, This is a familiar result. Induced Power = Thrust times Induced Velocity! © Lakshmi Sankar 2002

In-Plane Forces In addition to thrust, that act normal to the rotor disk (or along the z-axis in the coordinate selected by the user), the blade sections generate in-plane forces. These forces must be integrated to get net force along the x- axis. This is called the H-force. These forces must be integrated to get net forces along the Y- axis. This is called the Y-force. These forces will have inviscid components, and viscous components. © Lakshmi Sankar 2002

Origin of In-Plane Forces
L’sinb=L’b L’ L’b Y b X One source of in-plane forces is the tilting of the Thrust due to the blade coning angle. © Lakshmi Sankar 2002

Origin of In-Plane Forces-II
V∞ Radial flow causes radial Skin friction forces A component of the free-stream flows along the blade, in the radial direction. This causes radial skin friction forces. This is hard to quantify, and is usually neglected. © Lakshmi Sankar 2002

Origin of In-Plane Forces III
D=Di+D0 Sectional drag (which is made of inviscid induced drag, and viscous drag) Can give rise to components along the X- direction (H-force), and Y- direction (Y-Force). Engineers are interested in both the instantaneous values (which determine Vibration levels), as well as azimuthal averages (which determine force balance). © Lakshmi Sankar 2002

Closed Form Expressions for CH and CY
Under our assumptions of constant chord, linear twist, linear aerodynamics, and uniform inflow, these forces may be integrated radially, and averaged azimuthally. The H- forces and the Y- forces are non-dimensionalized the same way thrust is non-dimensionalized. Many text books (e.g. Leishman, Prouty) give exact expressions for these coefficients. © Lakshmi Sankar 2002

Closed Form Expressions

Background In the previous sections, we developed expressions for sectional angle of attack, sectional loads, total thrust, torque, power, H-force and Y-force. These equations assumed that the blade flapping dynamics a priori. The blade flapping coefficients are determined by solving the ODE that covers flapping. © Lakshmi Sankar 2002

Solution Process Plug in the solution on both the left and right sides. The right side Can be integrated analytically, subject to usual assumptions. Equate coefficients on the left side and right, term by term. For example coefficient with sinψ on the left with the similar term on right. © Lakshmi Sankar 2002

Calculation of Trim Conditions Including Fuselage Aerodynamics
Level Flight Calculation of Trim Conditions Including Fuselage Aerodynamics © Lakshmi Sankar 2002

Background By trim conditions we mean the operating conditions of the entire vehicle, including the main rotor, tail rotor, and the fuselage, needed to maintain steady level flight. The equations are all non-linear, algebraic, and coupled. An iterative procedure is therefore needed. © Lakshmi Sankar 2002

Horizontal Force Balance
HT HM DF V∞ Total Drag= Fuselage Drag (DF) + H-force on main rotor (HM) + H-force on the tail rotor (HT) © Lakshmi Sankar 2002

Fuselage Lift and Drag These are functions of the fuselage geometry, and its attitude (or angle of attack). This information is currently obtained from wind tunnel studies, and stored as a data-base in computer codes. © Lakshmi Sankar 2002

Fuselage Angle of Attack
Extracted from Tip path angle Blade flapping dynamics Downwash felt by the fuselage from the main rotor Shaft inclination angle. © Lakshmi Sankar 2002

Relationship between Tip Path Plane Angle of Attack and Shaft Angle of Attack
V∞ TPP as b1c Shaft Plane aTPP © Lakshmi Sankar 2002

Angle of Attack of the Fuselage
Start with tip path plane angle of attack. Subtract b1c to get shaft angle of attack Subtract the inclination of the shaft Subtract angle of attack reduction associated with the downwash from the rotor © Lakshmi Sankar 2002

Iterative process Assume angle of attack for fuselage (zero deg).
Find LF and DF from wind tunnel tables. Compute needed T. during the first iteration, T is approximately GW-LF. Use this info. to find main rotor torque, main rotor H-force, tail rotor thrust needed to counteract main rotor torque, and tail rotor H- force. From blade trim equations, find b1c. Find tip path plane angle of attack. Recompute fuselage angle of attack. When iterations have converged, find main and tail rotor power. Add them up. Add transmission losses to get total power needed. © Lakshmi Sankar 2002

Autorotation in Forward Flight
The calculations described for steady level flight can be modified to handle autorotative descent in forward flight. Power needed is supplied by the time rate of loss in potential energy. © Lakshmi Sankar 2002

Descent V∞ HM HT LF DF Wsinχ W cosχ χ Rate of descent

Tip Path Plane Angle in Descent

Iterative Procedure The iterative procedure involves
assume a rate of descent Iterate on fuselage angle of attack to achieve forces to balance, as done previously in steady level flight. Compute the power needed to operate= main rotor+ tail rotor+ transmission losses. Equate this power needed with the power available from loss of potential energy= GW * Rate of descent. Iterate until power needed = power available © Lakshmi Sankar 2002

Steady, Level Forward Flight

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