Helicopter - Wikiparency

# Helicopter > [[PORTAL_Aviation|Aviation]] · [[PORTAL_Avionics|Avionics]] spine.  ## Microsims — three.js ### Helicopter: dissymmetry of lift (three.js) <div class="microsim-player"> <iframe src="https://wikitube-3d-microsims.netlify.app/Helicopter.html" width="100%" height="620" frameborder="0" loading="lazy" sandbox="allow-scripts allow-same-origin" title="Helicopter: dissymmetry of lift — three.js microsim"></iframe> </div> **Open it full-screen:** [Helicopter.html](https://wikitube-3d-microsims.netlify.app/Helicopter.html) · library `threejs` · route `microsim/threejs/` ### Related microsims Live sims on neighbouring articles: - [[Aircraft_flight_dynamics]] - [[Fixed-wing_aircraft]] - [[Aircraft]] - [[Turbojet]] - [[Avionics]] - [[Air_traffic_control]] *Sim hosted off-article; the article owns the reference, not the runtime (WIKI_RULES §10.4).*  ## Overview A helicopter is a rotorcraft whose lift and propulsion both come from powered rotors turning about substantially vertical axes. The rotor is a wing, but a wing that supplies its own airspeed by spinning, which is why a helicopter can hover, climb vertically, fly sideways and fly backwards — and why it is limited in forward speed in a way no fixed-wing aircraft is. Every one of those abilities, and that limit, follows from a single geometric fact. In a hover, a blade section at radius *r* meets the air at speed Ωr regardless of where the blade is around the circle, so the rotor disc is axisymmetric and the lift it produces is symmetric about the shaft. Move the aircraft forward at speed *V* and that symmetry is destroyed. The blade sweeping into the oncoming air — the **advancing blade**, on the starboard side of a rotor that turns counter-clockwise seen from above, which is the American and Russian convention — meets the air at Ωr + *V*. The blade sweeping downwind on the other side — the **retreating blade** — meets it at Ωr − *V*. Because the lift of a section goes as the square of the speed it meets, a rotor rigidly bolted to its shaft would produce far more lift on one side of the disc than on the other. That is **dissymmetry of lift**, and it is not a small effect: the microsim above computes a hub rolling moment of roughly 150 kN m at 40 knots and 370 kN m at 140 knots for a rotor of the size modelled, enough to roll the aircraft past ninety degrees of bank in well under a second. This is the wall every early experimenter hit. Rigidly mounted rotors either rolled the machine over or shook it to pieces. The aircraft that finally worked did so because Juan de la Cierva, developing his Autogiro in the early 1920s after a rigid-rotor prototype rolled over on take-off, hinged each blade at the root so it was free to **flap** up and down. Cierva's flapping hinge, flown successfully on the C.4 in January 1923, is the single most important mechanical idea in rotary-wing flight; every successful helicopter since — the Focke-Achgelis Fw 61 of 1936, Igor Sikorsky's VS-300 of 1939, and everything after — either uses a flapping hinge or a hingeless blade that bends to the same purpose. Because the rotor is driven by a shaft, the engine torque that spins it appears in equal and opposite measure on the airframe: left alone, the fuselage would rotate the other way. The conventional answer is a **tail rotor**, a small propeller on a boom thrusting sideways, whose moment about the main rotor axis cancels the main rotor's torque reaction and, being under the pilot's feet, also provides directional control in the hover. It typically absorbs of the order of ten per cent of installed power in the hover and brings a family of hazards: strike risk on the ground, loss of tail rotor effectiveness in certain crosswinds, and a single-point failure that leaves the aircraft with no yaw control. **NOTAR** (NO TAil Rotor), developed by Hughes and McDonnell Douglas and flown in production on the MD 520N, MD 600N and MD 902, replaces it with a fan that pressurises the tail boom; air bled through longitudinal slots makes the main rotor downwash follow the Coandă effect around the boom's circular section and deflect into a sideforce, while a rotatable direct-jet nozzle at the tip supplies the rest and the yaw control. Other answers exist: coaxial contra-rotating rotors (Kamov, Sikorsky X2), tandem rotors (Boeing CH-47), intermeshing "synchropter" rotors (Kaman K-MAX), and tip-jet drive, which produces no shaft torque at all. The microsim models a conventional single-main-rotor machine of medium size. Its assumed rotor is **radius *R* = 8.18 m** (16.36 m diameter), four blades of 0.53 m chord, giving a solidity σ = *N*b*c*/π*R* = 0.0825. At the nominal **258 rotor rev/min** the tip speed Ω*R* is **221 m/s (725 ft/s)**, a tip Mach number of 0.65 in ISA sea-level air. These are the dimensions and tip speed of a Sikorsky UH-60-class utility helicopter, although the blade itself is idealised as rectangular with a linear −10° washout and a single thin symmetric section. The design gross weight is taken as 11.0 t, a disc loading of 513 N/m² and a blade loading *C*T/σ of 0.104 — a well-loaded but entirely ordinary operating point. ## The physics ### The velocity field seen by a blade Fix an azimuth angle ψ measured from the downstream (tail) position and increasing in the direction of rotation, so that ψ = 90° is the advancing side and ψ = 270° the retreating side. Non-dimensionalise radius by *R* and all velocities by the tip speed Ω*R*, and define the **advance ratio** > μ = *V* / (Ω*R*) which is the single number that governs edgewise rotor aerodynamics. The tangential (chordwise) velocity at station *x* = *r*/*R* is then > *u*T = *x* + μ sin ψ and this is where everything begins. At the tip, *u*T runs from 1 + μ on the advancing side to 1 − μ on the retreating side. At the reference rotor at 140 knots, μ = 0.33, so the advancing tip meets the air at 1.33 Ω*R* = 293 m/s (Mach 0.86) while the retreating tip meets it at 0.67 Ω*R* = 148 m/s (Mach 0.44). Dynamic pressure differs between the two by a factor of (1.33/0.67)² ≈ 3.9. A further consequence appears wherever *u*T goes negative. Setting *x* + μ sin ψ < 0 and converting to Cartesian disc coordinates gives a circle of diameter μ*R* lying tangent to the hub on the retreating side: the **reverse-flow region**, inside which the air meets the blade from the trailing edge. It occupies a fraction μ²/4 of the disc — three per cent at μ = 0.35 — and the microsim draws it as a violet disc that grows as the airspeed comes up. ### The rolling moment a rigid rotor would produce The lift per unit span of a section is d*L*/d*r* = ½ρ*U*²*c C*l, and the rolling moment about the aircraft's longitudinal axis is the lateral moment arm *r* sin ψ integrated over span and averaged over azimuth: > *M*x = −(*N*b/2π) ∫₀²π ∫₀ᴿ (d*L*/d*r*) *r* sin ψ d*r* dψ For a hovering rotor this integral vanishes by symmetry. In forward flight, with the blade pitch fixed, it does not, because the *u*T² weighting is much larger on the advancing side. The minus sign records that lift on the starboard side rolls the aircraft to port — **towards the retreating side**, which is the direction every textbook and every accident report gives. Lock the flapping hinge in the microsim and this integral is exactly what the heads-up display reports. ### The flapping hinge Hinge the blade at the root so it can rotate in a vertical plane through the shaft, and it becomes a pendulum whose restoring force is not gravity but centrifugal force. For an idealised blade hinged on the axis with no hinge offset and no spring, the equation of motion is > *I*b β̈ + *I*b Ω² β = *M*aero where β is the flap angle, *I*b the blade's flapping moment of inertia about the hinge and *M*aero the aerodynamic flapping moment. Divide through by *I*b Ω² and change the independent variable from time to azimuth ψ = Ω*t*. Introducing the **Lock number** > γ = ρ *a*₀ *c R*⁴ / *I*b which measures aerodynamic forces against inertial ones and lies near 8 for an articulated metal rotor, the equation collapses to the form the microsim integrates: > d²β/dψ² + β = (γ / 2*a*₀) ∫₀¹ *u*² *C*l *x* d*x* Two structural facts fall straight out of the left-hand side. First, the coefficient of β is exactly 1, so the **natural frequency of flapping is exactly one per revolution**, set by centrifugal stiffening alone and therefore independent of blade weight, rotor speed and material. Second, an oscillator driven at its own natural frequency responds with a **90° phase lag**. The peak aerodynamic forcing occurs where the velocity is highest, at ψ = 90° on the advancing side; the peak flapping *displacement* therefore occurs a quarter of a turn later, over the nose. This is the phase lag that rotor rigging and swashplate geometry exist to accommodate, and it is why a pilot's control input appears on the disc ninety degrees around from where the blade pitch actually changed. ### Why flapping cures the dissymmetry The cure is in the velocity normal to the disc. Including the flapping motion, the through-disc velocity at a section is > *u*P = λ + *x* (dβ/dψ) + μ β cos ψ where λ is the inflow ratio. The middle term is the whole mechanism. A blade flapping upwards has an upward velocity *r*β̇ at each station, which appears to the section as an extra *downward* component of relative wind, which reduces the inflow angle's complement and hence the **angle of attack** > α = θ − arctan(*u*P / *u*T) So the advancing blade, presented with excess lift, does not transmit it to the hub as a rolling moment; it simply flaps up, loses angle of attack, and sheds the excess. The retreating blade flaps down, gains angle of attack, and makes up its share. The rotor trims itself, continuously, once per revolution, with no pilot input and no feedback loop. The same *x*β̇ term provides the flap damping — approximately γ/8 per unit β̇, a damping ratio of γ/16 ≈ 0.5 — which is why the motion settles within a couple of revolutions rather than ringing. The resulting motion is described to good accuracy by its first three harmonics: > β(ψ) = β₀ − β₁c cos ψ − β₁s sin ψ β₀ is the **coning** angle, the steady cone the blades take up under the balance of lift and centrifugal force; the microsim's reference rotor cones at about 6° in the hover. β₁c is the **longitudinal flapping**, the aft tilt of the tip-path plane commonly called **blowback**, which grows from zero in the hover to about 10° at 190 knots with no cyclic applied — the reason a helicopter accelerated with collective alone pitches nose-up and decelerates. β₁s is the **lateral flapping**, which tilts the disc towards the advancing side and arises mainly from the coning interacting with forward speed and from the fore-and-aft variation of the induced inflow. ### Flapping and feathering are the same thing There is a deep and useful equivalence here. Flapping and cyclic feathering differ only by the choice of reference plane. Adding a cyclic pitch θ₁s sin ψ produces exactly the same disc tilt that a corresponding amount of flapping would; conversely, viewed from a plane that follows the tip path, a rotor which is flapping is not flapping at all. This is why the pilot's cyclic stick, which tilts a swashplate and thereby feathers each blade once per revolution, can command the tip-path plane to point wherever the pilot wants, and why the microsim's cyclic slider and its flapping display are two views of one phenomenon. In steady forward flight the pilot holds forward cyclic to cancel blowback and keep the disc where it is wanted. ### The inflow The through-disc velocity λ is not a free parameter. Glauert's extension of momentum theory to edgewise flight gives > λ = μ tan α_disc + *C*T / (2 √(μ² + λ²)) where the first term is the freestream passing through the tilted disc and the second is the induced inflow. In forward flight the disc must tilt nose-down by roughly *D*/*T* to pull the aircraft against parasite drag; the oncoming air then enters the disc from above, so that term adds to the downwash. (The bookkeeping checks: rotor power *T*(*V* sin τ + *v*i) is then exactly parasite power *D V* plus induced power *T v*i.) The microsim solves this with an assumed equivalent flat-plate drag area *f* = 2.3 m², a realistic figure for a utility fuselage, and superimposes the linear fore-and-aft and lateral inflow gradients of the Drees (1949) wake model, which is what makes the computed lateral flapping come out at a believable magnitude. The inflow is distributed radially like √(*r*/*R*) rather than assumed uniform. This matters more than it sounds: a uniform inflow makes the inflow angle λ/*x* diverge at the root and hands the inboard stations a spurious *negative* angle of attack. The √*x* shaping reproduces the closed-form blade-element momentum result for a linearly twisted blade to within a few per cent while leaving the disc total unchanged. ### Retreating blade stall, and why it is a hard speed limit Flapping solves the rolling moment, but it cannot conjure dynamic pressure. The retreating blade must still carry its share of the aircraft's weight, and it must do so at a local dynamic pressure that falls as (*x* − μ)². The only variable left is angle of attack, and it must rise roughly as 1/(*x* − μ)². Push the speed up far enough and the retreating blade runs out of angle of attack: the section stalls. Two effects sharpen the limit. The stall angle itself is not a constant. Below about Mach 0.3 a thin symmetric section such as the NACA 0012 stalls near 14°; by Mach 0.75 shock-induced separation has pulled that down to roughly 8°. And at the same time the *advancing* tip is running towards its own compressibility limit: at 190 knots the microsim reports an advancing tip Mach number of 0.94, well into drag divergence and the region where shock noise and vibratory loads become the design driver. The conventional helicopter is squeezed between the two, and the practical ceiling on advance ratio is about μ = 0.4. The microsim marks stalled sections in red on the disc and reports the stalled fraction. Holding the reference rotor at 11.0 t, nothing stalls up to about 90 knots, stall begins in the outboard retreating quadrant near 110 knots, reaches 4 per cent of the working disc at 140 knots, 12 per cent at 165 knots and 28 per cent at 190 knots. Real onset arrives earlier at high gross weight, at high density altitude and in manoeuvres, since all three raise the blade loading *C*T/σ that the retreating blade must produce. The symptoms a pilot feels are characteristic: a low-frequency vibration at the blade-passage frequency, a pitch-up, and a roll towards the retreating side, all of which are relieved by lowering collective, reducing speed and increasing rotor rpm — which is exactly the set of levers the microsim exposes. Escaping the limit means changing the geometry, not refining it. The advancing-blade-concept coaxial rotor, flown on the Sikorsky XH-59A and revived on the X2, S-97 Raider and SB-1 Defiant, unloads the retreating side of each rotor entirely and lets the two advancing blades, on opposite sides, carry the roll balance between them. Compound helicopters add a wing to offload the rotor and a propulsor to supply thrust, as on the Eurocopter X³. Tilt-rotors sidestep the problem by converting to a propeller-driven aeroplane. ## Controls -> what each maps to | Control | Symbol | Range and units | Physical meaning | | --- | --- | --- | --- | | Forward airspeed | *V* | 0–200 kt (0–103 m/s) | True airspeed of the aircraft. Enters the model only through the advance ratio μ = *V*/(Ω*R*), shown live in the heads-up display. It sets the size of the velocity asymmetry, the diameter μ*R* of the reverse-flow region and, through parasite drag ½ρ*V*²*f*, the nose-down disc tilt needed for propulsion. | | Collective pitch | θ (at 0.75*R*) | 0–17 degrees | Blade pitch applied equally at all azimuths, quoted at the 75 per cent radius as is conventional. The root pitch is θ − 0.75 θtw with the linear washout θtw = −10°. Collective is the rotor's thrust lever: it moves *C*T and hence the lift the machine makes, and raising it into a high-μ condition is the fastest way to provoke retreating blade stall. | | Cyclic pitch | θ₁s | −8 to +8 degrees (longitudinal) | Once-per-revolution feathering, θ₁s sin ψ, applied by the swashplate. Positive values raise pitch on the advancing side and lower it on the retreating side; because of the 90° phase lag the disc responds a quarter-turn later, tilting aft. Negative (forward) cyclic is what a pilot holds in cruise to cancel blowback. | | Rotor speed | *N* (Ω) | 170–290 rev/min | Shaft speed, nominal 258 rev/min. Tip speed Ω*R* follows directly, and so do the advancing and retreating tip Mach numbers and the advance ratio. Reducing rpm raises μ at the same airspeed and forces a higher blade loading, bringing retreating blade stall on much earlier — which is why rotor rpm is a limit a pilot guards closely. | | Flapping hinge | β free / β locked | on / off | With the hinge free, β(ψ) is obtained by integrating the flapping equation round the azimuth. With it locked, β is held at the steady coning angle a very stiff blade would take, the rotor becomes rigid, and the rolling moment the hinge normally cancels appears at the hub and rolls the aircraft. This is the central experiment. | | Show stall region | — | on / off | Draws the red markers on every disc cell whose angle of attack exceeds the Mach-dependent static stall boundary, and the violet reverse-flow circle of diameter μ*R*. | | Hold gross weight | *W* = 11.0 t | on / off | When on, the collective is trimmed on every solve so that rotor thrust equals weight, which is what a pilot does. This is what makes airspeed a clean single variable: without it, thrust grows with speed at fixed collective and the stall onset is contaminated. Moving the collective slider by hand turns it off. | | Pause / play | — | button, or **P** | Freezes the blade animation so a single azimuth can be studied. The solution keeps updating. | | Reset | — | button, or **R** | Restores 100 kt, nominal rpm, free hinge, weight hold and level attitude. | | Camera | — | drag, wheel | Orbits and zooms. The load surface is a genuinely three-dimensional object and reads differently from above, from the side and from ahead. | Arrow keys duplicate the airspeed and collective sliders, and **H**, **S**, **P** and **R** duplicate the toggles and buttons, so the whole sim is reachable without a pointer. ## Learning objective **After using this microsim, the learner should be able to explain why a rigidly mounted rotor cannot fly forward — deriving the rolling moment from *u*T = *x* + μ sin ψ and the *u*T² dependence of section lift — to explain how the flapping hinge cancels that moment through the *x*β̇ term in the through-disc velocity, and to predict, from the same geometry, the airspeed at which the retreating blade must stall.** ## Limits and connections The model is a blade-element analysis with a rigid, centrally hinged blade, and everything it leaves out is worth naming. The blade is treated as **rigid in bending and torsion**, hinged exactly on the shaft axis with no hinge offset and no restraining spring. Real articulated rotors put the flap hinge several per cent of the radius outboard, which raises the flapping natural frequency slightly above 1/rev, reduces the phase lag below 90°, and lets the rotor transmit a genuine hub moment — which is where a helicopter's control power in pitch and roll comes from. Hingeless and bearingless rotors achieve the same by elastic bending. The sim therefore under-states control power and cannot distinguish articulated from hingeless handling. The aerodynamics are **quasi-steady and two-dimensional**: each section gets the lift coefficient it would have in steady flow at the local angle of attack and Mach number, with a Prandtl–Glauert correction to the lift-curve slope and a crude linear collapse past a Mach-dependent static stall angle. Real retreating blades experience **dynamic stall** — because the angle of attack is changing rapidly, a leading-edge vortex forms and sheds, lift briefly overshoots the static maximum by a large margin, and a violent nose-down pitching moment follows. Dynamic stall delays onset by several degrees and dominates the control loads once it starts. Marking stall by the static boundary therefore flags it a little early and says nothing about the torsional loads that actually limit the aircraft. The **inflow is prescribed**, not computed from the wake. A uniform Glauert value with the Drees linear gradient and a √*x* radial shape is a serviceable engineering model, but it cannot represent blade–vortex interaction, the rolled-up tip vortices that dominate low-speed descent, or the strongly non-uniform inflow near transition. There is no tip-loss factor, so lift is carried right to the tip and thrust is slightly optimistic. Lift in the **reverse-flow region is set to zero**; real sections there produce a small, usually negative lift with a sharp edge forward. The region is inboard where dynamic pressure is low, so the thrust error is small, but the sim says nothing useful about reverse-flow aerodynamics. Sections whose local speed is under a quarter of tip speed are excluded from the stall statistic for the same reason: their geometric angle of attack diverges as *u*T → 0, a real feature of the flow that carries no load. The trim is a **rotor trim, not an aircraft trim**. Thrust is set equal to weight and the disc is tilted enough to overcome parasite drag, but there is no tail-rotor sideforce, no fuselage download, no lateral cyclic, no pitch or yaw balance and no fuselage aerodynamics. The rolling moment reported with the hinge locked is the moment the rotor delivers to the hub, and the roll it produces is integrated against an order-of-magnitude roll inertia of 6.3 × 10³ kg m² purely so the departure can be animated; the animation runs, like the blades, at one twentieth of real time. In reality the departure takes a fraction of a second. Finally, the sim runs at ISA sea level with fixed density and speed of sound, so it cannot show the density-altitude sensitivity that is one of the main practical drivers of retreating blade stall, and its blade is rectangular and linearly twisted rather than swept, tapered and built from several aerofoils as modern rotors are. For the wider picture, [[Aircraft_flight_dynamics]] covers the six-degree-of-freedom rigid-body problem this rotor sits inside; [[Fixed-wing_aircraft]] gives the comparison case where the wing's airspeed is the aircraft's airspeed and none of this arises; [[Turbojet]] and [[Aircraft]] cover the powerplants and airframes; [[Avionics]] and [[Air_traffic_control]] cover the systems and the airspace in which rotorcraft operate. ## References - Abbott, Ira H.; von Doenhoff, Albert E. *Theory of Wing Sections, Including a Summary of Airfoil Data*. Dover Publications: New York, 1959. ISBN 978-0-486-60586-9. (Source of the NACA 0012 section characteristics idealised here.) - Bramwell, A. R. S.; Done, George; Balmford, David. *Bramwell's Helicopter Dynamics*, 2nd ed. Butterworth-Heinemann: Oxford, 2001. ISBN 978-0-7506-5075-5. - Coleman, Robert P.; Feingold, Arnold M.; Stempin, Carl W. *Evaluation of the Induced-Velocity Field of an Idealized Helicopter Rotor*. NACA Advance Restricted Report L5E10. National Advisory Committee for Aeronautics: Washington, DC, 1945. - Drees, Jan Meijer. "A Theory of Airflow Through Rotors and its Application to Some Helicopter Problems." *Journal of the Helicopter Association of Great Britain* **1949**, *3* (2), 79–104. - Federal Aviation Administration. *Helicopter Flying Handbook*, FAA-H-8083-21B. U.S. Department of Transportation, Flight Standards Service: Oklahoma City, 2019. - Gessow, Alfred; Myers, Garry C., Jr. *Aerodynamics of the Helicopter*. Macmillan: New York, 1952. (Reprinted by Frederick Ungar, 1967.) - Glauert, Hermann. *A General Theory of the Autogyro*. Aeronautical Research Committee Reports and Memoranda No. 1111. His Majesty's Stationery Office: London, 1926. - Harris, Franklin D. *Introduction to Autogyros, Helicopters, and Other V/STOL Aircraft. Volume I: Overview and Autogyros*. NASA SP-2011-215959. NASA Ames Research Center: Moffett Field, CA, 2011. - Johnson, Wayne. *Helicopter Theory*. Princeton University Press: Princeton, NJ, 1980. (Dover reprint 1994, ISBN 978-0-486-68230-3.) - Johnson, Wayne. *Rotorcraft Aeromechanics*. Cambridge Aerospace Series 36. Cambridge University Press: New York, 2013. ISBN 978-1-107-02807-4. - Leishman, J. Gordon. *Principles of Helicopter Aerodynamics*, 2nd ed. Cambridge Aerospace Series 18. Cambridge University Press: New York, 2006. ISBN 978-0-521-85860-1. (The blade-element, flapping and inflow formulations used in the microsim follow this text.) - McCroskey, William J. *The Phenomenon of Dynamic Stall*. NASA Technical Memorandum 81264. NASA Ames Research Center: Moffett Field, CA, 1981. - Padfield, Gareth D. *Helicopter Flight Dynamics: The Theory and Application of Flying Qualities and Simulation Modelling*, 2nd ed. Blackwell Publishing: Oxford, 2007. ISBN 978-1-4051-1817-0. - Prouty, Raymond W. *Helicopter Performance, Stability, and Control*. PWS Engineering: Boston, 1986. (Reprinted with additions by Krieger, 1995, ISBN 978-1-57524-209-4.) - Seddon, John; Newman, Simon. *Basic Helicopter Aerodynamics*, 3rd ed. John Wiley & Sons: Chichester, 2011. ISBN 978-0-470-66501-5. **On the spine:** [[Aircraft]] · [[Aircraft_flight_dynamics]] · [[Fixed-wing_aircraft]] · [[Helicopter]] · [[Turbojet]] · [[Jet_engine]] · [[Sonic_boom]] · [[Contrail]] · [[Air_traffic_control]] · [[Avionics]] · [[Aviation]].  **Part of the [[Aviation]] hub** — main article for section A10, *Rotorcraft*. Related sections: Unmanned aerial vehicle.  ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Helicopter) : [Wikitube](https://en.wikitube.io/wiki/Helicopter) --- *PORTAL_Aviation three.js batch · 2026-08-05 · sim staged in `_3d_deploy_stage/`.*