TRANSPIRE DYNAMICS · FLIGHT DYNAMICS CONCEPT SERIES · PART I SIKORSKY UH-60 BLACK HAWK TRANSPIREDYNAMICS.COM

How a Black Hawk
stays in the air

The rotor drives air down; the reaction holds the aircraft up. This simulator shows you how — from first principles, in real time. Fly a simplified UH-60 right here in your browser, watch the forces at work in every phase of flight, and (if you want them) open the full engineering derivations underneath.

No background required Live physics simulator Engineer mode: derivations inside
▶ Start the simulator
THRUST T WEIGHT W DRAG D DOWNWASH vᵢ MAIN ROTOR · 4 BLADES · R = 26.8 FT
SEC 01

The rotor is a wing that never stops

An airplane bolts its wings on and drags them through the air with engines. A helicopter spins its wings instead — four blades, each one a long thin wing, sweeping a 54-foot circle over the cabin about 4.3 times every second. Spinning wings make lift even when the helicopter itself is standing still. That single trick is what buys hovering — and it creates almost every quirk of helicopter flight you'll meet below.

Collective — up & down

MORE LIFT MORE DOWNWASH

The lever in the pilot's left hand. Pull it up and every blade tilts to a steeper bite at the same time ("collectively"). Steeper bite → more lift → the helicopter rises. Lower it to descend. It's the throttle of vertical motion — the engines automatically feed in fuel to keep the rotor speed constant.

Simulator: slider or W/S.

Cyclic — tilt to travel

THRUST TILTS …SO IT MOVES

The stick in the right hand. Helicopters don't have a "go forward" engine — they lean their lift. Push the cyclic forward and each blade's bite is changed once per revolution ("cyclically"), tilting the whole rotor disk. Part of the thrust now points forward and drags the aircraft along with it.

Simulator: slider or /.

Pedals — fighting the twist

ROTOR: CCW BODY SPINS BACK TAIL ROTOR

Newton's third law never sleeps: the engines twist the rotor one way, so the rotor twists the fuselage the other way. The small tail rotor pushes sideways at the end of a long boom to cancel that twist. Foot pedals adjust its bite — that's how the pilot points the nose, and why losing a tail rotor is a genuine emergency.

Simulator: pedal slider or A/D.

For engineers — where rotor thrust comes from: momentum theory

Model the rotor as an actuator disk of area \(A=\pi R^2\): an infinitely thin disk that adds momentum to the air passing through it. Far above, air is at rest; at the disk it has induced velocity \(v_i\); far below it has accelerated to the wake velocity \(w\). Mass flow through the disk is \(\dot m = \rho A v_i\), and thrust equals the rate of momentum added:

\[ T = \dot m\, w = \rho A v_i w \]

Bernoulli applied above and below the disk (the disk adds a pressure jump \(\Delta p = T/A\)) gives \(w = 2 v_i\) — the wake ends up moving twice as fast as the air at the disk. Substituting:

\[ T = 2\rho A v_i^2 \quad\Longrightarrow\quad v_i = v_h = \sqrt{\frac{T}{2\rho A}} \]

For a UH-60 at a typical mission weight \(m \approx 7{,}700\,\mathrm{kg}\) (so \(T = mg \approx 75.5\,\mathrm{kN}\)), \(R = 8.18\,\mathrm m\), \(A \approx 210\,\mathrm{m^2}\), sea level \(\rho = 1.225\,\mathrm{kg/m^3}\):

\[ v_h = \sqrt{\frac{75{,}500}{2(1.225)(210)}} \approx 12.1\ \mathrm{m/s} \approx 40\ \mathrm{ft/s} \approx 24\ \mathrm{kt\ of\ downwash} \]

The ideal power to hover is thrust times induced velocity, \(P_i = T v_h\); real rotors need \(\kappa \approx 1.15\) more for tip losses and non-uniform inflow, plus profile power to drag the blades through the air (Sec 06). This is why heavy + high + hot is the enemy: \(v_h \propto \sqrt{T/\rho}\), so power climbs steeply with weight and altitude.

SEC 02

Flight simulator — fly the physics

A real longitudinal flight-dynamics model of a UH-60 (momentum-theory rotor, drag, torque, rotor-speed dynamics) running live. Use the phase buttons for guided demonstrations, or just grab the controls. You can crash — gently, please.

UH-60A · LONGITUDINAL MODEL · ρ = 1.225 kg/m³ · GW 17,000 LB
Feedback v1.17.11

Disclaimer

This is an independent educational concept demonstrator by Transpire Dynamics LLC. It is not affiliated with, endorsed by, or certified by the U.S. Army, Sikorsky, Lockheed Martin, or the U.S. Department of Defense.

All aerodynamic parameters are derived from publicly available, open-domain literature. Models are physics-based and derived from first principles, but optimized for real-time computation and not intended for full engineering analysis. High-fidelity work requires dedicated offline simulation tools.

transpiredynamics.com →

Phase demonstrations — press one, then watch the message bar
KEYS W / S — pull up to climb
KEYS ◀ / ▶ — forward = nose down = accelerate
KEYS A / D — keep the nose from swinging
Press START / RESET, raise the collective slowly past ~55%, and the wheels will leave the ground.

The fine print: this is a 2-D teaching model — real Black Hawks add lateral dynamics, flapping cross-couplings, a stabilator schedule and a flight-control computer. The trends you feel here (power bucket, ETL shudder, autorotation energy trade, VRS escape) are the real ones.

SEC 03

The power curve — why forward flight is cheap

Live chart — the amber marker is the simulator's current airspeed. Curves computed from the same model that flies the sim.

Hovering is the most expensive thing a helicopter does. Standing still, the rotor must hurl air downward hard to hold the aircraft up, and all that acceleration costs power ("induced power", the red curve).

Start moving forward and the rotor constantly meets fresh, undisturbed air — like a shovel scooping from a new pile instead of re-shoveling the same one. Induced power collapses. Meanwhile fuselage drag ("parasite", blue) grows with speed cubed. Add the cost of simply spinning the blades ("profile", green) and the total forms a bucket: a valley around 70–80 knots where the UH-60 flies on roughly half its hover power.

That bucket speed is where you get maximum endurance, best rate of climb, and the safest place to be if an engine quits.

Translational liftCrews feel the left side of this curve as a little shudder and surge around 16–24 kt — effective translational lift (ETL). The rotor outruns its own recirculating downwash and suddenly performs better. Try the takeoff demo and watch torque drop as speed builds.
For engineers — deriving the power bucket

Total shaft power in steady forward flight at speed \(V\):

\[ P = \underbrace{\kappa T v_i}_{\text{induced}} \;+\; \underbrace{\tfrac{\sigma C_{d0}}{8}\,\rho A (\Omega R)^3\bigl(1+4.6\mu^2\bigr)}_{\text{profile}} \;+\; \underbrace{\tfrac12 \rho V^3 f}_{\text{parasite}} \]

Induced term — Glauert's extension

In forward flight the induced velocity satisfies the implicit momentum relation (rotor treated as a circular wing):

\[ v_i = \frac{v_h^2}{\sqrt{(V\cos\alpha)^2 + (V\sin\alpha + v_i)^2}} \]

Solved by fixed-point iteration in the sim each frame. For \(V \gg v_h\) it tends to \(v_i \approx v_h^2/V\) — induced power decays like \(1/V\), which is the falling red curve.

Profile term

From blade element theory (Sec 06), drag of the blades themselves; \(\sigma = N_b c/\pi R \approx 0.083\) is rotor solidity, \(C_{d0}\approx 0.01\) mean blade drag coefficient, and \(\mu = V/\Omega R\) the advance ratio. The \((1+4.6\mu^2)\) factor (Stepniewski) accounts for the asymmetric velocity field sweeping the disk.

Parasite term

\(f \approx 3\,\mathrm{m^2}\) equivalent flat-plate area for the UH-60's fuselage, hub and gear. Cubic in \(V\) — the right-hand wall of the bucket and the ultimate speed limiter (together with retreating-blade stall, Sec 05).

Minimum-power speed follows from \(dP/dV = 0\); with the induced term \(\propto 1/V\) and parasite \(\propto V^3\), \(V_{mp} \approx (2 v_h^2\, \kappa T /(3\rho f))^{1/4}/\!\sqrt[4]{1}\) — order 35 m/s ≈ 70 kt for these numbers, matching the chart.

SEC 04

The phases of flight

Every helicopter sortie moves through different physical regimes. Each card explains one in plain language; each has a matching demo button in the simulator above.

Hover

Thrust exactly balances weight; the rotor recycles its own column of disturbed air, so this is peak power demand. Near the ground (within about one rotor diameter) the downwash piles up against the surface and forms a supporting cushion — ground effect — trimming the power bill by up to ~10–15%. Pilots hold a hover with constant tiny corrections; it's famously like balancing on a beach ball.

Takeoff & translational lift

Lift the collective, ease the cyclic forward, and as the aircraft accelerates through roughly 16–24 knots the rotor escapes its recirculating wake. The disk bites clean air, efficiency jumps, the nose shudders slightly, and the helicopter wants to climb on the same power. That free performance is ETL — effective translational lift — and crews plan takeoffs to reach it as early as possible.

Cruise

In the power bucket (~70–145 kt for a UH-60, max cruise about 150 kt) the helicopter is at its most airplane-like: disk tilted forward, fuselage slightly nose-down, stabilator trimming the tail. Speed is capped not by engine power alone but by the rotor itself — the retreating blade running out of airspeed (Sec 05) and the advancing tip flirting with the speed of sound.

Approach & landing

The reverse trip down the power curve — and the dangerous direction, because slowing down while descending steers you toward the vortex ring corner (below). Standard technique: keep some forward speed all the way down, decelerate near the ground, then trade the last of the speed for a cushioning pull of collective just before touchdown.

Autorotation — the engine-out glide

Helicopters do not fall when engines fail. Drop the collective and the upward rush of air through the descending rotor spins the blades like a sycamore seed — the rotor becomes its own windmill, storing energy as RPM. The aircraft descends fast (~2,000+ ft/min) but under full control. Near the ground the pilot flares, then pulls the stored rotor energy as one final cushion of thrust. Practiced by every helicopter pilot, regularly.

Vortex ring state (settling with power)

Descend nearly straight down, with power, at a rate close to your own downwash speed, and the rotor re-ingests its own wake: a doughnut of recirculating air forms around the disk. Pulling more collective just feeds the doughnut — the helicopter sinks faster. Escape is counter-intuitive: push forward (or sideways) to fly into clean air, or chop the power. The simulator's VRS demo lets you feel both the trap and the escape.

For engineers — autorotation energy balance & VRS boundary

Autorotation

Steady autorotation is a power balance: the rate at which gravity supplies energy equals the rotor's total power demand (no shaft input):

\[ T\,V_d \;\approx\; W V_d \;=\; P_{induced} + P_{profile} + P_{parasite} \]

so the descent rate is \(V_d = P_{req}(V)/W\). Since \(P_{req}\) has its bucket near \(V_{mp}\), minimum-descent autorotation happens near bucket speed — for the UH-60, roughly 70–80 kt and on the order of 2,000–2,500 ft/min. Pure vertical autorotation costs more: momentum theory gives \(V_d \approx 1.8\,v_h \approx 22\ \mathrm{m/s} \approx 4{,}300\ \mathrm{ft/min}\).

Along the blade, the inflow angle tilts the lift vector forward in a mid-span driving region (torque-producing) and aft in tip/root driven regions (torque-consuming); equilibrium RPM is where they cancel. Collective is the RPM governor: too much pitch → decay, too little → overspeed. Rotor kinetic energy \(\tfrac12 I_R \Omega^2\) (UH-60: order 4–5 MJ) is the savings account spent in the landing flare.

Vortex ring state

Momentum theory breaks down when descent rate and induced velocity are comparable — there is no clean slipstream direction. Empirically VRS occupies roughly

\[ 0.3 \lesssim \frac{V_d}{v_h} \lesssim 1.5 \quad\text{with}\quad \frac{V_{fwd}}{v_h} \lesssim 1 \]

With \(v_h \approx 12\,\mathrm{m/s}\) (≈40 ft/s), that's descent rates of ~700–3,500 ft/min at low airspeed — exactly the corner of a steep, slow approach. The sim models it as a thrust deficit plus stochastic buffeting inside that boundary, removed once \(V_{fwd}/v_h \gtrsim 1\) (the Vuichard-style escape).

SEC 05

Dissymmetry of lift — the rotor's speed limit

Forward speed 80 kt
Advancing blade (fast air) Retreating blade (slow air) Reverse-flow region

Look down on the rotor in forward flight (animation left — drag the slider). The helicopter's own speed adds to the blade swinging toward the nose on the right side, and subtracts from the blade retreating on the left. At 150 kt, the advancing tip slices air at ~580 kt while its retreating twin sees only ~290 kt — and near the left-side root, air actually flows backwards over the blade (the gray crescent).

Lift grows with speed squared, so untreated, the right side would lift far harder than the left and roll the aircraft over. The fix is elegant: blades are hinged to flap. The advancing blade, lifting harder, flaps up — which reduces its angle of attack; the retreating blade flaps down and bites harder. The rotor equalizes itself automatically, every revolution, ~4.3 times a second.

Push faster and the retreating blade eventually stalls — the true speed limit of every conventional helicopter, and the reason compound designs add wings and propellers.

For engineers — blade velocity field, flapping equation, gyroscopic 90° lag

Velocity over a blade element

For a blade at azimuth \(\psi\) (zero over the tail, advancing side \(\psi = 90^\circ\)), the in-plane tangential velocity at radius \(r\) is

\[ U_T(r,\psi) = \Omega r + V\sin\psi = \Omega R\left(\frac{r}{R} + \mu\sin\psi\right),\qquad \mu = \frac{V}{\Omega R} \]

UH-60 at 150 kt: \(\mu \approx 77/221 \approx 0.35\). Reverse flow (\(U_T<0\)) fills a circle of diameter \(\mu R\) on the retreating side — the gray region in the animation, drawn from this exact expression.

Flapping dynamics

A centrally hinged blade with flap angle \(\beta\) behaves as a pendulum in the centrifugal field. Taking moments about the flap hinge (inertia \(I_b\), aerodynamic moment \(M_A\)):

\[ I_b\ddot\beta + I_b\Omega^2\beta = M_A(\psi) \quad\Longrightarrow\quad \frac{d^2\beta}{d\psi^2} + \beta = \frac{M_A}{I_b\Omega^2} \]

The undamped natural frequency is exactly \(\Omega\) — 1/rev. The dissymmetry forcing is also 1/rev (it goes as \(\sin\psi\)). A resonant system driven at its natural frequency responds 90° after the forcing: maximum upward flap occurs not on the advancing side where lift peaks, but over the nose. This is the famous gyroscopic-like phase lag, and it's why cyclic control rigging applies blade pitch ~90° of rotation before the place you want the disk to tilt.

Why flapping cancels the roll

Flapping up at rate \(\dot\beta\) adds a downward relative wind component, changing local angle of attack by \(\Delta\alpha \approx -\dot\beta r/U_T\). The steady solution sets first-harmonic lift variation to zero: the rotor trims its own moment, leaving longitudinal flapping \(\beta_{1c}\) (disk tilt-back with speed) that the pilot counters with forward cyclic — the source of the speed-stability "stick gradient".

For engineers — blade element theory: thrust from geometry

Lift on a blade element of chord \(c\) at radius \(r\) (hover, small inflow angle \(\phi \approx \lambda R / r\), where \(\lambda = v_i/\Omega R\)):

\[ dL = \tfrac12\rho U_T^2\, c\, a\,(\theta - \phi)\, dr, \qquad U_T=\Omega r \]

with \(a \approx 5.7\,\mathrm{rad^{-1}}\) the lift-curve slope and \(\theta(r)\) the geometric pitch set by the collective. Summing \(N_b\) blades and integrating root to tip with constant chord and linear twist absorbed into an effective \(\theta_{.75}\):

\[ C_T \equiv \frac{T}{\rho A (\Omega R)^2} = \frac{\sigma a}{2}\left( \frac{\theta_{.75}}{3} - \frac{\lambda}{2} \right) \]

Pairing this with momentum theory's \(\lambda = \sqrt{C_T/2}\) closes the system: given a collective setting \(\theta_{.75}\), solve for \(C_T\) — which is precisely the mapping the simulator uses between your collective slider and rotor thrust. Numbers check: UH-60 hover needs \(C_T \approx 0.0065\), giving \(\theta_{.75} \approx 9.5^\circ\) — squarely in the real ship's rigging range.

SEC 06

The tail rotor — a sideways helicopter

Why it exists

The engines push ~1,500–3,000 horsepower of twist into the main rotor; the airframe receives the equal-and-opposite twist. The UH-60's 11-ft tail rotor, mounted on a 9.9 m arm, pushes sideways with up to several kN to cancel it. More collective → more main-rotor torque → more left pedal needed. It consumes 5–15% of total power purely as a tax on physics.

The canted trick

Look closely at a Black Hawk: its tail rotor is tilted 20° upward. The sideways thrust gains a vertical component — about 400 lb of free lift — letting Sikorsky shift the usable center-of-gravity range aft. A rare case of getting paid back some of the anti-torque tax.

When it fails

Loss of tail-rotor effectiveness (LTE) or a drive failure leaves the fuselage spinning against the rotor. The survival tools: forward airspeed (the vertical fin weather-vanes the nose straight) or, ultimately, chopping the power and autorotating — with no engine torque, there's nothing to cancel.

For engineers — sizing the anti-torque

Main-rotor torque follows directly from shaft power: \(Q = P/\Omega\). At a 1,500 kW hover with \(\Omega = 27\,\mathrm{rad/s}\):

\[ Q = \frac{1.5\times10^6}{27} \approx 55.6\ \mathrm{kN\,m} \]

Tail rotor thrust with moment arm \(l_t \approx 9.9\,\mathrm m\):

\[ T_{tr} = \frac{Q}{l_t} \approx 5.6\ \mathrm{kN} \;(\approx 1{,}260\ \mathrm{lb}) \]

That thrust costs its own induced power, \(P_{tr} = \kappa_{tr} T_{tr} \sqrt{T_{tr}/2\rho A_{tr}}\) with \(A_{tr} = \pi(1.68)^2 \approx 8.9\,\mathrm{m^2}\) — about 100–150 kW, the 5–15% tax. The 20° cant gives vertical lift \(T_{tr}\sin 20^\circ \approx 1.9\,\mathrm{kN}\) at hover power. Yaw control derives from \(\dot r = (T_{tr} l_t - Q)/I_{zz}\), which is the (simplified) relation behind the sim's heading tape and pedal slider.

SEC 07

UH-60 reference data — the numbers in this model

ItemValue usedWhy it matters
Main rotor radius R26.8 ft / 53.7 ft dia (8.18 m)Sets disk area A ≈ 2,260 ft² (210 m²) — bigger disk, cheaper hover
Blades / chord / solidity σ4 · 1.73 ft chord (0.53 m) · 0.083Blade area fraction; drives profile power & stall margin
Rotor speed Ω258 RPM (27 rad/s), tip 725 ft/s (221 m/s)Held nearly constant by the engines' governor
Engines2 × GE T700, ~1,900 shp ea.Hover takes ~60–75% of it; the rest is climb & speed margin
Model gross weight17,000 lb (7,700 kg)Typical mission weight; MTOW 22,000 lb
Hover downwash vh≈ 40 ft/s / 24 kt (12.1 m/s)Sets the VRS danger zone & ETL speed scale
Equivalent flat plate f≈ 32 ft² (3.0 m²)Fuselage drag; the V³ wall of the power bucket
Tail rotor11 ft dia, canted 20°, arm 32.5 ft (9.9 m)Anti-torque + ~400 lb of free lift
Max cruise / never-exceed~150 kt / 193 ktRetreating-blade stall & advancing-tip Mach limits

Values are public-domain figures rounded for teaching; the simulator derives everything else (vh, power curves, torque) from these via the equations in the engineer panels.