Helicopter Structure, Aerodynamics and Controls

How does a helicopter work? This article explains the components of a helicopter and how it flies. We provide a detailed look at helicopter aerodynamics and control principles – essential knowledge before your first training flight. For clarity, the information is divided into sections.


1. Helicopter Rotor System

The rotor system is the heart of the helicopter. The main rotor provides lift and thrust, while a smaller tail rotor usually counteracts the main rotor’s torque. Depending on design, there are different types of helicopter rotors:

  • Fully articulated rotors – with mechanical flapping and lead-lag hinges
  • Hingeless rotors – no mechanical hinges, use flexible rotor blade roots instead
  • Bearingless rotors – no conventional bearings at the rotor hub, use elastomeric components
  • Semi-rigid rotors – two-bladed teetering rotors (a subset of articulated rotors)

Fully Articulated Rotors

In a fully articulated rotor system, each rotor blade is attached to the hub via mechanical hinges: flapping hinges (allowing blades to move up and down) and lead-lag hinges (allowing back-and-forth movement in the plane of rotation). Each blade also has a pitch hinge or bearing that lets it twist to change its angle of attack (see Fig. 2). This design allows the advancing blade to flap up and the retreating blade to flap down in forward flight, compensating for the difference in lift between them. Early autogiros by Juan de la Cierva in the 1920s first introduced such hinge systems to achieve stable flight. The drawback of fully articulated rotors is their mechanical complexity and high maintenance requirements, as many moving parts must be inspected and serviced regularly.

Mechanical pitch bearing and hinges of a fully articulated rotor head
Fig. 2: Schematic of a fully articulated rotor head with flapping and lead-lag hinges and a pitch bearing.

Hingeless Rotors

A hingeless rotor system eliminates the traditional flapping and lead-lag hinges. Instead, the rotor blades are made of flexible materials at the root that allow them to bend appropriately (Fig. 3). This became possible with the advent of modern composites. The MBB Bo-105 was the first helicopter to use such a hingeless rotor system​:contentReference[oaicite:95]{index=95}. The blade’s elasticity takes over the role of hinges, resulting in a “rigid” rotor that offers high agility and reduced maintenance (fewer moving parts). However, because the blades flex to absorb motions, very high forces act at the blade root. Thus, purely hingeless rotors are typically feasible only on smaller helicopters​:contentReference[oaicite:96]{index=96}, where those forces are manageable. (Large rotorcraft often use advanced designs that still include some flexibility but may not be fully hingeless.)

Flexible rotor hub without mechanical hinges (hingeless rotor)
Fig. 3: Hingeless rotor head – the blade root flexes instead of using mechanical hinges.

Bearingless Rotors

A bearingless rotor is an evolution of the hingeless concept. It removes even the pitch bearings by using special elastomeric joints for all motions. No traditional bearings or hinges are used, which greatly reduces maintenance and wear​:contentReference[oaicite:97]{index=97}. The entire rotor hub relies on elastic deformation to accommodate blade flap, lag, and pitch change. This simplicity comes with the penalty of extremely high stress at the rotor hub; hence bearingless rotors with elastomeric components are so far mainly used on smaller helicopters​:contentReference[oaicite:98]{index=98}. One example is the Aérospatiale Spheriflex rotor. Bearingless rotors offer lower vibration and upkeep but require advanced materials to handle the loads.

A special case of articulated rotors is the semi-rigid rotor system used on two-bladed helicopters (e.g. the Bell UH-1 Huey). The two blades are mounted on a teetering hinge (seesaw) that allows them to flap as a unit. Lead-lag motion is accommodated by blade flex or underslinging, so separate drag hinges aren’t needed. Semi-rigid rotors are simple and lightweight but limited to two-blade configurations.


2. Helicopter Aerodynamics


Basic Aerodynamic Principles

A helicopter, like any aircraft, must produce lift to overcome its weight. Lift is generated by the rotor blades, which act like rotating wings. In steady flight, lift equals the weight of the helicopter. Helicopter rotor blades have an airfoil cross-section (profile). As a blade moves through the air, its curved upper surface makes the airflow travel faster over the top than along the bottom. According to Bernoulli’s principle, higher airspeed above the blade means lower pressure, creating an upward suction force​:contentReference[oaicite:99]{index=99}. Meanwhile, the blade’s slight angle of attack deflects air downward (Newton’s third law: action and reaction), which increases lift by pushing the aircraft up​:contentReference[oaicite:100]{index=100}. The combination of pressure difference (Bernoulli) and downward deflection (Newton) produces the lift needed to fly.

The faster the rotor spins (or the higher the blade’s angle of attack), the more lift is generated – up to a limit. If the blade’s angle of attack is too large or the airflow too slow, the airflow can separate from the blade’s upper surface, causing a stall and loss of lift. This is analogous to an airplane’s wing stalling at too high an angle or too low a speed​:contentReference[oaicite:101]{index=101}. When the blade stalls, lift drops dramatically and the helicopter cannot maintain altitude. Reducing the angle of attack (or increasing airspeed) restores smooth airflow and lift.

Airflow split over a wing profile
Fig. 4: An airfoil splits airflow into a stream over the top and bottom. The upper curved path causes lower pressure (lift)​:contentReference[oaicite:102]{index=102}.
Low pressure (suction) above an airfoil generating lift
Fig. 5: Air moving faster over the curved top of a blade creates suction (low pressure) on the upper surface​:contentReference[oaicite:103]{index=103}.
Increased lift from positive angle of attack
Fig. 6: A positive angle of attack deflects air downward, which in turn pushes the wing up (Newton’s law). This increases lift but also drag​:contentReference[oaicite:104]{index=104}.
Airflow separation and stall at high angle of attack
Fig. 7: If the angle of attack is too large, airflow separates and vortices form (stall). Lift suddenly decreases and the rotor blade (or wing) loses effectiveness.

Helicopter vs Airplane

Helicopters differ fundamentally from fixed-wing aircraft. Both obey the same aerodynamic laws, but a helicopter’s rotating blades make things more complex. On an airplane, lift and control come from the wings and tail surfaces, and thrust comes from a propeller or jet engine. Each component has a distinct role: wings for lift, ailerons/elevator/rudder for control, engine for thrust. In a helicopter, the spinning rotor provides lift and control. By increasing the collective pitch of all rotor blades, the rotor acts like a giant fan and “blows” more air downward, increasing lift so the helicopter rises​:contentReference[oaicite:105]{index=105}. To move forward, the pilot tilts the rotor disc forward, so thrust is directed slightly rearward (pushing the helicopter ahead, see Fig. 9)​:contentReference[oaicite:106]{index=106}.

Forces and control surfaces on an airplane
Fig. 8: Airplane basics – wings generate lift upward, engines provide forward thrust, and control surfaces (ailerons, elevator, rudder) steer the aircraft.
Forces and tilt in helicopter forward flight
Fig. 9: Helicopter forward flight – tilting the rotor disc forward creates a horizontal thrust component. The lift vector (green) tilts forward, so the pilot must add collective to maintain altitude during acceleration.

The helicopter is controlled by tilting the rotor disk in the desired direction of travel. This sounds simple, but in reality the aerodynamics are complex. Moreover, Newton’s third law (“for every action, there is an equal and opposite reaction”) comes into play. As the main rotor spins, it tries to make the fuselage rotate in the opposite direction (torque reaction). To prevent this, most helicopters have a vertically-mounted tail rotor that pushes against the torque to hold the fuselage steady​:contentReference[oaicite:107]{index=107}. By varying tail rotor thrust with the foot pedals, the pilot can yaw the helicopter left or right about its vertical axis. Some helicopters do not have a tail rotor – for example, designs with two coaxial rotors or twin rotors (tandem) – because two counter-rotating main rotors cancel out each other’s torque.

Main rotor torque and compensation by tail rotor
Fig. 10: Main rotor torque effect (orange arrows) tends to spin the fuselage. A tail rotor provides a counter-thrust to neutralize this torque.

Hovering Flight

A helicopter’s ability to hover (remain stationary in the air) is its signature advantage over airplanes. In a hover, the main rotor pushes air straight down to generate lift. For simplicity, we can imagine the rotor as a circular disk pushing on the air. In a stable hover, the lift force exactly equals the helicopter’s weight (Fig. 11)​:contentReference[oaicite:108]{index=108}.

Lift equals weight in hover
Fig. 11: In a stationary hover, lift (blue) balances weight (red).

If the pilot increases the collective pitch of all blades, the blade angle of attack increases. The rotor accelerates more air downward (higher mass flow through the rotor disk), increasing lift – the helicopter rises (Fig. 12)​:contentReference[oaicite:109]{index=109}. Conversely, lowering the collective reduces overall lift and the helicopter descends (Fig. 13)​:contentReference[oaicite:110]{index=110}.

Helicopter climbing by increasing collective pitch
Fig. 12: Raising the collective pitch increases lift – the helicopter climbs.
Helicopter descending by lowering collective pitch
Fig. 13: Lowering the collective pitch decreases lift – the helicopter descends.

The spinning main rotor, as mentioned, creates a torque that wants to spin the fuselage opposite its rotation. In a hover, the tail rotor counteracts this torque (see Fig. 10). The more power the main rotor draws (e.g. when climbing), the more counter-thrust the tail rotor must produce​:contentReference[oaicite:111]{index=111}. However, the tail rotor’s sideways thrust tends to make the helicopter drift laterally. Depending on the main rotor’s rotation direction, the heli might drift slightly left or right. The pilot must compensate by tilting the rotor disk a bit opposite to the drift​:contentReference[oaicite:112]{index=112} to hold a stationary hover.

Often, a hovering helicopter will not be perfectly level but exhibit a slight tilt. This is because the thrust from the main and tail rotors act at different points, so the helicopter might lean slightly to one side (Fig. 14)​:contentReference[oaicite:113]{index=113}. The direction of lean (left or right) depends on the main rotor’s rotation direction. The pilot naturally adjusts for this.

Helicopter hovering out of ground effect
Fig. 14: Hovering out of ground effect (OGE). The helicopter often has a slight sideward lean in hover to balance forces.

Hovering generally requires more engine power than flying forward. Air density plays a key role in hover performance: the denser the air, the more lift the rotor can generate for a given power. In thin air (high altitude or high temperature), the helicopter can carry less weight in a hover or needs more power to maintain the hover​:contentReference[oaicite:114]{index=114}. Another factor is the ground effect. When hovering near a surface (within roughly one rotor diameter height), the downwash air is partially trapped and forms a cushion under the helicopter. This is called hovering in ground effect (IGE) and it increases lift efficiency. If the helicopter is high enough that downwash escapes freely, it’s out of ground effect (OGE). In OGE hover, more power is required than in IGE hover due to the lack of that cushioning effect​:contentReference[oaicite:115]{index=115}​:contentReference[oaicite:116]{index=116}.

Ground effect air cushion under a helicopter
Fig. 15: Hovering in ground effect (IGE) – The rotor’s downwash is partially trapped, forming an air cushion that supports the helicopter. Less power is needed to hover within ground effect.

Forward Flight

The great advantage of helicopters is that they can both hover and fly forward. The transition from hover to forward flight is called translational lift. As the helicopter gains speed, the rotor becomes more efficient and the required power decreases. Simplified, we can treat the rotor disk as a tilted wing during forward flight.

In a hover, air flows vertically down through the rotor disk (Fig. 16). To transition to forward flight, the pilot tilts the rotor disk forward. Now the airflow is directed slightly backward (Fig. 17), and the helicopter begins to accelerate forward. However, because the lift vector is tilted, it no longer points straight up. The pilot needs to add a bit of collective pitch (more power) during the transition to maintain altitude​:contentReference[oaicite:117]{index=117}.

Rotor downwash in hover
Fig. 16: In hover, air is pushed straight down through the rotor (downwash).
Rotor tilted forward for transition to forward flight
Fig. 17: Tilting the rotor disk forward pushes air backward (green arrow), creating forward thrust so the helicopter accelerates​:contentReference[oaicite:118]{index=118}.

Dissymmetry of Lift

Once the helicopter has forward speed, the rotor blades encounter different relative wind speeds on opposite sides. The blade moving forward relative to the air (the advancing blade) has a higher airspeed than the retreating blade on the opposite side​:contentReference[oaicite:119]{index=119}. For example, if the rotor tip speed is ~750 km/h and the helicopter flies at 200 km/h, the advancing blade’s tip effectively sees ~950 km/h wind (750+200, Fig. 18)​:contentReference[oaicite:120]{index=120}. The retreating blade’s tip sees only ~550 km/h (750–200, Fig. 19)​:contentReference[oaicite:121]{index=121}. This difference in airspeed means the advancing blade generates more lift than the retreating blade.

To prevent the helicopter from rolling due to this lift dissymmetry, the rotor system employs cyclic pitch. The blade’s pitch (angle) is automatically varied as it rotates: high pitch on the retreating side and low pitch on the advancing side. This equalizes lift around the disk​:contentReference[oaicite:122]{index=122}. Without cyclic feathering, a helicopter would bank uncontrollably in forward flight.

Relative airflow speeds on advancing vs retreating blade
Fig. 18: On the advancing side (right), a blade encounters higher relative wind speed than on the retreating side (left). In this example, the advancing blade’s tip sees ~950 km/h, whereas the retreating tip sees ~550 km/h​:contentReference[oaicite:123]{index=123}​:contentReference[oaicite:124]{index=124}.
Lift distribution on advancing blade
Fig. 19: The advancing blade generates more lift due to higher airflow (green area), while the retreating blade generates less (and near the hub might even stall, red area)​:contentReference[oaicite:125]{index=125}.

Conventional helicopters face an aerodynamic speed limit around 300–330 km/h. The fastest conventional helos (with rigid or coaxial rotors) can approach ~400 km/h under ideal conditions. Beyond that, two phenomena occur: parts of the advancing blade approach the speed of sound (causing shockwaves and drag rise), and the retreating blade experiences stall due to insufficient relative airflow​:contentReference[oaicite:126]{index=126}​:contentReference[oaicite:127]{index=127}. No single rotor airfoil can handle such extremes of Mach and low speed simultaneously​:contentReference[oaicite:128]{index=128}. This fundamental barrier is known as retreating blade stall​:contentReference[oaicite:129]{index=129}. It essentially limits the top speed of conventional helicopters to around 400 km/h​:contentReference[oaicite:130]{index=130}. The long-standing world record (about 400 km/h, ~216 kn) was set by a modified Westland Lynx in 1986​:contentReference[oaicite:131]{index=131}. Pushing a helicopter faster requires creative designs (like compound helicopters or tiltrotors).


Autorotation (Engine Failure)

If a helicopter’s engine fails in flight, the aircraft can still land safely using autorotation. Autorotation is a flight condition where the main rotor is driven solely by the upward flow of air through the rotor, instead of engine power​:contentReference[oaicite:132]{index=132}. It’s analogous to gliding in an airplane. When the engine quits, a freewheeling clutch automatically disengages the engine from the rotor, allowing the rotor to spin freely​:contentReference[oaicite:133]{index=133}. The pilot immediately lowers collective pitch, reducing the blade angle so the rotor won’t slow down too quickly. The helicopter enters a descent, and air begins to flow up through the rotor disk from below​:contentReference[oaicite:134]{index=134}. This upward airflow keeps the rotor turning (like a windmill) and maintains rotor RPM.

Airflow during autorotation after engine failure
Fig. 20: In autorotation, air flows upward through the rotor (bottom image) instead of downward (top image). A freewheeling unit lets the rotor spin freely when the engine stops​:contentReference[oaicite:135]{index=135}.

In Fig. 20 above, the top shows normal powered flight (air pushed down), and bottom shows autorotation (air entering from below). To understand autorotation, consider a rotor blade during vertical descent. We can divide the blade into three zones: an inner driving region, a middle neutral region, and an outer driven (drag) region. In the driving region, the aerodynamic force on the blade is slightly ahead of the rotor axis, pulling the blade around and driving the rotor (Fig. 23)​:contentReference[oaicite:136]{index=136}. In the neutral zone, the force is aligned with the axis, neither speeding nor slowing the rotor (Fig. 24)​:contentReference[oaicite:137]{index=137}. Near the tip (driven region), the force is behind the axis, acting to slow the rotor (Fig. 25)​:contentReference[oaicite:138]{index=138}. The very center of the rotor disk gets little airflow and contributes no lift (often stalled).

Rotor blade regions in vertical autorotation
Fig. 21: Blade regions in vertical autorotation – inner green region drives the rotor, middle gray region is neutral, outer red region drags (absorbs energy).
Driving region in autorotation
Fig. 22: In the driving region, the lift force is ahead of the rotor axis, accelerating the rotation (autorotative force)​:contentReference[oaicite:139]{index=139}.
Neutral region in autorotation
Fig. 23: In the neutral region, the force is aligned with the axis – rotor speed remains constant​:contentReference[oaicite:140]{index=140}.
Driven (braking) region in autorotation
Fig. 24: In the outer (driven) region, the force is behind the axis, tending to slow the rotor​:contentReference[oaicite:141]{index=141}.

Helicopters are designed so that, in autorotation, the driving and driven regions reach a balance and the rotor RPM stays within limits through a range of airspeeds​:contentReference[oaicite:142]{index=142}. If the helicopter flies too fast during autorotation, the driving region shifts too far and the rotor may not sustain RPM​:contentReference[oaicite:143]{index=143}. Therefore, pilots are trained to autorotate at an optimal glide speed (often around 60–80 knots depending on the model). To land from autorotation, the pilot must dissipate the descent rate and forward speed just before touchdown. This is done with a maneuver called a flare: as the ground nears, the pilot pulls the nose up, trading forward speed for a brief increase in lift and a reduction in descent rate. This also increases rotor RPM slightly. Then the pilot levels the helicopter and raises collective to cushion the landing. If done correctly, the helicopter can touch down smoothly with little or no forward speed. Autorotation landings are a standard emergency procedure taught to helicopter pilots​:contentReference[oaicite:144]{index=144}, but they require timing and skill.


v

Vortex Ring State (Settling with Power)

The vortex ring state is a hazardous aerodynamic condition that can occur when a helicopter descends too fast straight down under power with low forward speed. In this situation, the air that the rotor is pushing downward doesn’t move away; instead, it recirculates back up through the rotor from below, forming a toroidal vortex ring around the rotor disk​:contentReference[oaicite:145]{index=145}. Essentially, the helicopter is descending into its own downwash. This often happens during steep, powered descents (for example, a vertical landing approach) when the descent rate exceeds roughly 2.5 m/s (500 ft/min) and airspeed is low​:contentReference[oaicite:146]{index=146}. The rotor’s thrust is then wasted in circulating airflow, and despite high power, the helicopter continues to sink – hence the term “settling with power.” Pilots notice a sudden loss of lift, an increased sink rate, and often a shuddering or vibration in the rotor as the airflow becomes turbulent.

Disturbed airflow in hover with high descent rate
Fig. 25: Vortex ring formation – With a high descent rate and low speed, some of the downward airflow (green) is turned back upward into the rotor (blue), disrupting the normal thrust.
Formation of a closed vortex ring in the rotor
Fig. 26: The inner portion of the rotor airflow recirculates upward, forming a ring vortex. The helicopter may start to descend rapidly despite added power​:contentReference[oaicite:147]{index=147}.

In a developed vortex ring state, adding power only feeds the vortex. The helicopter can experience an accelerating descent and heavy vibration. To recover, a pilot has two main options (which must be executed quickly due to rapid altitude loss): (1) Push forward cyclic and gain airspeed – by flying out of its own downwash, the helicopter can escape the vortex and resume normal flight​:contentReference[oaicite:148]{index=148}. (2) Enter autorotation by lowering collective pitch to eliminate the downwash – essentially “resetting” the airflow and breaking the vortex ring​:contentReference[oaicite:149]{index=149}. In practice, if altitude is low (near the ground), the forward flight recovery is preferred to avoid hitting the ground. If altitude is sufficient, entering autorotation can stop the vortex. In any case, the key is prevention: pilots are trained to avoid high rates of vertical descent at low speeds (by monitoring the “height-velocity diagram” and keeping within safe combinations of altitude and speed).

Helicopter in vortex ring state
Fig. 27: A helicopter caught in a vortex ring state (settling with power). Despite high power, it descends rapidly and may exhibit strong vibrations. The pilot must quickly recover by pushing into forward flight or entering autorotation to break the vortex.


FAQ – Frequently Asked Questions


How does a helicopter work?

A helicopter generates lift with its rotor blades, which act like spinning wings. In a hover, the main rotor pushes air downwards to hold the helicopter up. To fly forward, the rotor disc is tilted forward, creating a thrust that propels the helicopter. Control is achieved via three main inputs: the cyclic stick (tilts the rotor disk), the collective lever (changes the overall blade pitch/lift), and the tail rotor pedals (yaw the helicopter by offsetting torque). By coordinating these controls, a pilot can maneuver the helicopter in any direction and even hover in place. In summary, how a helicopter works is by continuously adjusting rotor thrust and direction to oppose weight and drag, allowing vertical takeoff, hovering, and flight in any direction.


What are the helicopter’s controls and how are they used?

Helicopters have three primary controls: the cyclic, the collective, and the anti-torque pedals. The cyclic (joystick) tilts the rotor disk forward, backward, or sideways, which controls the helicopter’s pitch and roll (direction of travel). The collective (lever on the left) raises or lowers the pitch of all main rotor blades together, controlling overall lift (up/down). The pedals control the tail rotor (or anti-torque system) to yaw the helicopter left or right. These helicopter controls, explained simply, adjust the rotor thrust vector. In practice, the pilot must coordinate all three: for example, increasing collective (upward lift) often requires adding pedal to counter torque and a bit of cyclic to prevent drif​:contentReference[oaicite:150]{index=150}​:contentReference[oaicite:151]{index=151}】. Mastering this coordination allows precise helicopter control.


What is the purpose of the tail rotor?

The tail rotor’s primary purpose is to counteract the torque of the main rotor and prevent the fuselage from spinning opposite the main rotor’s rotatio​:contentReference[oaicite:152]{index=152}】. According to Newton’s third law, as the main rotor turns the air, the air exerts an opposite force trying to spin the helicopter’s body. The tail rotor produces a sideways thrust that balances this torque, keeping the helicopter’s nose pointed in the desired direction. The tail rotor (controlled by the pedals) also allows the pilot to rotate or yaw the helicopter in hover. In short, without a tail rotor (or an alternative anti-torque system), a single-rotor helicopter would uncontrollably spin. (Some helicopters use other anti-torque methods, like coaxial rotors or NOTAR, instead of a tail rotor.)


What happens if the engine fails during flight?

If the engine fails, a helicopter can perform an emergency landing using autorotation. When the engine quits, a freewheeling clutch disengages it from the rotor so the rotor can spin freel​:contentReference[oaicite:153]{index=153}】. The pilot immediately lowers the collective pitch, which reduces drag and allows the rotor to continue spinning due to air flowing up through the rotor dis​:contentReference[oaicite:154]{index=154}】. The helicopter then glides downward like a maple seed. By managing airspeed and rotor energy, the pilot can flare near the ground to slow down and then cushion the landing with remaining rotor energy. This procedure lets helicopters land safely without engine powe​:contentReference[oaicite:155]{index=155}】. All trained helicopter pilots practice engine-out landings (autorotations) as part of emergency training.


What is vortex ring state (settling with power)?

Vortex ring state, also known as “settling with power,” is a condition where a helicopter can start to sink into its own downwash. It typically happens when descending quickly (over ~500 ft/min) with little or no forward speed while still under powe​:contentReference[oaicite:156]{index=156}】. The rotor’s down-flowing air circulates back upward in a donut-shaped vorte​:contentReference[oaicite:157]{index=157}】, causing a sudden loss of lift. The helicopter may shudder and continue to descend even if the pilot raises collective. To recover, the pilot must break the vortex by either moving into forward flight (so the rotor gets clean air​:contentReference[oaicite:158]{index=158}】 or by briefly reducing power (entering autorotation) to stop the recirculatio​:contentReference[oaicite:159]{index=159}】. Vortex ring state is dangerous if not corrected immediately, but avoiding it is simple: do not attempt steep, slow descents at high power – instead, maintain some forward airspeed or limit vertical descent rate.


Are there helicopters without a tail rotor?

Yes. There are several designs that eliminate the tail rotor. Coaxial helicopters (like the Russian Kamov series) use two main rotors spinning in opposite directions on the same axis. These counter-rotating rotors cancel out each other’s torque, removing the need for a tail roto】. Tandem rotor helicopters (like the CH-47 Chinook) have two main rotors (front and rear); they rotate opposite each other and likewise balance out torque. Another solution is the NOTAR system (No Tail Rotor) on some MD Helicopters models, which uses an enclosed fan that blows air through tail boom slots to provide anti-torque thrus​:contentReference[oaicite:160]{index=160}】. All these designs achieve directional control without an exposed tail rotor. They improve safety (no tail rotor strike risk) and can reduce noise, but often at the cost of added mechanical complexity or weight.