# Rotor Design

The main rotor is the most important and complex component of a helicopter.  Main rotor design balances many contradictory goals and constraints.  Examples include minimizing cost, maximizing performance, and keeping vibration and noise below thresholds.  In this article, we cover some of the basics of helicopter main rotor design.

## Main Rotor Size

Size matters.  The size of the main rotor has a widespread impact on design, affecting almost all criteria.  We focus on performance here, but briefly discuss some simple, practical matters first.

The size of the main rotor may be constrained so that it can fit in certain spaces.  For example, military helicopters may be required to fit in an elevator on an aircraft carrier.  Of course, this constraint may be avoided by designing folding rotor (e.g. see this video of a V-22), but with increased complexity and cost.  Downwash can constrain main rotor size from the other end.  When carrying the same weight, smaller rotors have larger downwash.  This can prevent landing in certain areas due to risk of damaging structures and vegetation.  The V-22 has not been used on several missions for this reason.

Now let's talk performance.  Larger main rotors are more efficient than smaller ones.  Larger rotors can lift more weight per unit power / fuel.  Why?  In simple terms, a larger rotor doesn’t have to work on the air as hard as a smaller one.  Rotors create thrust by throwing air downward, producing downwash.   According to Newton, this thrust equals the product of the mass and acceleration of the downwash.  To lift the same weight, a smaller rotor must impart more acceleration to a smaller volume (mass) of air.  This generates larger downwash velocities, albeit in a smaller area.  The energy imparted to this air is proportional to the mass and the square of the velocity: $$mv^2$$.  The $$v^2$$ term means that, when the same momentum is added to the air, more energy is required to do this with a smaller mass and more acceleration.  Hence a larger rotor, accelerating a larger volume of air by a smaller amount, is more efficient.  For more detail on this see our article on momentum theory which derives the relevant equations.

Helicopter cost and vibration are particularly sensitive to the number of main rotor blades.  If you blindfold an experienced pilot and take them for a ride, they can probably tell you if the main rotor has two versus four blades.  Perhaps surprising, performance is less influenced by blade count.   There are performance pros / cons that roughly balance out.  For example, fewer blades require large chord length to produce the same thrust, which provides L/D benefits via Reynolds effects.  However, such blades have smaller aspect ratio and hence suffer larger tip losses.

Vibration is a potent enemy of helicopters.  It is more than an annoyance to pilots and passengers.  Vibration causes real fatigue that can limit flight times.  It also ages all the parts on a helicopter, shrinking their lifetime and increasing maintenance costs.  Excess vibration can cause a lethal crash by breaking critical components in flight.  Designers must carefully limit vibration.

The largest source of vibration occurs at the frequency $$Nf$$, where $$N$$ is the number of blades and $$f$$ is the frequency of the main rotor (its rotational speed).  Given a similar rotor speed, this means a 4-bladed rotor’s vibration is twice the frequency of a 2-bladed rotor.  Ill affects of vibration vary with amplitude and frequency.  For more information on how vibration affects humans read this.

Larger blade counts are less desirable structurally.  As mentioned before, the chord length is inversely proportional to the blade count.  With twice as many blades, the chord length of each will be half.  Each blade will be required to carry half as much load, but the structural stiffness will be less than half (all else equal).  This is because the stiffness varies with the square of the chord.  Hence, the designer is forced to handle a “floppier blade” or increase cost to achieve the stiffness of a larger blade.

## Main Rotor Speed

Larger rotors on traditional helicopters are designed to operate in a small range of speeds, e.g. 290 to 310 RPM.  Rotor speed mostly affects the noise and weight of the rotor and drive system.  Rotor noise is difficult to predict, but roughly scales with the 5th power of the rotor tip speed!  Tip speeds above 750 ft/s are generally too loud for practical aircraft.  On the flip side, faster rotor blades experience more dynamic pressure and hence need less chord to produce the same lift.  This means faster rotors facilitate smaller blades.  Larger rotor speeds also reduce torque (per unit power), facilitating cheaper, lighter drive systems.

Rotor speed is constrained on the high and low side by several other factors.  In forward flight the advancing blade tip speed (relative to the air) is the hover tip speed plus the forward flight speed.  If this nears the speed of sound, regions of air around the blade will compress, shock waves will develop, and aerodynamic loads become unmanageable.  For instance, a strong pitch down moment called Mach tuck can twist the advancing blade nose down.  On the flip side, the hover tip speed needs to be at least double the maximum desired forward speed.  Otherwise, at max forward speed the retreating blade will stall, lacking enough velocity relative to the air.  This is referred to as retreating blade stall or RBS.

The compressibility effect requires a lower rotor speed at higher aircraft speed, while retreating blade stall requires higher rotor speed at higher aircraft speed.  There’s a limit, around 200kts, where no rotor speed can satisfy both criteria.  This is shown by the intersecting lines in the diagram below.  It’s not practical to fly a traditional helicopter beyond this speed.

## Blade Shape and Airfoil Design

Performance and cost are sensitive to blade shape.  The shape of a cross section of a blade is called the "airfoil shape" and is shown in the figure below.  A blade designer must choose airfoil shapes along the length of the blade along with their size (chord length) and orientation (twist).  We'll discuss this briefly here.

Airfoil shapes are designed for airplane wings, wind turbine blades, and helicopter blades among other things.  There are many varying design constraints / objectives for each.  We won’t go into the details of helicopter rotor airfoil design here.  Suffice it to say that airfoils are selected based on structural concerns, manufacturability, acoustic properties and lift / drag properties.  Typically CFD and wind tunnel tests will be performed to determine lift and drag coefficients for the airfoils as a function of angle of attack.  These values will be used heavily in the (3-D) blade design process.

Once the airfoil(s) are chosen, the 3-D blade shape can take form.  This shape is governed by the chord length, twist and airfoil distribution along the blade.  If multiple airfoils will be used, designers must choose which airfoils will be used where on the blade.  For example, airfoil A will be used from the tip to 18’, airfoil B will be used from 16’ down to 12’, ….  This is what is meant by airfoil distribution.  Some method of interpolating airfoil shapes will be used to transition between airfoils, e.g. from 16’ to 18’ in the example.

The airfoil shape is non dimensional, but real blades have a specific size, or chord length, at each location along their length.  E.g. the airfoil at 10’ radius will have a 1.1’ chord.  This chord distribution or taper must be selected to provide enough thrust at the design rotor speed.  Lift is proportional to chord length, so the chord will need to be large enough to lift the weight of the helicopter and provide thrust for acceleration and overcoming drag at high forward speed.  However, If the chord is too large there will be a performance penalty.  A larger chord airfoil needs to operate at a smaller angle of attack, which results in excess drag (operating at a smaller L/D).

Simply making a rectangular blade with the same chord length everywhere is simple and cheap to manufacture but comes with a performance cost.  Optimal efficiency requires a uniform induced velocity over the entire rotor, which calls for a tapered blade with a very large chord inboard, shrinking towards the tip.  Most designs are neither square nor performance-optimal; tradeoffs result in something in between.

Blades are also twisted – the airfoils are rotated leading edge down going from the root to the tip of the blade.  Like grabbing the tip of the blade and twisting it leading edge down.  Why?  Rotor rotation makes the outer portion of the blade move much faster than the inboard section – the speed due to rotation is the product of the rotor angular speed and the distance from the axis of rotation $$\Omega r$$.  However, the vertical flow of air from induced velocity (downwash) is about the same along the length of the blade.  As a result, the “inflow angle” of air decreases along the blade from root to tip.  The diagram down below may help you see this.  Without twist, inboard airfoils would operate at a small or negative angle of attack while outboard airfoils would stall.  To make more of the blade operate at a desirable angle of attack (blade pitch angle minus inflow angle), it must be twisted / pitched up from the tip to the root.  Optimal twist would require a 90 deg (vertical) airfoil at the root and is not practical, so again a tradeoff results in some twist, but not optimal twist.  To keep things simple, many helicopters use a linear twist – the angle varies linearly with span.  See the diagram below showing cross sections of a blade at 5’ and 15’ from the rotor center.  The outer section moves 3x faster and hence the inflow angle is about 10 degrees smaller (calculations assume 320 RPM).

Ideal airfoils, chord and twist are more important on the outer portion of a blade.  Inboard, the designer can violate aerodynamic idealities more for cost, structural or other practical benefit.  This is because the outer portion of the blade is more critical for performance.  Assuming thrust is uniformly distributed over the main rotor area, the section of the blade from 19’ to 20’ covers an area of $$\pi (20^2-19^2)\approx 122.5 ft^s$$ and hence accounts for about 9% of the thrust of a 21’ blade, while the portion of the blade from 2’ to 3’ covers an area of just $$\pi(3^2-2^2) \approx 15.7ft^2$$ and provides only 1% of the thrust.  Likewise, the outer portion of the blade creates more torque per unit length that must be overcome by the engine.  This brings us to our last topic on blade shape – tip shape.

Blade tip design is a humbling activity for the best engineers.  Even in hover, blade tips experience complex, 3D airflow.  This is because the aerodynamic forces must drop to zero at the tip, causing vortices to form and spanwise flow along the blade.  Many designs just use a square blade tip and avoid complicated tip analysis.  However, the tip is critical because, per unit length, it (1) covers the most rotor area, (2) has the strongest effect on vortex formation, (3) is the most susceptible to compressibility, and (4) has the largest impact on rotor noise and elastic blade bending.    So, challenging as it may be, alternate tip shapes have been designed.  Examples include Sikorsky’s Black Hawk and Westland’s AW101.  For more information about these check out the BERP program.

The primary feature provided by nonstandard tips is compressibility mitigation.  By sweeping a blade tip (or the leading edge thereof) aft, the speed of air normal to the leading edge may be reduced – the speed of air relative to a blade section with rotational speed $$\Omega$$ at a distance $$r$$ from the hub is $$\Omega r$$.  However, if the leading edge is swept at an angle $$\delta$$, the speed normal to the leading edge is reduced to $$\Omega r \cos\delta$$.  Swept blades may also exhibit much more elastic twist.  Forces pushing up on an aft-swept tip tend to twist the blade nose down and vice versa.  This can be leveraged for improved performance and even longitudinal stability.

## Rotor Hub Design

Rotor hub design may be overlooked by outsiders, but experts know the hub type is critical for cost/maintainability, safety and flying qualities.  Unlike most components, hub designs vary greatly between helicopters, even at the fundamental level.  For example, some designs like teetering rotors work well for two bladed helicopters but are inapplicable for rotors with more than two blades.   Hubs do have common goals like keeping the blades attached to the shaft while allowing them to feather, flap and sometimes lead/lag.  Before discussing hub designs, we’ll investigate these goals in a bit more detail.

Feathering is the rotation of a blade about its length.  It can partially be seen in this videoThe pilot controls the feathering in order to control the flight path as explained here.  The main rotor hub must allow each blade to feather independent of the other blades.  This is typically done with a swashplate as described here.

Feathering changes a blades angle of attack, which changes the aerodynamic lift force and causes flapping.  Flapping is a vertical movement of a blade relative to the hub and is shown in the figure below.  The hub may facilitate flapping in a few different ways as we'll describe shortly.  Flapping allows rotor thrust to be directed forward, aft or even sideways to create roll and pitch moments.  Flapping may also (more directly) induce pitch or roll moments via an offset flap hinge or hub restraint.  These effects change the helicopters attitude, speed and direction of flight.  See this page for more information on how a helicopter is controlled via rotor feathering and flapping.

When a blade flaps up (or down) its center of mass moves closer to the hub.  Like a spinning ice skater drawing in her arms, this causes the blade to want to rotate faster, via conservation of angular momentum (the Coriolis effect).  Depending on the number of blades, the next blade may be trying to “slow down” at the same time this one is trying to “speed up.”  This puts significant, high frequency, periodic stress on the blades.  Rather than beef up a blade to handle this stress, it’s often more efficient to allow the blades to move in this “lead/lag” direction to some extent.

The main rotor hub facilitates all of these blade motions while keeping the blades attached to the shaft.  This is accomplished in a difficult environment.  Huge centrifugal forces are pulling each blade away from the hub while an engine is applying up to thousands of horsepower to the shaft.  Highly oscillatory 20,000 lb+ airloads are lifting the blades up.  The thin, aerodynamic blades are not rigid either.  In flight, they vibrate in different modes like a guitar string.  This transmits complex vibration and loads to the hub.  In this environment, the hub must work with essentially 0 chance of failure.  Unlike an engine failure, a failure in feathering control or any of these hinges / flexures almost definitely spells death.

### Flapping

Unlike feathering, there are significantly different designs for flapping.  One approach is to add a flap hinge to each blade as shown in the diagram below.  Each blade flaps up/down independently of the others.  The distance of the hinge from the center of rotation, called the hinge offset, plays a role in the rotor’s dynamics and the aircraft’s handling qualities.  There is typically some resistance to flapping a la a torsion hinge spring which also plays a role in rotor dynamics and handling qualities.  This article has more information about the impact of hinge offsets and flap springs on rotor dynamics.  Note that a flap spring is not required - centrifugal forces when the rotor is spinning automatically pull the blades out, taming flap motion.  However, some type of flap stop or static stop is needed to limit flapping when the rotor is not spinning at full speed or otherwise experiencing extreme flapping.

The flap hinge described above may be skewed so that it naturally pitches a blade nose down when it flaps up.  The reduces the angle of attack and therefore lift on a blade as it flaps up, providing a passive mechanism to limit flapping.  This design is often referred to as delta 3 or a delta hinge.   These hinges are more typically used on tail rotors – the reduced flapping allows them to be mounted closer to the tail boom.  Click here for a video of a real tail rotor with a delta hinge.

Since the dominant flap motion is 1/rev, 2-bladed rotors can be made with just one “flap hinge” in the center of the rotor so that the two blades flap together as a unit about this hinge.  When one blade goes up, the other goes down.  This is called a teetering rotor and behaves like a child’s seesaw as shown in the diagram below.  The hinge is called a teeter hinge.

A teetering rotor may also have a coning hinge and undersling.  The coning hinge further relieves flapping stress from the blade root by giving each blade its own flapping hinge as described above.  This primarily relieves coning stresses but can also help with any other non 1/rev flap motion.  The teeter hinge cannot relieve coning stresses because it must flap blades to the opposite direction (1 up, 1 down), while coning consists of all blades flapped up by the same amount.

Undersling reduces the Coriolis effect that makes the blades lead/lag.  It is accomplished by placing the teeter hinge above the blades as shown in the diagram below.  As a blade flaps up and it’s center of gravity moves in toward the hub, the blade root is effectively pushed out away from the hub.  We’ll do a quick calculation below to demonstrate the reduction in Coriolis forces.

Let’s say a rotor is operating with a coning angle $$\beta$$ with cyclic flapping $$+/- a$$ at 1/rev.  Let $$r$$ denote the distance of the blade’s center of mass (CM) from the root and that the amount of undersling is $$\lambda r$$.  When a blade flaps up it’s center of mass still moves from $$r\cos\beta$$ to $$r\cos (\beta+a)$$ toward the blade root, but the root moves away from the shaft by the amount $$\lambda r \sin a$$.  So the movement of the CM without undersling is $$\delta = r\cos\beta - r\cos (\beta+a)$$ while the movement with undersling is $$\delta - \lambda r \sin a$$.  So if there’s 2deg coning, 3deg flapping and an underlsing of $$r/20$$ then $$1-\frac{\delta-\lambda r\sin a}{\delta} \approx 82$$% of the CM movement has been removed by undersling.

The last flapping mechanism we’ll discuss here is hingeless or rigid design.  The blade is mounted directly to the hub but contains flexible material near the root which acts as a "virtual flap hinge."  The alleviates the maintenance and failure risk associated with a hinge, but increases the transmission of vibration / loads to the fuselage.