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I am trying to understand how adding offset hinges would provide a self-feathering force. Does anyone have more information on Delta hinges on helicopters? I am a bit confused with the information I have available. A video would be nice, but I couldn't fidn much on YouTube.

Here are two references illustrating the concept, and a picture from the first link (b). Delta-3 Hinge Delta-3 Hinge - AOPA Explanation Delta-3 Hinge

The AOPA explanation does not have the offset/angled blade: AOPA Delta-3 hinge explanation

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It's because the blade's span-wise axis is "swept" relative to its flapping hinge line, so when it flaps, the sweep angle results in the blade's effective AOA changing somewhat (for the advancing blade, reducing; that's what the feathering part means).

To picture it, imaging you are standing directly in front of the tail rotor disc watching the advancing blade of the plain rotor come over the top or bottom straight toward you. All you see is its thin leading edge. If the plain blade flaps left or right, still all you see is the thin leading edge because the blade's axis is perpendicular to the feathering hinge axis.

Now imagine the Delta three rotor blade from the same vantage point, and the hinge flaps left or right. Because the blade's span-wise axis is offset, or "trailing" the flapping hinge axis, you will be able to see some of the upper or lower surface of the blade from your vantage point when it flaps. To the airstream, this is effectively a slight reduction in AOA ("feathering") compared to the AOA of the unflapped blade as set by the pitch change linkage (or an increase when flapping while retreating).

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    $\begingroup$ I think that makes sense! Thank you! $\endgroup$
    – Gerry
    Commented Jan 13, 2020 at 22:15
  • $\begingroup$ There's a tiny flaw in the explanation. There exist both negative and positive delay 3 hinges Notably on Bell helicopters the blade "in" feathers with flapping. See my explanation below $\endgroup$
    – George Williams
    Commented May 12 at 21:26
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Delta-3 (also spelled $\delta_3$) is a simple yet difficult to visualise concept.

I suppose that the best way to understand it is by comparing a rotor with a built-in delta-3 with one without it. The following two rotor heads are both taken from Wikipedia (plus a couple of colorful lines from my side): the first one is the main rotor head of a Bell 222, the second one is the tail rotor of a Robinson R44.

main rotor of Bell 222

Main rotor of Bell 222. Source: Wikimedia Commons

tail rotor of Robinson R44

Tail rotor of Robinson R44. Source: Wikimedia Commons

For the Bell 222, the red line of the pitch hinge is perpendicular to the blue line of the flapping hinge (also called teetering hinge in this particular design). For the R44, the red line and the blue one are not perpendicular but intersect at some 45°. Hopefully this difference is visibile despite the different perspective of the pictures. This not-perpendicular intersection like in the R44 is the delta-3 angle.

Now, when a blade of the B222 flaps upward... well it just flaps upward. When the blade of the R44 flaps upward (actually "outward" is maybe a more correct term for a tail rotor but for ease of comparison I stick with upward) it flaps upwards... and it reduces it's pitch, exactly because these two axis are not perpendicular due to delta-3.

That's not very easy to represent but it could become clearer looking at an extreme case: starting with the B222 configuration and its 90° shifted blue and red line, let's reduce this angle (i.e. increase delta-3); we get to the R44 configuration with its 45° between blue and red line; now let's reduce this angle again (i.e. further increase delta-3) till we get to the extreme case where blue and red line overlap (i.e. delta-3=90°): now flap and pitch have become the same thing, they both lie on the same axis, they are coincident, 1° of flap coincides to 1° of pitch. To sum up:

  • blue and red lines perpendicular (B222) $\Leftrightarrow$ $\delta_3=0°$ $\Rightarrow$ flap and pitch fully decoupled
  • blue and red lines coincident $\Leftrightarrow$ $\delta_3=90°$ $\Rightarrow$ flap and pitch fully one-to-one coupled
  • $\delta_3$ between 0 and 90° (R44) $\Rightarrow$ flap and pitch partially coupled.

(This latter partial coupling is mathematically translated as $\theta_{\delta_3}=\beta tan \delta_3$, where $\theta_{\delta_3}$ is the change in pitch due to the flapping $\beta$. Maybe if you have some Lego technic laying around you could try it out and verify it live).

What's the benefits of that? As said, when the blade flaps upward (outward) the delta-3 makes its pitch reduces which, in turn, reduces the aerodynamic thrust generated by the blade which, in turn, reduces its flapping. The opposite is true when the blade flaps downward (inward) i.e. the delta-3 makes the pitch increases, increasing its thrust and therefore its flapping:

  • blade flaps $\Rightarrow$ due to $\delta_3$ pitch reduces $\Rightarrow$ lift reduces $\Rightarrow$ flap reduces

So delta-3 is a simple cinematism used to limit the amount of flapping of the blades of the rotor.

This has a twofold outcome:

  1. clearance between tail rotor and tailboom is reduced giving a shorter and lighter shaft;
  2. flapping produces also lead-leg movement via the Coriolis force; if flap is reduced also lead-leg movement is reduced and it can be reduced to the point that the lead-leg hinge can be disposed of, as in the R44.

Free weight saving is always good in the aerospace world.

Bonus material

Delta-3 is used exactly for the same reason also on the main rotor, anyway in a milder way since flapping is the physical effect which permits the control of the helicopter, so a compromise between shorter/lighter mast and controllability is needed here.

Since on the red line only 2 blades can lie, for rotors with 3 or more blades the delta-3 effect is obtained with a different cinematism: if someone is interested in it, I could expand this answer.

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  • $\begingroup$ I wanted to add that these “dynamic stabilization” examples occur all throughout aerodynamics, even the way an arrow flies. Perturbations happen throughout each rotation of the rotor. I can have a more rigid mast that imposes structural force on the rotor to counter the perturbation. Or I can have a lighter mast with less vibration transferred to the aircraft if I design the hub in such a way that the “correcting” force is NOT applied to the blades by the mast, but rather by aerodynamic forces acting on the blade. I hope this adds the “why?” Context to sophit’s “how” answer. $\endgroup$
    – Max R
    Commented Aug 7, 2022 at 15:52
  • $\begingroup$ @sophit, isn't the equation, $\theta_{\delta_3}=\beta tan \delta_3$, only valid when the delta-3 angle for pitch flap coupling is implemented by the control system geometry? I believe relationship you describe above (in words before giving the equation) is true when the delta-3 angle for pitch flap coupling is implemented by the flap hinge geometry. Note, in your example, when $\delta_3=90$, $tan \delta_3=\infty$, not 1. (Ref: Wayne Johnson, Helicopter Theory, pg 239) $\endgroup$
    – eball
    Commented Nov 17, 2022 at 14:09
  • $\begingroup$ Also note that the formula $\theta_{\delta_3}=\beta tan \delta_3$ is approximate because it assumes $\theta_{\delta_3}$ and $\beta$ are small angles. $\endgroup$
    – eball
    Commented Nov 17, 2022 at 14:25
  • $\begingroup$ @eball: yes correct, thanks for spotting it. If think I'll remove the equation, nobody is anyway going to use it :) $\endgroup$
    – sophit
    Commented Nov 17, 2022 at 15:05
  • $\begingroup$ @eball: have you spot any other error or needed improvement? $\endgroup$
    – sophit
    Commented Nov 17, 2022 at 16:46
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Bell 429 Tail Rotor Hub

Unfortunately the other explanations are wrong because they do not cater for for the negative delta 3 angles found on Bell 407, AW139 and Bell 429. Here's a different perspective.

Hinges on rotor blades do not necessarily have to be perpendicular to the blade axis and the behaviour of the rotor can be tuned by angling hinges or moving control rods to provide coupling between the drag, feathering and flapping of blades.

Juan De La Cierva

Juan De La Cierva experimented with various axes during development of the autogyro in the 1930s and produced a naming convention to describe the hinges. The delta 3 hinge is a hinge where flapping and feathering motions are coupled with the hinge lying in the plane of rotation (delta refers to the feather/flap cross-couple and 3 refers to the plane of rotation).

Cierva hinge model

Without a delta 3 hinge a degree of cross coupling is still likely on the rotor blades as the pitch rods remain fixed while the rotor blade flaps, inducing pitching or dragging motions. These motions can be undamped and oscillatory leading severe vibration. Cierva found that by change the angle of the hinge of the blade he could reduce the maximum flapping angle of his unpowered rotor which has a lot in common with the tail rotors of today. The delta 3 hinge can be either at a positive angle (increased flapping results in decreased feather angle) or a negative angle (increased flapping results in increased feather angle). A positive delta 3 angle is shown below. Note the pitch change rod is in front of the blade for this positive delta 3 unlike the Bell 429 shown at the start where the pitch change rod is behind the blade for a negative delta 3 angle.

Delta 3 hinges - Tail Rotor Design Part 1 - Bell Helicopter - October 1970

Each option provides several advantages for the designer, particularly with reference to the tail rotor: a. Simpler layout of control rods in tail rotors with greater than 2 blades. b. Reduced vibration as the point of maximum flapping and maximum feathering can be phase shifted away from each other. c. Reduced flapping angles due to phase shifting the point of maximum flapping from maximum feathering with either positive or negative delta 3 hinges. d. Damping of tail boom oscillations dependent on the structure of the tail boom Of course, as with everything on helicopters it is not a free ride, so the choice of hinge could also increase vibration. The choice of hinge is also heavily dependent on whether the blade has a hinge offset. A negative delta 3 with a hinge offset is a recipe for high vibration levels according to the research.

Delta 3 produced by hinges

The delta 3 hinge can be created by introducing a change in the hinge itself or by locating the pitch change rod such that the motion of the blade as it flaps introduces the appropriate motion. Typically tail rotor pitch control rods are located in front of the blade for positive delta 3 effects or aft of the blade for negative delta 3 effects. The lead angle of positive delta 3 inputs tends to be larger than the lag angle of negative delta inputs due to the potential for further vibration effects at large negative delta 3 angles.

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  • $\begingroup$ Thank you for the informative response! $\endgroup$
    – Gerry
    Commented Mar 25, 2023 at 7:22

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