#### C. Beaty

##### Gold Supporter
The Australian Aborigines figured out rotor physics several thousand years ago when they developed the boomerang. It flies a curved path as a result of aerodynamic forces causing gyroscopic precession but does it flap and lead/lag while in flight?

A helicopter/gyro rotor neither flaps or lead/lags when viewed from the real axis of rotation, the tip plane axis. Viewed off axis, we can imagine all sorts of motion taking place; for example, the valve stem of a rotating bicycle wheel moves nearer and farther away from the viewer if not viewed along the real axis of rotation so we could say it is flapping but that’s only an illusion.

JC’s plank on a rope should have brought clarity to lead/lag/flap confusion but maybe not.

#### Jean Claude

##### Junior Member
As always, coming from Chuck, this is an excellent example.
Seen from an observer on the ground, the accelerations of the valve stem are a very complicated mathematical exercise,

While for the physicist, it is nothing complicated: Only the centrifugal acceleration

#### Smack

##### Re-member?
My physics professor said that "centrifugal" is, at best, misleading; there is only 'centripetal' (positive and negative signs) force.

#### WaspAir

##### Supreme Allied Gyro CFI
Your professor was expressing a perfectly reasonable preference for inertial reference frames and the avoidance of "false" apparent forces. An astronaut candidate getting a 9-g centrifuge ride will have difficulty imagining the force as "false".

I used to use the cycloid path of a bike valve stem as an example when teaching pre-calculus students how to write parametric equations (it produces a nice pair with a sine on one axis and a cosine on the other). If you start a demo with a bicycle held stationary at the front of the classroom, with the valve stems on both wheels at the bottom of their travel, and then ask which way the stems will move when you start to push the bike forward, most will predict that it goes backwards at first, and they are surprised to see the real shape.

#### Jean Claude

##### Junior Member
Brian, To dissociate centrifugal force and centripetal force is wasted time. Simply impossible.

#### kolibri282

##### Active Member
Very interesting paper Javier, it seems to support the view held by naca investigators, that it is sufficient to calculate the Fourier sieries for flapping up to second order. How did you come across a German report, do you speak the language? Actually the author makes things even more complicated by using fairly complicated dependent clauses, well them Germans, you know....;-)

#### XXavier

##### Member
Very interesting paper Javier, it seems to support the view held by naca investigators, that it is sufficient to calculate the Fourier sieries for flapping up to second order. How did you come across a German report, do you speak the language? Actually the author makes things even more complicated by using fairly complicated dependent clauses, well them Germans, you know....;-)
I read German well enough for simple, specially technical, texts. I find Kant impossible, but I'm able to read, and with delight, the discussions between Naphta and Settembrini in 'Zauberberg'. There was a time when I also spoke it fluently, but that was long ago...

I'm not sure now, but I probably found that paper in https://b-ok.cc/ searching by 'Tragschrauber' or 'Hubschrauber'. Or perhaps I found it in the website 'ANNO' http://anno.onb.ac.at/ , where one can find really surprising things, like the 'Völkischer Beobachter' (Vienna edition)... Some websites are really fascinating...

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#### bryancobb

##### Junior Member
Whoever it was that coined the term "flapping," was putting fertilizer in the soil in the garden of argument. I suspect it was Cierva or maybe he just discovered it happening and may Art Young gave it the messed-up name?

No one disagrees with the fact that a tilted rotor disc has it's own rotational axis. No one disagrees with the fact that if that rotating rotor is analyzed from that invisible axis, (like Chuck's plank on a rope illustration) no flapping is occurring and the mass centroid of every blade is the same distance from that invisible axis which means constant rotational velocity thus no lead/lag.

Now, at the same time all that eqilibrium is present about the invisible axis when a plank and a sting represents the rotor, we need to look at the helicopter rotor like a universal joint (with only two rotational axes and a single coupling), not a plank on a rope.

There exists a second axis. The rotor mast or input axis. It is not invisible. The flight controls, delta hinges, pitch links, swashplate, and some of the other components that affect blade pitch are constrained by that mast and rotate about its visible input axis in a circular path, some of those components and rotate about the tilted, invisible output axis in an elliptical path.

When analysis is done in relation to the axis of the rotor mast and the body of the helicopter, there is absolutely no equilibrium. Watch this constant-velocity rotating thingy and tell me if equilibrium exists about the visible rotational mast.

In a constant RPM drive system, universal-joints are usually used in pairs to cancel the accel/decel of the output shaft. A helicopter rotor is a drive system that only has a single universal-joint. The rotational axes of input and output rarely coincide. Any time the invisible axis of a tilted rotor disc (output) is not along the axis of the rotor mast (input), there IS a change in rotational velocity of any component of the rotor head that travels in an elliptical path.

The fact that some rotating components that are connected to each other are traveling in a circular path and some are traveling in an elliptical path, necessitates hinges and a dampers be installed.

Based on the videos below, list the components in a typical helicopter's entire rotor system that belong in the INPUT group and the OUTPUT group.
All the input items have a constant rotational velocity. All the output items are constantly seeking a different rotational velocity (subject to Newtons 2nd).

My argument is: The "plank-on-a-rope" principle does not apply. The "2-axis universal joint, single coupling" principle does apply.

INPUT (circular) OUTPUT (elliptical)
Rotor Mast Flapping Clevis or Blade Mount Yoke
Hub Feather Bearings
Flapping Bolts Retention Nuts
Flapping Bearings Grease
MRGB Driven Gear(s) Rotating Star of Swashplate
Hardware Elastomeric or Hydraulic Lead/Lag Dampers

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#### XXavier

##### Member
Very good example... The red-tipped rod of the cross represents the longitudinal axis of the blades of a semi-rigid rotor, as usual in our gyros. Hence, and as seen from the left-hand, 'input' axle, the blades flap, but don't feather. On the other hand, and as seen from the right-hand, 'output' axle (perpendicular to the tip-path plane) the blades feather, but don't flap.

When the input and output axles are parallel, both feathering and flapping are, as expected, zero...

#### bryancobb

##### Junior Member
Now I have always been taught that a 2-bladed tetering rotor does not need lead/lag hinges and dampers because both blades speed up and slow down the same amount.

Really, if we take each rotating component of a 2-bladed rotor system and pigeon-hole it into ELLIPTICAL or CIRCULAR, we will quickly see that the rotor mast has a constant rotational velocity and all the components that rotate about an invisible tilted axis are always seeking a different rotational velocity due to their elliptical paths.

My thought is...this is why we just accept by faith that all 2-bladed rotors have an unavoidable 2-per-rev? The teter bolt or pin is constant velocity and rotates with the mast. The center of mass of each of the components that rotate about the invisible tilted axis (output) travels in an ellipse and at a varying rotational velocity. That difference "thumps" or transmits forces onto the teter bolt or pin into the body of the aircraft.

I can't help but believe some talented designer could eliminate 2-per in two bladed rotors. It occurs to me that if a two bladed rotor had TWO properly-phased universal joints and three rotational axes instead of the current two, a smooth two bladed rotor could exist.

#### All_In

##### Gold Supporter
I can't help but believe some talented designer could eliminate 2-per in two bladed rotors. It occurs to me that if a two bladed rotor had TWO properly-phased universal joints and three rotational axes instead of the current two, a smooth two bladed rotor could exist.
Interesting, concept. Think I'll add this to see if the colleges would like to tackle it for us with an open source project??

#### XXavier

##### Member
Now I have always been taught that a 2-bladed tetering rotor does not need lead/lag hinges and dampers because both blades speed up and slow down the same amount.

Really, if we take each rotating component of a 2-bladed rotor system and pigeon-hole it into ELLIPTICAL or CIRCULAR, we will quickly see that the rotor mast has a constant rotational velocity and all the components that rotate about an invisible tilted axis are always seeking a different rotational velocity due to their elliptical paths.

My thought is...this is why we just accept by faith that all 2-bladed rotors have an unavoidable 2-per-rev? The teter bolt or pin is constant velocity and rotates with the mast. The center of mass of each of the components that rotate about the invisible tilted axis (output) travels in an ellipse and at a varying rotational velocity. That difference "thumps" or transmits forces onto the teter bolt or pin into the body of the aircraft.

I can't help but believe some talented designer could eliminate 2-per in two bladed rotors. It occurs to me that if a two bladed rotor had TWO properly-phased universal joints and three rotational axes instead of the current two, a smooth two bladed rotor could exist.
The 'two per rev' vibration has nothing to do with the rotor head articulation, but is due to the fact that our semi-rigid rotors have two blades. Hence, in normal flight the relative wind strikes the blades 'full frontal' twice per revolution, and that 2/rev vibration disappears in vertical autorotation, when the horizontal velocity is zero...

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#### C. Beaty

##### Gold Supporter
Birds’ wings flap, rotor blades rotate.

How anyone can view JC’s plank and claim it’s flapping is beyond me. It is doing exactly the same thing as a seesaw rotor in forward flight.

Javier is correct inasmuch as 2/rev is a result of aerodynamic input, ie, 2 hits/revolution when the rotor is broadside to the airstream and nothing can be done about it except to isolate the vibration via a limber mast or something similar such as a slider.

Lead-lag hinges or inplane flexibility make the 2/rev vibration worse by allowing the rotor CG to move away from the axis of rotation. A soft inplane rotor/stiff mast assembly is likely to be resonant at or near the aerodynamic input; if so, watch out!

The combination of a seesaw rotor and teeter hinge forms a Cardan or Hookes U joint which causes a 2/rev velocity variation of the rotor drive shaft which must be accommodated by either a torsionally flexible shaft or in the case of the Bell-47 by mounting the engine/gearbox on soft rubber mounts.

#### bryancobb

##### Junior Member
Chuck ...in the rope and plank video, there's no driven and driving members. Inertia is the driving member. The universal joint is the rope and nothing is moving in an ellipse. The rope represents the rotor mast and it's not rigid. In my opinion, the plank is flapping in the classic sense of helicopter rotors and they way they have always been taught and understood.

#### C. Beaty

##### Gold Supporter
Bryan, rotor blades, whether seesaw or fully articulated, rotate in a perfect circle at uniform angular velocity because Mother Nature, constrained by the laws of physics, says they must. Flap and lag hinges are a mechanical necessity where the hub is not aligned with the tip plane. Floating hub rotors where the hub axis is aligned with the tip plane axis don’t require hinges other than feathering hinges.

The Doblhof WNF-342 Austrian helicopter from WWII was the first hingeless rotor helicopter, followed by the Doman LZ-5 and the McDonnell XV-1 convertiplane.

I'm perfectly aware of how rotor theory is taught in helicopter ground schools, taking the terminology used in scientific papers literally because that’s the convention.

Gessow and Myers, authors of one of the first really useful textbooks on helicopter theory repeatedly point out that flapping is an illusion resulting from viewing the rotor along an imaginary axis rather than along its true axis, the tip plane axi

The drawing in the previous reply is a scan from Gessow and Myers showing all the axes from which a rotor may be viewed but the tip plane axis in the real one. One can pretend a bicycle wheel is flapping when viewed off axis.

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#### Jean Claude

##### Junior Member
Bryan,
Conventional studies choose to assume a constant rotation of the drive shaft instead of assuming a constant rotation of the blade in the tip plane .
This is a misleading point of view, because the rotating inertia of the shaft (and engine) is much lower than that of the blades.
It make sense that 3 or more blades are articulated in flapping if it impose to the others a different rotation via the shaft. Then,
their own inertias are fighting each other.
Doman's Rotor or Seesaw rotor does not do it

#### bryancobb

##### Junior Member
Bryan,
Conventional studies choose to assume a constant rotation of the drive shaft instead of assuming a constant rotation of the blade in the tip plane .
This is a misleading point of view, because the rotating inertia of the shaft (and engine) is much lower than that of the blades.
It make sense that 3 or more blades are articulated in flapping if it impose to the others a different rotation via the shaft. Then,
their own inertias are fighting each other.
Doman's Rotor or Seesaw rotor does not do it
I'll agree. It's really some of the components of the hub that follow an elliptical path and speed up and slow down. The blades do not belong in that group.

My main gripe with this thread is, for decades...helicopter pilots have been taught rotor system aerodynamics in a certain way that may not pass muster scientifically when hairs are split, but it gives pilots a way of scuba diving in a cave called "Understanding Rotor System Aerodynamics" without stirring up the silt. This forum is not a scientific forum. It's a forum of recreational aviators who may very well be trying to pass an FAA Rotorcraft Written and those questions are not scientific either.

#### XXavier

##### Member
I'll agree. It's really some of the components of the hub that follow an elliptical path and speed up and slow down. The blades do not belong in that group.

My main gripe with this thread is, for decades...helicopter pilots have been taught rotor system aerodynamics in a certain way that may not pass muster scientifically when hairs are split, but it gives pilots a way of scuba diving in a cave called "Understanding Rotor System Aerodynamics" without stirring up the silt. This forum is not a scientific forum. It's a forum of recreational aviators who may very well be trying to pass an FAA Rotorcraft Written and those questions are not scientific either.
I believe that most pilots (and non-pilots) here wish to have a better understanding of the (aero)dynamics of rotorcraft, specially of gyros, and the rational, scientific answers given here by a few respected experts are highly valuable for most readers. In this forum, one learns a lot... Knowing the 'standard answers' for an exam may be important, but knowing the 'real', scientific answers is far more important...