Hooke's joint

[Arm out the window]

Maybe Hooke was smoking one when he came up with the concept, I'm not sure.

All I remember about it from principles of flight lessons was that the guy giving the talk had 2 rulers held together with a bit of a coning angle, then he tilted them forwards and said "there you go, like that...Hooke's joint effect!"

Now I never understood that, funnily enough...would anyone like to have a go at explaining what it is and how it works in layman's terms, if possible?

[Lu Zuckerman]

This space reserved. I have to go to a medical appointment. When I return I will respond.

I just got back. Much of what I was going to say was said in the next post however the below listed paragraph taken from that post is not correct.

“When the rotor disc is tilted by blade flapping, the two athwartship blades, in order to maintain a constant velocity in the plane of rotation, must move on their drag hinges to the position shown. If drag hinges were not fitted, the blades would be forced to accelerate and decelerate with every revolution”.

It states that if drag hinges were not fitted the blades would accelerate and decelerate with every revolution. The blades do in fact accelerate and decelerate with every revolution. That is called leading and lagging. What the author of the quoted article should have said was this. If the drag hinges were not fitted the mechanical forces would cause the blades to bend spanwise moving forward and backward at the tip with the bending taking place throughout the length of the blade. This is what happens on a rigid rotor system such as that used on the BO 105 and BK117 as well as others that do not have lag hinges. It even happens on Bell Helicopters but the bending is limited by the underslinging of the rotorhead.

The term Hooke’s joint effect is not a good description of what is happening. The technical term is Conservation of Angular Momentum. Also, it should be noted that the advancing blade leads and the retreating blade lags. Another incorrect point made by the author is that the blades are 90-degrees apart during hover. (Assuming a four-blade system). This would indicate that in hover the blades form a cross. This is not true. The blades because of their inertia hang back slightly from the pure radial position. All leading and lagging takes place behind the radial position and, the only time the blades are ahead of the radial position is when you are autorotating or, applying the rotor brake.


[This message has been edited by Lu Zuckerman (edited 21 June 2001).]

[sling load]

I am quoting from a book by John Fay,"The helicopter and how it flies" Test Pilot and Instructor, Westland Helicopters 1967,

"Hookes Joint Effect, The inclination of the rotor disc is obtained by the blades moving up and down about flapping hinges during rotation. Thus although the rotor disc is inclined, the drive shaft remains fixed in the fuselage. When the rotor disc is inclined at any angle other than that normal to the drive shaft, the blades tend to move on their drag hinges in order to maintain constant speed. In the case of a hovering helicopter, theblades are at 90 degrees to each other with respect to the plane parallel to the plane of the hub.

When the rotor disc is tilted by blade flapping, the two athwartship blades, in order to maintain a constant velocity in the plane of rotation, must move on their drag hinges to the position shown. If drag hinges were not fitted, the blades would be forced to accelerate and decelerate with every revolution.

The situation occuring when the plane of the rotor disc and the drive shaft are not normal to each other is often referred to as "Hookes joint effect". A hookes joint is a universal joint, but is not a constant velocity joint. Thus when the two shafts, with the joint in between, are not in line, although one shaft is rotating at constant speed, the other is rotating in a series of accelerations and decelerations."

End of Quote.....

Hope this helps, cant get much better than that i guess.

[Arm out the window]

Thanks for your replies.
Our POF lecturer talked about Hooke's Joint Effect and Conservation of Angular Momentum (which I understood) as if they were two different things, but it seems that he was confused himself.

[Lu Zuckerman]

This will most likely result in my being jumped on by everyone that has attended a helicopter POF course.

A common analogy often used in describing lead and lag is the speeding up and slowing down of an ice skater as he/she moves their arms in or out while spinning around. They say as the skater brings his/her arms inward the center of mass is more closely concentrated and the speed of rotation increases and, when the arms are moved outward the center of mass moves outward and the spinning becomes slower.

Once this point is made they include a diagram of a rotor system where the blades are flapping above and below the pure radial position. The point made with this diagram is that the upward flapping blade has its’ mass closer to the center of rotation and as a result increases speed (leads). It is further pointed out that the downward flapping blade has its’ center of mass further from the center of rotation and it slows down (lags). There are inconsistencies in this theory. If you look at the diagram in flapping up the blade mass moves closer to the center of rotation and according to the laws of physics it should speed up. The downward flapping blade moves its’ center of mass also but it is also moving closer to the center of rotation and it too should also speed up. If this is the case, there is only leading and no lagging.

There is another major inconsistency in using this diagram, as it is not the correct diagram to use. This diagram represents a sideward view of the rotor system and it should be not a sideward view but a head on view of the rotor system. In looking at this view it can be assumed that the blade on the left is the advancing blade and the blade on the right is the retreating blade. Since the advancing blade is diving and the retreating blade is climbing it can be seen that both blades are at the same point in relation to the pure radial position and as such the respective masses of the blades are equal. Granted, the advancing blade at one point is higher than the retreating blade but the blades move up and down and are always equally distributed above and below the radial position and therefore the mass distribution is equal. The only conclusion that can be reached assuming I am correct is that there are the laws of conservation of angular momentum but the ice skater should be left out of it.


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The Cat

[Dave Jackson]

To: Arm out the window;

"the guy giving the talk had 2 rulers held together with a bit of a coning angle,"

I assume that the two rulers were held together end-for-end, with a slight coning angle between the two.

The concept of 'universal joint' [Hooke's joint] and 'knuckle joint' are only used, to my knowledge, in reguard to 2-blade teetering rotors. These joints represent the teetering hinge. The Bell 47 had an 'X joint' in its hub and I think this is where the expression 'universal joint' [Hooke's joint] was applied. I do not know of any currently made helicopter that now incorporates the second hinge, with the possible exception of the Safari (mini belle). All teetering hinges are now 'knuckle joints' but some people still refer to them as [Hooke's joints].

The speeding up and slowing down of the blades of a teetering rotor are the result of the Coriolis effect. The two blades accelerate and decelerate at the same time, therefor there is no requirement for lead-lag hinges.

Some people refer to this teetering action and the associate speed change as being the result of Coriolis [Conservation of Angular Momentum]. Others refer to it as being the result of the Hooke's joint action. They are really one and the same.

If he had the rulers crossed as an X, then I don't know what the h--l I'm talking about :)


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Project: UniCopter.com (http://www.synchrolite.com/)

[This message has been edited by Dave Jackson (edited 22 June 2001).]

[Arm out the window]

To Dave Jackson: The rulers in question were end to end, as you said. The lesson was in relation to 2 bladed Bell helicopters, although I think the reference material was the good old English AP3456 RAF series of books which the RAAF used too.

To Lu Zuckerman: I see what you're getting at regarding both blades having their centre of mass displaced an equal amount regardless of whether they were flapping above or below the axial plane, but thinking a bit further about this I figured that the reference that slingload used was probably talking about a rotor that has a significant coning angle.
In that case, in forward flight the C of G of the advancing blade would be continuously moving outwards from the axis until the blade reached the front of the disc; then it would move in again.
I'll have to do a bit more pondering about this point. Thanks for your responses.

Cheers!

[John Farley]

Lu

<font face="Verdana, Arial, Helvetica" size="2">The downward flapping blade moves its’ center of mass also but it is also moving closer to the center of rotation</font>

Only true if it flaps into a negative cone angle?

JF

[212man]

John,
Quite! (nice to see someone is reading)

Bell try and alleviate this problem by the use of pre-coning and underslinging the feathering axes relative to the teetering axis. This reduces the degree of movement of the blade C of G position relative to the mast and hence the lead/lag tendencies. Those that remain are absorbed by the head.

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Another day in paradise

[Lu Zuckerman]

To: John Farley

I checked the explanatory diagrams in several training texts issued by the service departments of two helicopter manufacturers. In one case the rotor system is depicted as a flat “V” with the blades connected to the center point. This indicates the hover-coning angle. I believe this would represent a rotor system similar to that used on the Sikorsky S-51 or possibly a Vertol CH-47 that both employ a spider hub with minimal offset. In the case of this illustration flapping is indicated by the tipping of the “V” with one part of the cone going below the hover position and the opposite side of the cone going above the hover cone angle.

In the other illustration the rotor is indicated by a flat plate with the blades coning about an offset hinge. In this illustration one blade drops below the flat plate (rotorhead) and the other rises above the flat plate (rotorhead). In both illustrations there is negative and positive flapping which means that the blade center of mass is moving equally towards the center of rotation.

Granted there is an interlock and the rotorhead and fuselage will tip in the direction of movement but in most helicopters the horizontal fin is controllable and this forces the tail down for a better ride. When this happens, the angular differences between the hover cone angle and the blade-flapping angle would appear to increase the flapping angle relative to the rotorhead while still moving the blade center of mass equally towards the center of rotation.


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The Cat

[The Nr Fairy]

As an aside, and off the top of my head, the only time I can thing ANY helicopter would have a negative coning angle is immediately after a hard landing, prior to the blades chopping the tailboom off.

In all other circumstances the rotor system is suspending the rest of the helicopter by the head, and therefore coning ( tips upwards ) is an inevitable consequence of increasing pitch to get the machine to fly.

Having said that, anyone out there managed otherwise ?

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For the last bloody time, it's "The En Ar Fairy" . . .

[4dogs]

Lu,

Is your axis of rotation the shaft axis or the coning axis? They are different and to my simple mind the crux of the confusion typifying this current debate.

For the rest,

The motion of individual blades in a dynamically interactive rotor system is extremely complex. For easier understanding of low order interactions, we tend to employ explanations based on models of other more simplistic systems. For example: when we are discussing the effects of coning in relation to the coning axis, we use Coriolis and the Ballet/Ice Dancer model; whereas when we are discussing the effects of tilting the coning axis in relation to the shaft axis, we use Hookes Joint Effect and the reversing semi-trailer model (for those of you who have ever wondered why the motion and the engine noise is sinusoidal at low RPM).

When you attempt to deal with the system and all of its interactions as a simultaneous observation, it is just too hard. If, however, you doubt me, get yourself a simple dynamics text like Bramwell, Gessow & Myers or Stepniewski & Keys and look at the equations of motion and the stress/vibration equations.

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Stay Alive,

[email protected]

[Arm out the window]

4Dogs,

Would you mind elaborating on the 'reversing semi trailer' model, and the comment about sinusoidal engine noise? I haven't come across these ideas before.
Thanks.

[Lu Zuckerman]

To: 4dogs

In the initial description of the rotor diagrams the blades were in the cone position with no introduction of cyclic. The diagrams were of the rotor system as viewed from the left-hand side.
In this diagram the rotating and driven axes are coincident with each other. With the introduction of forward cyclic the disc tilted forward and at this point the driven and rotating axes separated and lead and lag began. The diagrams showed the rotor disc in both conditions so there is no confusion.

To: The Nr Fairy

Someone above used the term negative flapping. This is not the same as negative coning. If you can visualize the rotor system in pure hover the rotor system is coned and is supporting the weight of the helicopter. With the introduction of forward cyclic the disc tilts taking the tip path past what would be termed the pure radial position of the blades. I believe this is what negative flapping meant.

To: John Farley

The basis of my stating that the Ice Skater analogy is not applicable is because the blade mass of the upward flapping blade is closer to the center of rotation and therefore, the blade leads. Assuming negative flapping to the same extent of upward flapping the blade mass is just as close to the rotational axis as the downward flapping blade. That is, if you view the disc from the left side. (See above). However if you view the disc head on you can see that the laterally disposed blades are on an equal plane and thus there is no difference between the blades. This is not to say that the mechanical forces are not there (conservation of angular momentum). I just feel that with adequate investigation it can be proven that the Ice Skater analogy is not consistent with what is happening in the rotor system. The only other alternative theory is that the upward flapping blade over the tail and the downward flapping blade over the nose react 90-degrees later much like gyroscopic precession as lead and lag are effected in the advancing blade and the retreating blade. (This is a joke).


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The Cat

[Dave Jackson]

It appears that there might be some confusion in differentiating between the 2-blade rotor, with a teetering hinge, and the rotor with 3 or more blades, which has both flapping and lead-lag hinges.

2-blade teetering rotor:
It can be said that these blades do not flap, because they do not have flapping hinges. They teeter about their teetering hinge. The hub of this rotor has a pre-coning angle of approximately 3-degrees. At hover, (average loading conditions) the teetering hinge is located on the line that passes between the center of masses of the two blades. Therefore, as the blade-yoke-blade assembly teeters the masses of both blades will move in toward the mast, in unison. Therefor they accelerate and decelerate in unison.

3+ -blade rotor:
These blades flap independently of one another. These blades cannot maintain an equal distance between their center of masses and the mast, therefor they for they must have lead-lag hinges so that they may accelerate and decelerate independently.

The coning angle on a teetering hinge must never be allowed to go negative, because this implies the rotor has lost its lift and there is the risk of mast bumping. The blades on both type of rotor can teeter/flap into a negative angle during maneuvers.

-------------
The above has not been proven, qualified, confirmed and possibly not even requested.


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Project: UniCopter.com (http://www.synchrolite.com)

[This message has been edited by Dave Jackson (edited 25 June 2001).]

[212man]

DJ,
If one blade lags as the other leads ) that is quite a torque on the mast/hub. What actually happens(in an underslung teetering head)(draw a simple diagram) is that as the blades flap about the teetering hinge, the c of g does not move relative to the mast.

Of course this is nothing to do with Hooke's Joint effect and everything to do with coriolis effect.

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Another day in paradise

[Lu Zuckerman]

To: 212 Man

You said," If one blade lags as the other leads ) that is quite a torque on the mast/hub. What actually happens(in an underslung teetering head)(draw a simple diagram) is that as the blades flap about the teetering hinge, the c of g does not move relative to the mast".

This is exactly what happens on a Robinson head. Since the blades are free to flap on the cone hinges the blades will lead and lag. Since there are no lead lag hinges the lead lag action is reacted by the cone hinges which will wear in an elliptical pattern. This load is further reacted by the teeter hinge and transmitted directly to the mast.

While all this is going on, the blades are flexing in plane which in my mind can lead to instability and/or divergence.



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The Cat

[Arm out the window]

To DJ:

Surely with a 2-bladed teetering head, there is still a coning angle from preconing and from blade flexing, so whenever the disc is tilted away from the mast axis the C of G of the blades must move radially, causing the blades to try to lead and lag?

For example, if the disc is tilted forward, wouldn't the front blade's C of G move outwards from the mast and the rear blade's move inwards, no matter what sort of hinge arrangement the aircraft had?

[Dave Jackson]

To: 212man

I think you'll find that the act of one blade leading and another blade lagging is an in-plane force, and without lead-lag hinges it would attempt to bend the mast, not necessarily attempt to change its rotational speed.

I agree that in a rotor with a teetering hinge, and under the 'average' operating condition (say ~ hovering out of ground effect), the center of the rotor's mass will be concentric with the center of the teetering hinge. Exceptions to this 'average', such as a different gross weight, flight maneuvers etc. will cause this center of mass to move away from the center of the teetering hinge. This means that all the 'centers' that relate to a rotor hub are in a constant state of motion around and about each other, during flight.

I understand that a teetering-hinge rotor does experience a very small amount of lead lag, but it is too small for helicopter manufactures to be concerned with and they let the mast take care of it. The Robinson, with its flapping/coning hinges, in addition to its teetering hinge, no doubt presents a different scenario. I have no idea whether the triple hinge is an advantage or disadvantage, and this, of course, is the subject that Lou is concerned about.


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Project: UniCopter.com (http://www.synchrolite.com/UniCopter_Index.html)

[Lu Zuckerman]

To: Arm out the window

The design of the Bell two blade systems (post 47) underslings the rotorhead so that when the blade disc is tilted by cyclic input the driven axis is behind the driving axis the sole purpose of which is to minimize if not eliminate the tendency to lead and lag. On the larger Bell models any tendency to lead and lag is reacted by the drag brace and transmitted into the rotorhead.

There is another force at work and that is spanwise bending. This is caused by the chordwise CG not being in line with the pitch change axis. The mechanical forces inherent in the rotor system will cause the blade to bend spanwise so that the tip of the blade is in alignment and the two points are coincident with each other (at least at the tip). On the Bell blades this is reacted by the drag brace or on the 206 series by the pillow blocks. This is common on most helicopters and all manufacturers to combat the problem use a common solution. What they do, is to design the rotorhead so that the pitch change axis is slightly ahead of the rotating axis. If you were to diagram this out the blade pitch change axes are not in alignment with each other. On some helicopters this offset can be ¼” or more on larger rotor systems. On those systems that do not use this design solution the blades are designed with stiffeners to absorb the spanwise bending.


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The Cat

[Dave Jackson]

To: Arm ot the window;

The last few posts may have provided you with an answer; but if not, here goes.

The 2 blades form a shallow 'V' of about 3-degrees each off the horizontal.
The teetering hinge is 2" above the bottom of the 'V' and this 2" is called the undersling.
Let's say that the tip of each blade is 4" above the bottom of the 'V'.
Lets also say that the GofG is half way out each blade.
At half way out each blade, the CofGs will be 2" above the bottom of the 'V'.
The two GofG's and the teetering hinge will be in a line since they are all at the same 2" elevation.
The blade tips are another 2" higher at 4" elevation.

As the rotor teeters; one GofG will move UP AND IN toward the mast's centerline and the other GofG will move DOWN AND IN towards the mast's centerline.

Meanwhile, unlike the CofG's, one of the tips will move UP AND IN toward the mast's centerline and the other tip will move DOWN AND OUT away from the mast's centerline.

If this is hard to visualize, then just sketch it out, making sure that the CofG-teetering_hinge-GofG line is at right angles to the line representing the centerline of the mast. This way you will see that the CofG's can only swing inward.

The mass of both blades is moving inward toward the mast an equal amount, therefor(e) because of the conservation of angular momentum (Coriolis or Hookes Joint Effect ~ your pick) there is no lead-lag between the two blades.


To: 212man;

Just a bit of useless trivia.
The universal joint was originally known as a Cardan or Hooke's coupling.


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Project: UniCopter.com (http://www.synchrolite.com/UniCopter_Index.html)

[imabell]

the c of g of the blades on a teetering system move in toward the centre of the disc and outboard to the tip as the blades flap up and down. this causes the blades to accelerate and decelerate. this is called coriolis effect.
this occurs most noticeably in the flare for an increase in speed and in a bunt or pushover for a decrease in speed.
underslinging the head reduces the stresses on the head due to hunting for the centre of pressure by the blades again as they accelerate and decelerate nomally in forward flight. underslinging is one of those great little inventions we would all like to have come up with, like flapping hinges and delta 3 hinges.
hookes joint effect describes a theoretical axis of rotation that occurs mainly on rigid and fully articulated heads. as the hub is fixed firmly to the mast and the blades are free to flap and drag individually they are at different phases of acceleration and deceleration at exactly the same time. they are also flapping at different rates at the same time as they are in different positions around the disc, unlike the teetering head where everything is directly opposing. when one blade flaps up the other blade flaps down, when one blade speeds up the other does exactly the opposite.
the result of these differing phases on the blades means that at different positions on the disc that the blade tips are not equidistant from the mast axis, the mathematical result is a new theoretical centre of rotation somewhere in space not too far from the mechanical centre.
this point may in fact move about due to unequal damper condition and other factors.





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your too high,your too low, your too fast your too slow

[Arm out the window]

Thanks again for your thoughtful replies.

Imabell, I'd have thought that airflow changes would contribute to flare effect much more than C of G movement, especially if what Dave Jackson says about the C of G movements of a 2-bladed underslung teetering head design is correct.

For a bunt, for instance, the C of Gs presumably move in when the disc is tilted forwards, but the rotor slows down.

(or should that be Cs of G?!)

[This message has been edited by Arm out the window (edited 26 June 2001).]

[This message has been edited by Arm out the window (edited 26 June 2001).]

[Dave Jackson]

To: Arm out the window;

What Imabell is saying is correct. My posts are about the teetering of a 2-blade rotor. He is talking about a separate but related subject, the changes in the coning angle.

All rotors will cone [except for the (theoretical) UniCopter :)]. The pre-cone in a teetering hub, of approximately 3-degrees, is the manufacture's estimate of what the average coning angle will be.

As Imabell says;
During a flare; 1/ the load on the rotor disk is increased, 2/ all the blades and/or flapping/coning hinges bend upward, 3/ this causes all the blades' CofGs to move inward, 4/ this causes the rotor to accelerate (Coriolis).
During an unloading of the rotor disk; 1/ the centrifugal force will try to take the conning angle of all the blades to zero, 2/ the CofGs will move outward, 3/ this causes the rotor to decelerate (Coriolis).

To: Imabell;

Your explanation of the Hooke's Joint Effect is different from what my understanding is/was, but yours certainly makes sense. Perhaps it has been used over time to describe two different activities, one related to the teetering rotor (Bell 47) and the other related to the rigid and fully articulated rotor's lead-lag & flapping hinges.



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Dave J. UniCopter.com (http://www.synchrolite.com)

[imabell]

to arm out the window

there are valid arguments to a lot of what is thought of as gospel in aviation.

remember that it is a theory of flight. the vectors that are drawn to show the foces at work are a mass of variables.

we know why lift is produced on an aicraft wing that is an aerofoil shape care of that great bloke bernouli and his fluid dynamics but most light helicopter blades are symmetrical to reduce centre of pressure problems so isaac newtons law makes just as much sense when applied to symmetrical helicopter rotor blades. equal and opposite and all that.

why for instance do we flare at the bottom of an autorotative descent?
it is primarily to reduce our rate of descent and the secondary effect is reduction of forward airspeed.

how does it do that? most would say that the angle of attack is increased and the thrust vector is tilted backwards slowing us down.
and thats it, is it?

we know that due to coriolis effect that the blades will spool up in the flare, that is why we carry out forward speed autos', to conserve rotor rpm till it's needed at the bottom then convert the forward speed to more rotor rpm.

as we are slowing down due to the flare our rate of descent is also slowing , then obviously our rate of descent airflow is decreasing therefore our angle of attack is also decreasing.

as the speed of the blades is increasing due to our mate coriolis the angle of attack on the blades is being reduced even further. as a rotor blade accelerates it will flap up inducing an airflow from above reducing the angle of attack. and we still have the collective on the floor.

the change in the angle of attack during the flare is minimal and then two actions take place immediately that constantly reduce it till pitch is pulled.

never stop searching

just something else to ponder. http://www.pprune.org/ubb/NonCGI/confused.gif



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your too high,your too low, your too fast your too slow

[imabell]

to dave jackson

i have never actually seen hookes joint effect related to a teetering system it is usually only related to rotor systems that have three or more blades and articulated.

the concept of the teetering head rather than flapping hinges on each blade as juan de la cierva pioneered {and frank now uses) was a leap forward in helicopter flight back in those days.

the gimbal ring is attached to the mast but you can't attatch the blades to the gimbal
because the hunting stresses would fracture the grips in no time.

the underslung yoke with the grips attached at right anles to the gimbal teeter bearings allows the yoke to rock back and forth.

as the blades are rotating at six times a second approx. the hunting factor is minor and a small amount of rocking of the yoke and teetering of the gimbal ends up as a small twisting motion that alleviates all stress problems.

all very interesting

[Arm out the window]

Imabell, I agree with most of what you say, except the part about the angle of attack changes in the flare being minimal.

I always thought (open to correction though!) that it happens like this:

As you commence and develop the flare, the aircraft's attitude is raised significantly with respect to the relative airflow.
For example, if you fly level at a highish speed and enter autorotation, you can maintain your height by trading off speed. The flight path is essentially straight, but as the attitude goes from level-ish to quite nose up, there must be a reasonably big change in the angle at which the airflow strikes the disc (coming more and more from the bottom, so to speak).

You can maintain rotor rpm as you do this, and as the Nr is essentially staying constant, the coning angle should also stay the same. What then is maintaining your Nr?
I say it is the changing angle of the relative airflow with respect to the disc which allows the total reaction to tilt more and more forward, maintaining autorotative force.

Therefore, I don't think that conservation of angular momentum is playing such a big part here as airflow is.

Could be total rubbish I'm talking, of course!

[Helmet]

Let me wade in here and make a fool of myself.

Hookes Joint Effect is not anything to do with blade conning.

Blade conning is the term applied to the rotor tip path plane as a whole, not the path of an individual blade (which is called flapping). It is the angle between the tip path plane and the axis of rotation.

Conservation of Angular Momentum is not Coriolis Force, although Coriolis Force is related to the theory of Conservation of Angular Momentum. (is that why the blades go the other way in France?)

The ice skater demo is accurate. As the chick moves her arms in, she is moving mass inwards, and, in accordance with the theory of conservation of angular momentum, rotational speed increases. If you flare, the helicopter pulls harder on the main rotor system as the main rotor system attempts to deccelerate the fuselage. This causes the blades to cone (as distinct from flap). Blade coning causes the CofG of each blade (regardless of blade numbers) to move toward the centre of rotation and RRPM increases. This has nothing to do with Hookes Joint Effect.

And so we return to the original question....

As blades lead, lag or flap, the CofG moves either toward or away from the center of rotation at the hub. As each blade pivots at the hub, the greatest distance the CofG can be from the hub is when the blade is exactly at right angles vertically to the rotor mast, and horizontally in plane. Accordingly, as the rotor disk will have a conning angle in powered flight as a "steady state", a flap up of the blade will cause CofG to move in, and a flap down will cause a move out until it goes past the exact right angle position. In lead and lag, it depends upon movement of the blade away from (CofG moves inward) or toward (CofG moves outward)the steady state right angle position in plane. Blades in a multibladed system can move far more independantly than a two bladed system, thus causing overall CofG to be constatntly shifting in flight.

But all of this is semantics. Now for a simple explanation: As blades lead, lag, and flap, the CofG of the blade moves in relation to the center of rotation. As each blade moves differently, the CofG balance across the rotor system is constantly changing, causing moentary out of balance conditions and the resulting rotor vibration through the airframe. This is known as Hookes Joint Effect. To counter this effect, various systems have been employed. Sikorsky have used a Bifilar system of weights, the BK-117 has pendulum weights on each blade, etc. These systems are designed to react to the harmonics caused by Hookes, shift position and thus readjust the overall CofG of the system to reduce/minimise vibrations.

Hope this helps Arm out window.

My Helmet is now on fire.....aaagggghh

[Grey Area]

Absolutely, remember this is a dynamic system, the demonstration using rulers convinces the questioning tyro but leaves a seed of confusion for later in his career.

The most important part of helmets explanation is that it is the dynamic movement of the blade CofG that matters.

In an even bladed system the opposing 3 & 9 oclock blades will APPEAR to be in mirrored positions (about the longitudinal axis) BUT will have different, opposing velocities and should be examined independantly as you cannot introduce false symmetry and ignore the dynamics in the explanation. Hookes Joint Effect occurs in any head, but you may not see it. In a 2 bladed teetering head one could detect bending moments, in the Robinson you will see it, and so on.

[Arm out the window]

Yes mate, it does help, thankyou.
There's been some lack of agreement about what is or isn't Hooke's Joint effect, but the discussion has touched on a lot of points that I hadn't considered before, so good on all you learned contributors for your input!