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Posted

put up your educational background.

You too, Pokey, your "30 years around helicopters" appear to be somewhat unverified.

 

 

 

Eric, this is a discussion about RBS, that has turned into other worthwhile discussions, for the most part? most were very 'tactful' , i cannot say the same for you. However, if you want my qualification as a pilot/mechanic/engineer verified? Maybe you should start a new thread and title it "lets hear about how great we all are" I'm sure many would chime in.

Posted

And again Pokey dodges answering a relevant question.

 

Helonorth asked for my qualifications, I answered. If somebody wants to make fun of that, fill your boots.

 

Here comes a simple statement of fact:

 

There has never been ANY disagreement that flapping equalises lift. Not ever.

 

But that is only the first step. The second step is the disc tilting away from the relative airflow, flapping back. Pokey, your hero Juan knew all about this, because his disc was always flapping back. Look at your pictures again. It never tilted forward. The engine and propellor made it go forward, the disc was tilted back, giving lift and drag, not lift and thrust.

 

 

And here is the bit most of you are missing, because you are stuck on Figure 1.18 posted above (and what the heck is UNFLAP anyway?? You put your total faith on a book that doesn't even have proofreaders?) and never get past the bit about blades flapping away from the relative wind.

 

Third step, to STOP the disc flapping back, the pilot pokes in cyclic to MAKE THE DISC TILT FORWARD. Or else he isn't going forward.

 

Fourth step, the blade flaps down to the front and up to the back, or else the disc isn't tilting forward at all. Yes, it wants to flap away from the relative wind, but the cyclic feathering makes it do what the pilot wants it to do.

 

How hard is that to understand?

Posted (edited)

Let's keep the hilarity coming, Helonorth, put up your educational background.

You too, Pokey, your "30 years around helicopters" appear to be somewhat unverified.

 

UH-60, keep reading and get to the page that Chris referenced, where the pilot puts in forward cyclic to stop that flapping and continue in forward flight.

 

And I only taught Tom in a helicopter, he already knew how to fly planks, having his own P-51 Mustang.

I'm a pilot and former instructor, that's all. But I'm not the one challenging conventional wisdom on the subject. You are. As such, you need to bring a little more to the table than a 45 year old degree and having supposedly taught some crazy movie star how to fly. Interesting you're on a first name basis with "Tom". Call him up and discuss it with him and get back to us.

Edited by helonorth
Posted

S-55shadow_zps31b73075.jpg

 

The above still-photo (not a screenshot from a video) was taken by my cellphone while a buddy and I were cruising along in a Sikorsky S-55, doing a blistering 80 knots.

 

By the way, it is a three-blade rotor.

 

The rotor of the S-55 turns at 220 rpm, making it easy for just about any camera to "stop" the blades. The photo is interesting because it caught the rotor with one blade directly out in front of the ship. You can clearly see the advancing blade leading and the retreating blade lagging. This jibes with that illustration of a multi-blade rotor in forward flight as seen from the top - only this time we're seeing it from the bottom, in shadow. Same thing.

 

Dig the bending of the blades! It almost makes me not want to fly a helicopter ever again.

 

A rotor in forward flight is continually performing a delicate aerodynamic balancing act. The blades do all kinds of strange contortions as they make their way around the mast. To single one aspect out is not telling the whole story.

 

But...whatever the blades *want* to do, they most certainly do NOT flap up on the advancing side and down on the retreating side. We don't let them do that. We counter that natural tendency with cyclic pitch. Anyone who's ever gotten a direct side-view of a helicopter flying along would understand that. The tip-path plane is tilted *forward* with respect to the mast. It has to be.

Posted

Umm... NR, perhaps you are unaware of how a camera scans a picture, in this case top to bottom, so the blades are still moving as the focal plane moves down - hence the bending of the blades.

 

Helonorth,

there is no challenge to the "conventional wisdom" - your books explain Step 1 and Step 2 as above, and that's when you stopped reading. Read a bit further, and see that the wisdom is all there, you just stopped too soon. Older, more experienced dudes like NR have "got it", it will perhaps come to you too.

Posted

None of the other pictures that I've taken of S-55's in flight show the bending of the blades as drastically as the one of the shadow from directly above. It's possible that the camera is responsible for the blade bending illusion, but the lead and lag is still there even without the bending.

Posted

Bear in mind that the FAA Helicopter Flying Handbook is merely an introductory text, with overly simple explanations. Past editions have had numerous mistakes, and the current edition (from what I have skimmed) isn't much better. Case in point, the 'half venturi tube' lift theory...

 

When I was in flight school, the text was titled Rotorcraft Flying Handbook. It stated that a steep approach with a flare should be executed in a stuck right pedal situation. Would certainly not recommend doing that...

 

'Flapping to equality' in forward flight, with the greatest upflap occurring at the 3 o'clock position (counter-clockwise rotor), is yet another poorly explained f*ck-up.

 

You cannot base all of your knowledge on one poorly written text. And to use that one source against numerous other credible sources is merely ignorance.

Posted (edited)

Attention all flight instructors, student pilots and everybody else: tear out all the pages in any book that states that blade flapping neutralizes dissymmetry of lift and that the retreating side flaps down. Then burn them.

 

 

 

Since the dissenters insist on referencing it, I might as well show what the FM actually states. From FM 3-04.203 - Fundamentals of Flight, pages 1-13 and 1-14.

 

Huh, it says that in directional flight the advancing blade is flapping up and the retreating blade is flapping down. That's funny, who woulda thunk it?

 

You don’t have to tear out any pages, just understand the limits of the standard flight school textbook, with respect to aerodynamics.

 

Again, you’re looking at the basic teaching model of flapping, based around the early Cierva autogiro; the rotor was simply a lifting device. Roll and pitch control were obtained by ailerons on stub wings and by a conventional elevator. The autogiro’s thrust was obtained from an engine driven propeller, not the rotor.

 

The rotor was set at a permanent aft incline since the rotors turning force was from air inflow from below the rotor. In this format you had the advancing blade flapping up and the retreating blade flapping down, the entire flight. In the early Cierva autogiro the pilot did not have any mechanical device to control the rotor.

 

You need to read a bit more (pages 1-11 to page 1-14) as they tie in the pilot and the cyclic into the current day helicopter. The helicopter’s rotor provides lift, thrust, pitch, and roll, all controlled by the pilot.

 

Flapping is a way of partially relieving the unbalancing aerodynamic forces, it cannot automatically and perfectly balance the rotor because the flapping itself introduces pitching and rolling moments that upset the aircraft. Regardless of the flapping, if the pilot or stability system doesn’t continuously make the correct cyclic control inputs, the helicopter will pitch and roll, spiral out of control and destroy itself.

 

That old training format with the advancing blade flapping up and the retreating blade flapping down, the entire flight, is not how it works in forward flight. With the cyclic the pilot tilts the rotor disk forward, tilting the thrust vector forward as the cyclic feathering increases the pitch on the retreating side of the disk to trim the helicopter in forward flight (page 1-17 to page 1-23; page 1-41; Figure 1-33).

 

Download the manual and read a bit more.

 

Download Ref: Army Field Manual FM 3-04.203, Fundamentals of Flight

 

 

Cyclic Pitch can be used for two purposes: to trim the tip path plane with respect to the mast, and to produce control moments for maneuvering.

 

In the first case, the pilot can mechanically change the angle of attack of the blades by the same amount, as the flapping motion would have, thus eliminating the flapping.

 

This can be used to eliminate all of the flapping or leave just enough to balance pitching or rolling moments on the aircraft such as those due to an offset center of gravity.

 

Ref: Helicopter Performance, Stability, and Control (page 154); Raymond W. Prouty

 

Edited by iChris
Posted

And again Pokey dodges answering a relevant question.

 

 

 

again? (point out the 1st time) maybe had you not come across as so arrogant? & said my 30 years was unverified, i may have been willing to share with you my past 30 years experience and my educational background. Have you done a background check on me?

Posted

Are we almost to the bottom of this conundrum ?

 

dun,,dunnn dunnnn dunnn,,,,, just when ya thought it was safe to go back in the water

Posted

It's amazing how little some of you understand about how rotors work. Those of us who do (like iChris, Eric Hunt, etc.) realize that this "advancing blade flaps up" theory *ONLY* works in an autogyro and WILL NOT work in a helicopter...in fact does not work that way in a helicopter.

 

Let's say you had a helicopter that had no cyclic feathering avaliable to the pilot. And let's say the c.g. was trimmed for hovering flight. Now, say this helicopter does have a way of tilting the fuselage. (Look back at Igor Sikorsky's early versions of the VS-300. It had collective pitch control but no cyclic feathering, and horizontal props on outriggers at the rear for pitch and roll control because ol' Igor hadn't quite figured out phase lag yet.)

 

Okay, now you take your hovering helicopter and tilt the fuselage to a nose-down attitude. Sure enough, the helicopter will start to move forward. But the disk will quickly tilt back to a nose-up attitude and the speed of the aircraft through the air will be VERY limited due to what we now call "flapback." Or, if you continue to force the tail up/nose down you'll reach the limits of the flapping hinges.

 

Igor (who'd already stolen Cierva's flapping hinge thingee) found this out. He knew he had to overcome this unwanted flapping and somehow get the disk to stay tilted forward. The idea for a swashplate had been around since proposed by a fellow Russian inventor named Boris Badenov...sorry, Yuryev in 1911. Certainly Igor knew of Yuryev's work, and had actually installed a swashplate MR control system on the first VS-300 but it proved uncontrollable due to the aforementioned lack of understanding of phase-lag. So they installed the rear props for pitch and roll control.

 

Eventually in 1941, Igor modified the VS-300 to have a rotor with full cyclic feathering, did away with the three tail-rotor system in the back and...well, the rest is history.

 

So yeah, in an AUTOGYRO...or a rotor without cyclic feathering, the flapping hinge compensates for dissymmetry of lift and the advancing blade flaps up. But in a helicopter it is the cyclic feathering that overcomes the rotor's natural tendencies and forces it to not do that. If you want to say that the advancing blade of a rotor in forward flight flaps "up," fine. It just doesn't do that on a helicopter in which the rotor provides lift and thrust.

 

Here's a cute video of the VS-300 and Igor's vision of the future:

 

https://www.youtube.com/watch?v=SEAy2Yx_dY4

Posted (edited)

Once more into the breech...

 

1) I don't think anyone, anywhere in this thread has tried to imply that the disc is tilted to the rear during forward flight. I think it's pretty clear that's not the case (as someone has tried to use "it's pretty clear" as their argument).

 

2) I'm not saying those with dissenting opinions are wrong (or right), at all. But I would like for someone who has that opinion to actually try to teach and answer questions, instead of just going around and around in circles (no pun intended) and assuming that the information that they're putting out is informative, or understood. I've asked several questions on the matter, trying to learn, and get replies that are just a video someone else has done, that don't offer any different information than what has already been said (by myself, no less). Or "That's all that's necessary at the PTS level, but once you've been taken to task, you need to understand the rest of the story" without explaining the rest of the story, or if they are, without making sure the information has been communicated.

 

So, with that said, I'll try to ask some questions one last time (bolded for clarity in finding them), and phrase it differently. Don't take them as "gotcha questions". Take them as I'm a 5 hour pilot who's trying to learn. I'll also make assumptions, and try to state all of them that I'm making, and if they're wrong, again...please correct me. This offer is open to anyone.

 

A) We're sitting in a helicopter with a fully articulated rotor, 100% RPM, on the ground, cyclic centered, Collective increased only enough that we can all agree the blades are generating lift, in a no wind condition. I posit that we are not experiencing any of our previously mentioned aerodynamic conditions (dissymmetry of lift, RBS, flapping, or anything else.)

 

B) We push the cyclic forward, and in doing so, the rotor disc tilts forward. We can all agree on that? Further, can we agree that the advancing blade is flapping down, to its lowest at the nose, and up on the retreating side, to its highest at the tail? As has been said "it's pretty obvious", right? Let's make up a number to assign to these amounts of tilt and cyclic displacement, purely for demonstrative purposes. We'll say it's 10 degrees, and 2 inches of forward cyclic.

 

C) Now, let's introduce "forward" airflow. We're still sitting on the ground not moving, but directly off of our nose, we now have a wind of a steady state 20 knots. Question: What, if anything happens to the rotor disc if I don't move the cyclic? Second question: Why did that happen, if anything did?

 

D) Assuming something did happen... What would happen if we were to increase that wind to a steady state 40 knots?

 

 

My answers. Again, if I'm wrong, please try to explain, and help us all learn.

The disc will lose some of its forward tilt. Now instead of its "10" degrees, its only say, 7, with the cyclic still displaced 2" This happened because of the change in (tangential flow? The video posted in response to my last question on the phrase, while informative, had nothing to do with the subject, as I could tell) across the disk gave us an advancing and retreating blade, which we didn't have before. Advancing blade flaps up, retreating blade flaps down. DISC IS STILL TILTED FORWARD 7 DEGREES! No argument there. Net effect is that there's more flapping down on the advancing side than the retreating side, but it IS still flapping down on the retreating side. Otherwise, how could the disc have lost some of its forward tilt?

When we increase the wind to 40 knots, the disc once again loses some of its forward tilt. Now its at "5" degrees. If we now wanted to put it back to 10 degrees, we'd push the cyclic forward again.

 

This is what happens when we fly forward, except it happens on a curve. Assuming we take off into a no wind condition, we develop our own wind off the 12 o'clock in the form of forward airspeed. Every knot we move forward introduces another knot of differential airspeed (excuse me...that may be 2 knots, as you're adding to the advancing side and subtracting for the retreating side) between advancing and retreating sides. This creates the requirement to add more cyclic to increase airspeed. e.g. (warning, made up numbers ahead) to go from 0-10 kias requires 1 inch of total cyclic displacement. From 10-20 requires 2.25 total inches. 20-30 requires 3.75" and so on. Translation: cyclic displacement to forward airspeed is not a linear function.

 

Anyone?

Edited by CharyouTree
Posted (edited)

Once more into the breech...

 

DISC IS STILL TILTED FORWARD 7 DEGREES! No argument there. Net effect is that there's more flapping down on the advancing side than the retreating side, but it IS still flapping down on the retreating side. Otherwise, how could the disc have lost some of its forward tilt?

 

Translation: cyclic displacement to forward airspeed is not a linear function.

 

Anyone?

 

Are you still trying to get that retreating blade to flap down when the the disk is tilted forward?

 

If the disk is tilted forward the net results is:

 

the advancing blade is flapping down from its high point over the tail to its low point over the nose and the retreating blade is flapping up from its low point over the nose to its high point over the tail.

 

The only thing that's happened is the retreating blade is flapping up to a high of 7º over the tail vs. 10º

 

Until you get this clear in your mind, there's no need to deal with your other questions.

 

Translation: cyclic displacement to forward airspeed is not a linear function.

 

It was already shown in post #98 that it was not linear.

Edited by iChris
Posted

 

Are you still trying to get that retreating blade to flap down when the the disk is tilted forward?

 

If the disk is tilted forward the net results is:

 

the advancing blade is flapping down from its high point over the tail to its low point over the nose and the retreating blade is flapping up from its low point over the nose to its high point over the tail.

 

The only thing that's happened is the retreating blade is flapping up to a high of 7º over the tail vs. 10º

 

Translation: cyclic displacement to forward airspeed is not a linear function.

 

It was already shown in post #98 that it was not linear.

 

 

Great. I'll take that to mean you generally agree with the things I stated (that the wind would change the disk angle from 10 to 7). Could you now explain what's causing it to be 7 degrees instead of 10?

Posted (edited)

 

 

Great. I'll take that to mean you generally agree with the things I stated (that the wind would change the disk angle from 10 to 7). Could you now explain what's causing it to be 7 degrees instead of 10?

 

The change in forward flapping from 10º to 7º was due to the change in forward speed. The amount of flapping is proportional to the product of forward speed and rotor thrust (lift). In other words, if the blades have no lift to start with, the increase in forward speed will not produce flapping.

 

However, in most cases we can produce flapping under zero G conditions, zero lift, with cyclic inputs.

 

Your example requires the pilot not take an active part in the control of the helicopter.

Edited by iChris
Posted

 

The change in forward flapping from 10º to 7º was due to the change in forward speed. The amount of flapping is proportional to the product of forward speed and rotor thrust (lift). In other words, if the blades have no lift to start with, the increase in forward speed will not produce flapping.

 

However, in most cases we can produce flapping under zero G conditions, zero lift, with cyclic inputs.

 

Which is why I stated in the initial conditions that the blades were producing lift...

 

Ok...So why does forward airspeed reduce the disc from 10 to 7?

Posted (edited)

 

Which is why I stated in the initial conditions that the blades were producing lift...

 

Ok...So why does forward airspeed reduce the disc from 10 to 7?

 

The rotor disk pitches up 3º to an equilibrium point equal to the new airspeed and cyclic position, you have it fix at (helicopter on the ground). If the helicopter is in flight and the pilot doesn’t correct (cyclic remaining in its original position) the pitch up will also cause a roll moment.

 

The pitch and roll moments will increase in oscillation until the pilot takes control or the helicopter crashes. The dynamic characteristics (effects over time) of the helicopter are inherently unstable.

 

In your example the helicopter would end up in a constant state of acceleration due to the pitch and roll oscillations until the pilot takes control and regains equilibrium with corrective cyclic and collective.

Edited by iChris
Posted

 

The rotor disk pitches up 3º to an equilibrium point equal to the new airspeed and cyclic position, you have it fix at (helicopter on the ground). If the helicopter is in flight and the pilot doesn’t correct (cyclic remaining in its original position) the pitch up will also cause a roll moment.

 

The pitch and roll moments will increase in oscillation until the pilot takes control or the helicopter crashes. The dynamic characteristics (effects over time) of the helicopter are inherently unstable.

 

In your example the helicopter would end up in a constant state of acceleration due to the pitch and roll oscillations until the pilot takes control and regains equilibrium with corrective cyclic and collective.

 

I understand that the disc will tilt up. Through what mechanism is the blade caused to tilt back when introduced to an airflow coming from the 12 o'clock position (forward flight)?

Posted

 

I understand that the disc will tilt up. Through what mechanism is the blade caused to tilt back when introduced to an airflow coming from the 12 o'clock position (forward flight)?

 

The rotor will always flap away from the relative wind if not opposed, due to the change in lift caused by the increased airflow across the disk.

 

On the advancing side, due to the overall increase in lift, the advancing blade will start its flap down from the tail to the nose at a decreased flapping angle of 7° instead of 10°.

 

On the retreating side, due to the overall decrease in lift, the retreating blade will start its flap up from the nose to the tail at a decrease flapping angle of 7° instead of 10°.

 

In any case, the increase in airflow across the disk has caused a decrease in flapping angle from 10° to 7°; however, the rotor disk is still tilted forward.

 

We have to remember the starting and ending point when defining advancing or retreating blade flapping. The advancing blade starts its run from the tail and ends its run at the nose. The retreating blade starts its run from the nose and ends its run at the tail. The blades direction of movement from start to finish, will define if the net flapping is up or down.

 

Therefore, the net effect in your example is the advancing blade is still flapping down and the retreating blade is still flapping up, only the flapping angle has changed due to the increased airspeed across the disk.

Posted

 

The rotor will always flap away from the relative wind if not opposed, due to the change in lift caused by the increased airflow across the disk.

 

On the advancing side, due to the overall increase in lift, the advancing blade will start its flap down from the tail to the nose at a decreased flapping angle of 7° instead of 10°.

 

On the retreating side, due to the overall decrease in lift, the retreating blade will start its flap up from the nose to the tail at a decrease flapping angle of 7° instead of 10°.

 

In any case, the increase in airflow across the disk has caused a decrease in flapping angle from 10° to 7°; however, the rotor disk is still tilted forward.

 

We have to remember the starting and ending point when defining advancing or retreating blade flapping. The advancing blade starts its run from the tail and ends its run at the nose. The retreating blade starts its run from the nose and ends its run at the tail. The blades direction of movement from start to finish, will define if the net flapping is up or down.

 

Therefore, the net effect in your example is the advancing blade is still flapping down and the retreating blade is still flapping up, only the flapping angle has changed due to the increased airspeed across the disk.

 

Now you're using some of the same key words that I was earlier (or at least that I was thinking), and getting closer to acknowledging what I've been trying to say all along, and what (it seems like) everyone's feelings on the matter are.

 

Again...(to my knowledge) no one, anywhere, at all, has ever disputed that in forward flight, the blade has a forward tilt. The only dispute is whether in forward flight the retreating blade has a component of flapping downward. Yes, the blade still ultimately flaps up. Or, as we both said, "net flapping is up", but if that component wasn't there, the blade would be at our imaginary 10 degrees, instead of 7. The retreating blade is still flapping up. But not as much as if it didn't want to flap down.

 

Unless I'm missing something, this is what everyone has been saying, but the "dissenters" don't acknowledge that the blade has a tendency to want to downflap. (Not that it IS flapping down. But it's not flapping UP as much as it would otherwise, with a given cyclic displacement.)

 

I want to emphasize: the blade is NOT actively flapping down. But, as I understand it, and you seem to be agreeing with, it's not flapping up as much either, due to what we "less enlightened" call "flapping".

 

I think we're all on the same page, based on the sequence we've gone through here, Chris. But there's some misunderstanding due to the limitations of the medium (text).

Posted (edited)

 

Now you're using some of the same key words that I was earlier (or at least that I was thinking), and getting closer to acknowledging what I've been trying to say all along, and what (it seems like) everyone's feelings on the matter are.

 

Again...(to my knowledge) no one, anywhere, at all, has ever disputed that in forward flight, the blade has a forward tilt. The only dispute is whether in forward flight the retreating blade has a component of flapping downward. Yes, the blade still ultimately flaps up. Or, as we both said, "net flapping is up", but if that component wasn't there, the blade would be at our imaginary 10 degrees, instead of 7. The retreating blade is still flapping up. But not as much as if it didn't want to flap down.

 

Unless I'm missing something, this is what everyone has been saying, but the "dissenters" don't acknowledge that the blade has a tendency to want to downflap. (Not that it IS flapping down. But it's not flapping UP as much as it would otherwise, with a given cyclic displacement.)

 

I want to emphasize: the blade is NOT actively flapping down. But, as I understand it, and you seem to be agreeing with, it's not flapping up as much either, due to what we "less enlightened" call "flapping".

 

I think we're all on the same page, based on the sequence we've gone through here, Chris. But there's some misunderstanding due to the limitations of the medium (text).

 

Again, we have to remember the starting and ending point when defining advancing or retreating blade flapping. The advancing blade starts its run from the tail and ends its run at the nose. The retreating blade starts its run from the nose and ends its run at the tail.

 

The retreating blade always starts its run from its starting point, the nose. Due to the change in lift caused by the increased airflow the rotor disk position have changed from 10° to 7° forward tilt. In other words the flapping angle has changed 3°.

 

Also remember, the flapping is referenced with respect to the angle between the rotor disk (tip-path-plane) and the rotor mast. The cyclic is forward and the tilt is forward; moreover, the flapping angle is also forward with respect to the rotor mast.

 

The only down component is the advancing blade flapping down to its new 7° position.

 

The retreating blade has no effective component of flapping down in forward flight once the pilot is on the controls. The results of the net flapping are most relevant. The pilot would have cancelled out any such flapping with cyclic. In other words, flapping would have never reached 7°.

 

Your example is at best only relevant to, as you have placed it, when the cyclic is fixed in a set position. In which case, pitch and roll moments would further upset the helicopters equilibrium.

 

Cyclic Pitch can be used for two purposes: to trim the tip path plane with respect to the mast, and to produce control moments for maneuvering.

 

In the first case, the pilot can mechanically change the angle of attack of the blades by the same amount, as the flapping motion would have, thus eliminating the flapping.

 

This can be used to eliminate all of the flapping or leave just enough to balance pitching or rolling moments on the aircraft such as those due to an offset center of gravity.

 

In the second case, the pilot deliberately introduces an unbalanced lift distribution in order to make the rotor tilt for maneuvering. For example, if the helicopter is hovering and the pilot wishes to tilt the nose down, he pushes the stick forward, which tilts the swashplate down in front. The pitch of the blade at Ψ = 90º (advancing blade) is decreased and that at Ψ = 270º (retreating blade) is increased.

 

The resultant imbalance accelerates the right-hand blade down and the left-hand blade up. The rotor flaps down over the nose and up over the tail, tilting the rotor thrust vector forward to produce a nose down pitching moment about the center of gravity. The procedure is similar if the pilot wishes to pitch nose up or to roll in either direction.

 

Whether being used for trim or for control, the cyclic pitch is equivalent to flapping in that the changes in rotor conditions due to one degree of cyclic pitch are the same as those due to a one-degree change in flapping.

 

Ref: Helicopter Performance, Stability, and Control (page 154); Raymond W. Prouty

 

Edited by iChris
Posted

As a 200 hour CFII who has yet to instruct a student, I just want to thank everyone in this discussion for making me take a much closer look at a topic I thought I thoroughly understood.

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