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Flaps_50!
September 6th 09, 03:20 AM
I wonder, does anyone routinely recalculate limiting V speeds on the
basis of TOW? I assume that limiting Va speeds go with the square root
of the fraction MTOW loading so for 75% MTOW Va would drop to 86% Va.
But the question is, if Vne is limited by arodynamic issues such as
flutter or windshield how would that change with load? Put another
way, is Vne ever load dependent and/or does anyone use a rule like
that for Va?

A second sort of connected question is: is there any wing that can
produce more lift at 45 degrees AOA than at the stall point (I know
that most airfoils produce about the same lift at 45 AOA as at the
stalling point)? What I'm thinking about is wings with washout or drag
reducing devices that will reduce maximum lift at the stall point but
not the 'flat plate at 45 AOA' lift. Put another way, how much loss of
lift do we get from typical washout?

Cheers

Mike Ash
September 6th 09, 05:49 AM
In article
>,
"Flaps_50!" > wrote:

> I wonder, does anyone routinely recalculate limiting V speeds on the
> basis of TOW? I assume that limiting Va speeds go with the square root
> of the fraction MTOW loading so for 75% MTOW Va would drop to 86% Va.
> But the question is, if Vne is limited by arodynamic issues such as
> flutter or windshield how would that change with load? Put another
> way, is Vne ever load dependent and/or does anyone use a rule like
> that for Va?

I must admit that I never knew that Va went down, rather than up, with
weight. It makes sense now that I've read about the phenomenon, but this
is the first time I've heard of it.

As for Vne, my understanding of the causes of it (flutter, aerodynamic
loads, etc.) would indicate that it's not dependent on weight at all,
except for how weight might help you get to that speed, but I could very
well be wrong.

> A second sort of connected question is: is there any wing that can
> produce more lift at 45 degrees AOA than at the stall point (I know
> that most airfoils produce about the same lift at 45 AOA as at the
> stalling point)? What I'm thinking about is wings with washout or drag
> reducing devices that will reduce maximum lift at the stall point but
> not the 'flat plate at 45 AOA' lift. Put another way, how much loss of
> lift do we get from typical washout?

The coefficient of lift is at its maximum at the stall angle of attack,
by definition. It's not possible for 45 degrees to give the same lift as
stall unless the wing actually stalls at 45 degrees AOA, and that would
be really unusual. Don't know the answer to the washout question, but I
think it would be better framed as how much unnecessary drag is
produced, rather than "loss of lift", which is confusing because a wing
always produces the same amount of lift for a given weight in steady
level flight no matter what the speed, AOA, or wing configuration.

--
Mike Ash
Radio Free Earth
Broadcasting from our climate-controlled studios deep inside the Moon

Blanche
September 7th 09, 03:04 PM
For a complete discussion *why* Va changes with weight (and is the only
V speed that does, at least in our small pistons) go find Kershner's
book - any of Kershner's books. The equation is there, also, so you
can create a table of your favorite weights.

Peter Dohm
September 7th 09, 11:39 PM
"Blanche" > wrote in message
...
> For a complete discussion *why* Va changes with weight (and is the only
> V speed that does, at least in our small pistons) go find Kershner's
> book - any of Kershner's books. The equation is there, also, so you
> can create a table of your favorite weights.

Kerschner did give an excellent treatise on why and how Va (maneuvering
speed) changes weight, and Langeweische may have done so as well. But, Va
is most cetainly not the only V-speed that changes with wieght, and a
partial list should include:
Vx (best angle of climb speed)
Vy (best rate of climb speed)
Vso (stall speed--cleam)
Vc (design max cruising speed)
Vr (rotation speed) althought more commonly associated with heavy
aircraft operation and arguably related to Vsse and Vmc in multi-engined
aircraft.

There are more, but some of the more technically interesting examples, such
as Vh (maximum level flight speed at maximum power), might be considered
more appropriate for rec.aviation.homebuilt

I agree with you about Kerschner's calculation, and also suggest that it
could also be expecially usefull with regard to Vso; although the practice
is probably not approved for light aircraft and would be questioned it an
incident were to occur. Be aware, however, that weight based tables for all
of the common V-speeds are routinely used on transport aircraft.

Peter

Flaps_50!
September 11th 09, 09:49 AM
On Sep 6, 4:49*pm, Mike Ash > wrote:
> In article
> >,
>
> *"Flaps_50!" > wrote:
> > I wonder, does anyone routinely recalculate limiting V speeds on the
> > basis of TOW? I assume that limiting Va speeds go with the square root
> > of the fraction MTOW loading so for 75% MTOW *Va would drop to 86% Va..
> > But the question is, if Vne is limited by arodynamic issues such as
> > flutter or windshield how would that change with load? *Put another
> > way, is Vne ever load dependent and/or does anyone use a rule like
> > that for Va?
>
> I must admit that I never knew that Va went down, rather than up, with
> weight. It makes sense now that I've read about the phenomenon, but this
> is the first time I've heard of it.
>
> As for Vne, my understanding of the causes of it (flutter, aerodynamic
> loads, etc.) would indicate that it's not dependent on weight at all,
> except for how weight might help you get to that speed, but I could very
> well be wrong.
>
> > A second sort of connected question is: is there any wing that can
> > produce more lift at 45 degrees AOA than at the stall point (I know
> > that most airfoils produce about the same lift at 45 AOA as at the
> > stalling point)? What I'm thinking about is wings with washout or drag
> > reducing devices that will reduce maximum lift at the stall point but
> > not the 'flat plate at 45 AOA' lift. Put another way, how much loss of
> > lift do we get from typical washout?
>
> The coefficient of lift is at its maximum at the stall angle of attack,
> by definition. It's not possible for 45 degrees to give the same lift as
> stall unless the wing actually stalls at 45 degrees AOA, and that would
> be really unusual. Don't know the answer to the washout question, but I
> think it would be better framed as how much unnecessary drag is
> produced, rather than "loss of lift", which is confusing because a wing
> always produces the same amount of lift for a given weight in steady
> level flight no matter what the speed, AOA, or wing configuration.
>

The reason why I riase this is beacuse the variation of Cl with lift
is rarely shown at high AOA but I found some test diagrams and they
show that for typical foils, Cl at 45 is almost the same as the the
stall point. see:

http://www.aerospaceweb.org/question/airfoils/q0150b.shtml

Now, as I understand/see it, with washout, the overall max lift in a
wing must be less than that given by just max Cl at stall and
planaform. This would not be the case with the 'flat plate lift'. for
a hershey bar wing with say 3 degrees washout, I'd say that flat plate
at 45 could generate at least as much lift as at the stalling point.
This may be a bit esoteric but I think it's interesting and might
indicate an even lower Va if vertical winds are anticipated.

Cheers

Mike Ash
September 11th 09, 04:52 PM
In article
>,
"Flaps_50!" > wrote:

> The reason why I riase this is beacuse the variation of Cl with lift
> is rarely shown at high AOA but I found some test diagrams and they
> show that for typical foils, Cl at 45 is almost the same as the the
> stall point. see:
>
> http://www.aerospaceweb.org/question/airfoils/q0150b.shtml

Clearly my understanding of this subject was inadequate. Thank you for
this link.

> Now, as I understand/see it, with washout, the overall max lift in a
> wing must be less than that given by just max Cl at stall and
> planaform. This would not be the case with the 'flat plate lift'. for
> a hershey bar wing with say 3 degrees washout, I'd say that flat plate
> at 45 could generate at least as much lift as at the stalling point.
> This may be a bit esoteric but I think it's interesting and might
> indicate an even lower Va if vertical winds are anticipated.

Any vertical gust which approaches 45 degrees AoA is likely to destroy
your aircraft outright no matter what speed you're flying. We're talking
something near a 60kt vertical gust if you're flying at 60kts. That kind
of gust is beyond extreme. You're very unlikely to ever encounter such a
beast unless you're in a thunderhead or something of that nature.

Note that Va is set for a certain maximum vertical gust speed. For gusts
beyond that speed, no guarantees are made. My plane's manual explicitly
calls this out, saying:

"Note: According to the Regulations the term "severe turbulence" means
air movements which might be encountered in wave rotors, storm clouds,
visible whirlwinds and when overflying mountain ranges and ridges. As we
observed in Chapter II.1 this level of turbulence is reached when the
variometer indicates about 7m/s (+1378ft/min) momentary peak indication.
The experienced flyer knows that he can expect even more severe
turbulence in storms and in high mountain ranges."

So, yes, if you expect turbulence in excess of the numbers used to set
your Va, you should fly even slower yet. However I think this still
won't save you if it's as extreme as you describe, but fortunately such
extremes are very rare indeed.

--
Mike Ash
Radio Free Earth
Broadcasting from our climate-controlled studios deep inside the Moon

Morgans[_2_]
September 12th 09, 12:45 AM
> Clearly my understanding of this subject was inadequate. Thank you for
> this link.
>
>> Now, as I understand/see it, with washout, the overall max lift in a
>> wing must be less than that given by just max Cl at stall and
>> planaform. This would not be the case with the 'flat plate lift'. for
>> a hershey bar wing with say 3 degrees washout, I'd say that flat plate
>> at 45 could generate at least as much lift as at the stalling point.
>> This may be a bit esoteric but I think it's interesting and might
>> indicate an even lower Va if vertical winds are anticipated.

One other point that may add to your understanding, on Hershey Bar wings.
It is my understanding that HB wings have little to no need for washout, as
their stall characteristics are very controllable with zero washout. It has
been a while since I read on that subject, but I recall the distribution of
lift tends to make the stalls occur at the root and progress out to the tips
as the stall deepens, just like a "nice" wing should do.

But then, I could be wrong! <g>
--
Jim in NC

Flaps_50!
September 12th 09, 05:06 AM
On Sep 12, 11:45*am, "Morgans" > wrote:
> > Clearly my understanding of this subject was inadequate. Thank you for
> > this link.
>
> >> *Now, as I understand/see it, *with washout, the overall max lift in a
> >> wing must be less than that given by just max Cl at stall and
> >> planaform. This would not be the case with the 'flat plate lift'. for
> >> a hershey bar wing with say 3 degrees washout, I'd say that flat plate
> >> at 45 could generate at least as much lift as at the stalling point.
> >> This may be a bit esoteric but I think it's interesting and might
> >> indicate an even lower Va if vertical winds are anticipated.
>
> One other point that may add to your understanding, on Hershey Bar wings.
> It is my understanding that HB wings have little to no need for washout, as
> their stall characteristics are very controllable with zero washout. *It has
> been a while since I read on that subject, but I recall the distribution of
> lift tends to make the stalls occur at the root and progress out to the tips
> as the stall deepens, just like a "nice" wing should do.
>
> But then, I could be wrong! <g>
> --

Yes, quite right. I was using the term to imply constant chord which
makes it easier to think about Cl and lift variation along a wing with
washout. Some actual Hershey bar wings have very little washout
because they start their stall in the center and it may not rapidly
propagate, especially if fences or similar devices are added I
believe. That said, the constant chord wing with no washout introduces
other disadvantages...

Cheers

Stealth Pilot[_3_]
September 12th 09, 12:45 PM
On Fri, 11 Sep 2009 01:50:08 -0700 (PDT), "Flaps_50!"
> wrote:

>On Sep 7, 4:10*am, a > wrote:
>> On Sep 5, 10:20*pm, "Flaps_50!" > wrote:
>>

>>
>> Ever watch that video of the Air Tanker C-130 snap it's wings after
>> dropping retardant on a fire? Imagine pulling out of a dive (while
>> banking) in a valley and then within a second losing 10,000 lbs of
>> cargo... when the weight is reduced, load factor goes up for a given
>> value of lift. In the case of the C-130 there may have been other
>> factors such as metal fatigue but the increased load was the primary
>> cause.
>>
>> =======
>> I watched the video -- seems to me what happened is aoa was
>> appropriate for the load, when the load was lost the sudden excess
>> lift over weight acted pretty much the same *as if the pilot suddenly
>> yanked the yoke fully aft, but will let others more skilled make their
>> points.
>>

this diagnosis didnt make sense to me. how could an aircraft that has
just shed it's load fail? with the shedding of the load the airframe
gets relatively stronger.
the locheed reports indicate that the aircraft had an undetected crack
in the root of the mainspar that had grown to such an extent that the
structure was compromised. the events that you see on the video are
coincidental and not the cause of the crash.
the crack grew to the point that it broke up in flight. that was the
cause.
Stealth Pilot

Morgans[_2_]
September 12th 09, 02:25 PM
"Stealth Pilot" > wrote
>
> this diagnosis didnt make sense to me. how could an aircraft that has
> just shed it's load fail? with the shedding of the load the airframe
> gets relatively stronger.
> the locheed reports indicate that the aircraft had an undetected crack
> in the root of the mainspar that had grown to such an extent that the
> structure was compromised. the events that you see on the video are
> coincidental and not the cause of the crash.
> the crack grew to the point that it broke up in flight. that was the
> cause.

It shed its load of water, by dropping it on a fire. If you keep your
control surfaces in the same position, you will suddenly pull more G's when
the plane is much, much lighter. Those G's were more than a plane with an
already compromised wing could stand, so it broke up.

Does that make more sense?
--
Jim in NC

Stealth Pilot[_3_]
September 12th 09, 04:44 PM
On Sat, 12 Sep 2009 09:25:22 -0400, "Morgans"
> wrote:

>
>"Stealth Pilot" > wrote
>>
>> this diagnosis didnt make sense to me. how could an aircraft that has
>> just shed it's load fail? with the shedding of the load the airframe
>> gets relatively stronger.
>> the locheed reports indicate that the aircraft had an undetected crack
>> in the root of the mainspar that had grown to such an extent that the
>> structure was compromised. the events that you see on the video are
>> coincidental and not the cause of the crash.
>> the crack grew to the point that it broke up in flight. that was the
>> cause.
>
>It shed its load of water, by dropping it on a fire. If you keep your
>control surfaces in the same position, you will suddenly pull more G's when
>the plane is much, much lighter. Those G's were more than a plane with an
>already compromised wing could stand, so it broke up.
>
>Does that make more sense?

yes but I'm still not convinced.

Mike Ash
September 12th 09, 04:58 PM
In article >,
"Morgans" > wrote:

> "Stealth Pilot" > wrote
> >
> > this diagnosis didnt make sense to me. how could an aircraft that has
> > just shed it's load fail? with the shedding of the load the airframe
> > gets relatively stronger.
> > the locheed reports indicate that the aircraft had an undetected crack
> > in the root of the mainspar that had grown to such an extent that the
> > structure was compromised. the events that you see on the video are
> > coincidental and not the cause of the crash.
> > the crack grew to the point that it broke up in flight. that was the
> > cause.
>
> It shed its load of water, by dropping it on a fire. If you keep your
> control surfaces in the same position, you will suddenly pull more G's when
> the plane is much, much lighter. Those G's were more than a plane with an
> already compromised wing could stand, so it broke up.
>
> Does that make more sense?

Not really. You're pulling more gees because you're lighter. The wings
are exerting the same force as before, thus the spar is under the same
load as before.

The reason you have G limits as well as loading limits is because of
fixed-weight components in the structure. For example, it's my
understanding that the engine attachments in light singles are a major
factor in having G limits instead of just loading limits. Your wings
don't care if you're pulling 3 Gs at max gross or 6 Gs at half max
gross, but in the 6 G case your engine mounts have to bear twice the
load.

In a case like this, where it's the wings that failed, it can't be due
to attachments holding fixed-weight items. My totally uninformed guess,
since there was a crack, is that suddenly shedding this load caused the
wings to flex DOWN, and this flexing was the final straw that caused the
crack to fail catastrophically.

--
Mike Ash
Radio Free Earth
Broadcasting from our climate-controlled studios deep inside the Moon

a[_3_]
September 12th 09, 05:16 PM
On Sep 12, 11:58*am, Mike Ash > wrote:
> In article >,
>
>
>
>
>
> *"Morgans" > wrote:
> > "Stealth Pilot" > wrote
>
> > > this diagnosis didnt make sense to me. how could an aircraft that has
> > > just shed it's load fail? with the shedding of the load the airframe
> > > gets relatively stronger.
> > > the locheed reports indicate that the aircraft had an undetected crack
> > > in the root of the mainspar that had grown to such an extent that the
> > > structure was compromised. the events that you see on the video are
> > > coincidental and not the cause of the crash.
> > > the crack grew to the point that it broke up in flight. that was the
> > > cause.
>
> > It shed its load of water, by dropping it on a fire. *If you keep your
> > control surfaces in the same position, you will suddenly pull more G's when
> > the plane is much, much lighter. *Those G's were more than a plane with an
> > already compromised wing could stand, so it broke up.
>
> > Does that make more sense?
>
> Not really. You're pulling more gees because you're lighter. The wings
> are exerting the same force as before, thus the spar is under the same
> load as before.
>
> The reason you have G limits as well as loading limits is because of
> fixed-weight components in the structure. For example, it's my
> understanding that the engine attachments in light singles are a major
> factor in having G limits instead of just loading limits. Your wings
> don't care if you're pulling 3 Gs at max gross or 6 Gs at half max
> gross, but in the 6 G case your engine mounts have to bear twice the
> load.
>
> In a case like this, where it's the wings that failed, it can't be due
> to attachments holding fixed-weight items. My totally uninformed guess,
> since there was a crack, is that suddenly shedding this load caused the
> wings to flex DOWN, and this flexing was the final straw that caused the
> crack to fail catastrophically.
>
> --
> Mike Ash
> Radio Free Earth
> Broadcasting from our climate-controlled studios deep inside the Moon- Hide quoted text -
>
> - Show quoted text -

I am not an aeronautical engineer, but I think this 'model' makes
sense. The C130 was in coordinated flight with a heavy load of water.
that means it had to be trimmed for a lot of nose up to carry that
load. Now, drop the load. the airplane will pitch nose up because of
the trim setting, but its momentum will want it to continue straight
ahead. The wings now have a much greater angle of attack, much more
lift than was needed before. If you were flying straight and level
then yanked back on the yoke which I think is pretty much the same
thing aerodynamically, you might expect the wings to fail upward.

That's my take on explaining what I've seen in the video. Give me
enough speed and enough elevator authority and I might be able to fail
the wings of any airplane.

Mike Ash
September 12th 09, 07:42 PM
In article
>,
a > wrote:

> I am not an aeronautical engineer, but I think this 'model' makes
> sense. The C130 was in coordinated flight with a heavy load of water.
> that means it had to be trimmed for a lot of nose up to carry that
> load. Now, drop the load. the airplane will pitch nose up because of
> the trim setting, but its momentum will want it to continue straight
> ahead. The wings now have a much greater angle of attack, much more
> lift than was needed before. If you were flying straight and level
> then yanked back on the yoke which I think is pretty much the same
> thing aerodynamically, you might expect the wings to fail upward.
>
> That's my take on explaining what I've seen in the video. Give me
> enough speed and enough elevator authority and I might be able to fail
> the wings of any airplane.

Makes sense to me. Seems like there are several potential explanations:
sudden flexing of the wings like I said, sudden pitch up like you said,
or simply CG changes causing increased load on the wing. Lots of ways
for this failure to occur given a weakened wing, but the idea of the
wings failing under constant load with more Gs due to less weight
doesn't seem to make sense.

--
Mike Ash
Radio Free Earth
Broadcasting from our climate-controlled studios deep inside the Moon

a[_3_]
September 12th 09, 10:02 PM
On Sep 12, 2:42*pm, Mike Ash > wrote:
> In article
> >,
>
> *a > wrote:
> > I am not an aeronautical engineer, but I think this 'model' makes
> > sense. The C130 was in coordinated flight with a heavy load of water.
> > that means it had to be trimmed for a lot of nose up to carry that
> > load. *Now, drop the load. the airplane will pitch nose up because of
> > the trim setting, but its momentum will want it to continue straight
> > ahead. The wings now have a much greater angle of attack, much more
> > lift than was needed before. *If you were flying straight and level
> > then yanked back on the yoke which I think is pretty much the same
> > thing aerodynamically, you might expect the wings to fail upward.
>
> > That's my take on explaining what I've seen in the video. Give me
> > enough speed and enough elevator authority and I might be able to fail
> > the wings of any airplane.
>
> Makes sense to me. Seems like there are several potential explanations:
> sudden flexing of the wings like I said, sudden pitch up like you said,
> or simply CG changes causing increased load on the wing. Lots of ways
> for this failure to occur given a weakened wing, but the idea of the
> wings failing under constant load with more Gs due to less weight
> doesn't seem to make sense.
>
> --
> Mike Ash
> Radio Free Earth
> Broadcasting from our climate-controlled studios deep inside the Moon

Mike, if you look closely at the video I think you'll see the change
of pitch occur when the water is dropped. Of course if the airplane
stayed straight and level the reduced weight would reduce the wing
loading, but my theory is related to the dynamics, not the steady
state. We have all been taught to be gentle with the contols, this is
an argument that we have to be gentle with pseudo control changes too.
It would be like flying into a sharp edged updraft -- that can take
one's wings off too.

There are lots of theories here, but my bias is showing!

Morgans[_2_]
September 13th 09, 04:01 AM
"Mike Ash" > wrote

> Makes sense to me. Seems like there are several potential explanations:
> sudden flexing of the wings like I said, sudden pitch up like you said,
> or simply CG changes causing increased load on the wing. Lots of ways
> for this failure to occur given a weakened wing, but the idea of the
> wings failing under constant load with more Gs due to less weight
> doesn't seem to make sense.

It does seem counter-intuitive. I had problems with the concept when it
came to explaining max maneuvering speed.

I had it explained to me, something like this: You are cruising along at
low weight, and hit a strong upward air column, suddenly. With a light wing
loading, the strength of the updraft will make the machine move upward
rapidly, which will cause a G to register on your G meter.

Now, you take the same plane, loaded to max weight and going the same speed
as before. You hit the same updraft, but the plane has a higher wing
loading, and higher mass, but the same wing area, so it will accelerate
upwards more slowly. That will register a lower G on your meter. Same
force applied to the higher mass is equal to less acceleration, as shown in
F=MA.

In thinking about max maneuvering speed, the more gradually you move into an
updraft, the less force will suddenly be applied, and I think another factor
comes into play in this. The same wing with a higher wing loading will not
be as efficient at creating more lift. It will slip, or "mush" through the
air more at higher wing loading.

I believe the same factor took place in the fire fighting airplane that
pulled the wing off. With the lighter load, the wing slipped less, and
created more lift at the lighter weight. It changed direction much more
quickly, which converts to higher G's, which broke it's wing.

I don't know. I hope to always (usually?) explain things in the least
technical way possible. That is the teacher side of me trying to make
things make sense to people who are not experts in the subject that I am
attempting to explain. It makes sense to me, but maybe I'm all wet.
Something must make it true, because that is what people say who know how to
make fancy math work as related to aeroplanes.
--
Jim in NC

Morgans[_2_]
September 13th 09, 04:05 AM
"Mike Ash" > wrote

> Makes sense to me. Seems like there are several potential explanations:
> sudden flexing of the wings like I said, sudden pitch up like you said,
> or simply CG changes causing increased load on the wing. Lots of ways
> for this failure to occur given a weakened wing, but the idea of the
> wings failing under constant load with more Gs due to less weight
> doesn't seem to make sense.

What causes something to break more easily; a steady pull, or a sharp
impact (or pull)?

A steady bend is the result of more weight carried by the airplane. A
sudden G load causes the wing to flex rapidly.

At least that's my story, and I'm stickin' to it! <g>
--
Jim in NC

Orval Fairbairn[_2_]
September 13th 09, 06:49 PM
In article >,
"Morgans" > wrote:

> "Mike Ash" > wrote
>
> > Makes sense to me. Seems like there are several potential explanations:
> > sudden flexing of the wings like I said, sudden pitch up like you said,
> > or simply CG changes causing increased load on the wing. Lots of ways
> > for this failure to occur given a weakened wing, but the idea of the
> > wings failing under constant load with more Gs due to less weight
> > doesn't seem to make sense.
>
> What causes something to break more easily; a steady pull, or a sharp
> impact (or pull)?
>
> A steady bend is the result of more weight carried by the airplane. A
> sudden G load causes the wing to flex rapidly.
>
> At least that's my story, and I'm stickin' to it! <g>

In the case of the firefighting plane, if it was flying level and
dropped a large weight (slurry), the wings would have the same load,
either with or without the dropped weight. Other airframe components,
such as engine mounts, fixed equipment, crew, however, would experience
a sudden increase in G loading. If the plane was flying at too high
speed, sudden updrafts/gusts could overload the wings.

IIRC, the aircraft in question was an older C-130 and did not have the
proper, mandatory inspections performed on the wing spar box sections.

--
Remove _'s from email address to talk to me.

Morgans[_2_]
September 13th 09, 08:13 PM
"Orval Fairbairn" > wrote

> In the case of the firefighting plane, if it was flying level and
> dropped a large weight (slurry), the wings would have the same load,
> either with or without the dropped weight. Other airframe components,
> such as engine mounts, fixed equipment, crew, however, would experience
> a sudden increase in G loading. If the plane was flying at too high
> speed, sudden updrafts/gusts could overload the wings.

My point is that the plane had a load and probably had substantial up trim
in, and full power, so when it's load was released with the power kept the
same, it probably zoomed into a climb, mostly on its own. The pilot also
very well could have pulled back on the yolk after the load was released as
is normal practice, to gain altitude. They have to drop very close to the
ground, so gaining altitude after a drop is S. O. P.

> IIRC, the aircraft in question was an older C-130 and did not have the
> proper, mandatory inspections performed on the wing spar box sections.

Sadly, also true. The fact remains that the wing was overloaded (for it's
condition) soon after release, though some combination of plane's
aerodynamic characteristics and pilot actions.
--
Jim in NC

Robert Moore
September 13th 09, 08:43 PM
Orval Fairbairn wrote
> In the case of the firefighting plane, if it was flying level and
> dropped a large weight (slurry), the wings would have the same load,
> either with or without the dropped weight. Other airframe components,
> such as engine mounts, fixed equipment, crew, however, would
> experience a sudden increase in G loading. If the plane was flying at
> too high speed, sudden updrafts/gusts could overload the wings.

Say What!!

You may know what you are trying to say, but it sure didn't come out
making sense.

From Wikipedia:
The g-force experienced by an object is its acceleration relative to
free-fall. The term g-force is considered a misnomer, as g-force is not
a force but an acceleration.

You probably meant to say "wings would have the same LOAD FACTOR".
Clearly, the load supported by the wing of a loaded aircraft is more
than the wing loading of an empty aircraft even though both are
experiencing only 1g. If the pilot doesn't reduce the angle of attack
(amount of lift produced by the wing)as the load is dropped, the wing
root will experience an increase in g-force. G-force is equal to the
actual lift being produced by the wing (at that angle of attack and
airspeed) divided by the weight being lifted.

From a aerodynamic viewpoint, the smart thing to do would be to push-
over (reduce the angle of attack) just as the fire retardant is
released, thereby reducing the g-force on the wing root. This, however,
tends to prevent the retardant from exiting the aircraft. What the
pilots seem to be doing is pulling up AND turning at the point of drop
and thereby making a bad situation even worse.

Bob Moore

Orval Fairbairn[_2_]
September 14th 09, 01:48 AM
In article >,
Robert Moore > wrote:

> Orval Fairbairn wrote
> > In the case of the firefighting plane, if it was flying level and
> > dropped a large weight (slurry), the wings would have the same load,
> > either with or without the dropped weight. Other airframe components,
> > such as engine mounts, fixed equipment, crew, however, would
> > experience a sudden increase in G loading. If the plane was flying at
> > too high speed, sudden updrafts/gusts could overload the wings.
>
> Say What!!
>
> You may know what you are trying to say, but it sure didn't come out
> making sense.
>
> From Wikipedia:
> The g-force experienced by an object is its acceleration relative to
> free-fall. The term g-force is considered a misnomer, as g-force is not
> a force but an acceleration.
>
> You probably meant to say "wings would have the same LOAD FACTOR".
> Clearly, the load supported by the wing of a loaded aircraft is more
> than the wing loading of an empty aircraft even though both are
> experiencing only 1g. If the pilot doesn't reduce the angle of attack
> (amount of lift produced by the wing)as the load is dropped, the wing
> root will experience an increase in g-force. G-force is equal to the
> actual lift being produced by the wing (at that angle of attack and
> airspeed) divided by the weight being lifted.
>
> From a aerodynamic viewpoint, the smart thing to do would be to push-
> over (reduce the angle of attack) just as the fire retardant is
> released, thereby reducing the g-force on the wing root. This, however,
> tends to prevent the retardant from exiting the aircraft. What the
> pilots seem to be doing is pulling up AND turning at the point of drop
> and thereby making a bad situation even worse.
>
> Bob Moore

It is a paradox here.

Structurally, a wing really doesn't care WHAT the G-force is! All the
structure is concerned about is the amount of stress on its components.
A wing carrying, say 200K#, which suddenly drops 100K# will pull 2G, but
the STRESS on the wings remains 200K#. Other components of the aircraft
will experience 2G, but the wing's stresses remain the same.

--
Remove _'s from email address to talk to me.

Mike Ash
September 14th 09, 02:54 AM
In article >,
"Morgans" > wrote:

> "Mike Ash" > wrote
>
> > Makes sense to me. Seems like there are several potential explanations:
> > sudden flexing of the wings like I said, sudden pitch up like you said,
> > or simply CG changes causing increased load on the wing. Lots of ways
> > for this failure to occur given a weakened wing, but the idea of the
> > wings failing under constant load with more Gs due to less weight
> > doesn't seem to make sense.
>
> It does seem counter-intuitive. I had problems with the concept when it
> came to explaining max maneuvering speed.
>
> I had it explained to me, something like this: You are cruising along at
> low weight, and hit a strong upward air column, suddenly. With a light wing
> loading, the strength of the updraft will make the machine move upward
> rapidly, which will cause a G to register on your G meter.
>
> Now, you take the same plane, loaded to max weight and going the same speed
> as before. You hit the same updraft, but the plane has a higher wing
> loading, and higher mass, but the same wing area, so it will accelerate
> upwards more slowly. That will register a lower G on your meter. Same
> force applied to the higher mass is equal to less acceleration, as shown in
> F=MA.

Up to this point I agree with you. The more heavily loaded plane,
hitting the same updraft, will accelerate more slowly due to F=ma.

> In thinking about max maneuvering speed, the more gradually you move into an
> updraft, the less force will suddenly be applied, and I think another factor
> comes into play in this. The same wing with a higher wing loading will not
> be as efficient at creating more lift. It will slip, or "mush" through the
> air more at higher wing loading.

Here I disagree. A more heavily loaded wing is just as efficient at
creating lift as a lightly loaded wing.

The lift produced by a wing is dependent on factors like airspeed, angle
of attack, airfoil shape, density, etc., but it is not dependent on the
loading.

No matter what the loading, the maximum amount of lift that can be
produced at a certain airspeed is the same. In the more lightly loaded
airplane this translates to more gees, and more force on structural
members holding fixed objects, but it does *not* translate into more
force on the wings themselves. The more lightly loaded aircraft may well
experience structural failure at lower speeds than the published Va for
max gross weight but that structural failure will be in something other
than the wing spars, which by definition can take the load they're
experiencing.

> I believe the same factor took place in the fire fighting airplane that
> pulled the wing off. With the lighter load, the wing slipped less, and
> created more lift at the lighter weight. It changed direction much more
> quickly, which converts to higher G's, which broke it's wing.
>
> I don't know. I hope to always (usually?) explain things in the least
> technical way possible. That is the teacher side of me trying to make
> things make sense to people who are not experts in the subject that I am
> attempting to explain. It makes sense to me, but maybe I'm all wet.
> Something must make it true, because that is what people say who know how to
> make fancy math work as related to aeroplanes.

An explanation that's compatible with what I'm saying and that's mostly
compatible with what you're saying is that the wings were caused to
generate more lift after the airplane released its load, either because
of changes in CG or because the pilots pulled back in order to start a
climb.

--
Mike Ash
Radio Free Earth
Broadcasting from our climate-controlled studios deep inside the Moon

Dave Doe
September 14th 09, 01:42 PM
In article >,
says...
> In article >,
> "Morgans" > wrote:
>
> > "Mike Ash" > wrote
> >
> > > Makes sense to me. Seems like there are several potential explanations:
> > > sudden flexing of the wings like I said, sudden pitch up like you said,
> > > or simply CG changes causing increased load on the wing. Lots of ways
> > > for this failure to occur given a weakened wing, but the idea of the
> > > wings failing under constant load with more Gs due to less weight
> > > doesn't seem to make sense.
> >
> > It does seem counter-intuitive. I had problems with the concept when it
> > came to explaining max maneuvering speed.
> >
> > I had it explained to me, something like this: You are cruising along at
> > low weight, and hit a strong upward air column, suddenly. With a light wing
> > loading, the strength of the updraft will make the machine move upward
> > rapidly, which will cause a G to register on your G meter.
> >
> > Now, you take the same plane, loaded to max weight and going the same speed
> > as before. You hit the same updraft, but the plane has a higher wing
> > loading, and higher mass, but the same wing area, so it will accelerate
> > upwards more slowly. That will register a lower G on your meter. Same
> > force applied to the higher mass is equal to less acceleration, as shown in
> > F=MA.
>
> Up to this point I agree with you. The more heavily loaded plane,
> hitting the same updraft, will accelerate more slowly due to F=ma.

Well actually, F only equaled ma up to about 1905. Then some bright
spark discovered (when speed is reasonably relative to 299,792,458 m/s),
that mass and energy are entirely transmutable, and mass changes due to
speed - relatively speaking. OK, I'll bugger off now :)

--
Duncan

a[_3_]
September 14th 09, 02:44 PM
On Sep 13, 9:54*pm, Mike Ash > wrote:
> In article >,
>
>
>
>
>
> *"Morgans" > wrote:
> > "Mike Ash" > wrote
>
> > > Makes sense to me. Seems like there are several potential explanations:
> > > sudden flexing of the wings like I said, sudden pitch up like you said,
> > > or simply CG changes causing increased load on the wing. Lots of ways
> > > for this failure to occur given a weakened wing, but the idea of the
> > > wings failing under constant load with more Gs due to less weight
> > > doesn't seem to make sense.
>
> > *It does seem counter-intuitive. *I had problems with the concept when it
> > came to explaining max maneuvering speed.
>
> > I had it explained to me, something like this: *You are cruising along at
> > low weight, and hit a strong upward air column, suddenly. *With a light wing
> > loading, the strength of the updraft will make the machine move upward
> > rapidly, which will cause a G to register on your G meter.
>
> > Now, you take the same plane, loaded to max weight and going the same speed
> > as before. *You hit the same updraft, but the plane has a higher wing
> > loading, and higher mass, but the same wing area, so it will accelerate
> > upwards more slowly. *That will register a lower G on your meter. *Same
> > force applied to the higher mass is equal to less acceleration, as shown in
> > F=MA.
>
> Up to this point I agree with you. The more heavily loaded plane,
> hitting the same updraft, will accelerate more slowly due to F=ma.
>
> > In thinking about max maneuvering speed, the more gradually you move into an
> > updraft, the less force will suddenly be applied, and I think another factor
> > comes into play in this. *The same wing with a higher wing loading will not
> > be as efficient at creating more lift. *It will slip, or "mush" through the
> > air more at higher wing loading.
>
> Here I disagree. A more heavily loaded wing is just as efficient at
> creating lift as a lightly loaded wing.
>
> The lift produced by a wing is dependent on factors like airspeed, angle
> of attack, airfoil shape, density, etc., but it is not dependent on the
> loading.
>
> No matter what the loading, the maximum amount of lift that can be
> produced at a certain airspeed is the same. In the more lightly loaded
> airplane this translates to more gees, and more force on structural
> members holding fixed objects, but it does *not* translate into more
> force on the wings themselves. The more lightly loaded aircraft may well
> experience structural failure at lower speeds than the published Va for
> max gross weight but that structural failure will be in something other
> than the wing spars, which by definition can take the load they're
> experiencing.
>
> > I believe the same factor took place in the fire fighting airplane that
> > pulled the wing off. *With the lighter load, the wing slipped less, and
> > created more lift at the lighter weight. *It changed direction much more
> > quickly, which converts to higher G's, which broke it's wing.
>
> > I don't know. *I hope to always (usually?) explain things in the least
> > technical way possible. *That is the teacher side of me trying to make
> > things make sense to people who are not experts in the subject that I am
> > attempting to explain. *It makes sense to me, but maybe I'm all wet.
> > Something must make it true, because that is what people say who know how to
> > make fancy math work as related to aeroplanes.
>
> An explanation that's compatible with what I'm saying and that's mostly
> compatible with what you're saying is that the wings were caused to
> generate more lift after the airplane released its load, either because
> of changes in CG or because the pilots pulled back in order to start a
> climb.
>
> --
> Mike Ash
> Radio Free Earth
> Broadcasting from our climate-controlled studios deep inside the Moon- Hide quoted text -
>
> - Show quoted text -

Mike, dropping the load would have in itself resulted in a pitchup
even if the pilots did not pull back on the yoke. It was trimmed for
flight with one load, so the AoA would have been greater than needed
without that load.When the load was dropped that trim would have
suddenly cause a pitch up, and the sudden pitchup would have suddenly
increased lift, and the bending moment on the wing spar since the
airplane could not react immediately to new upward thrust.. That's my
story and I'm sticking to it.

Richard[_11_]
September 14th 09, 02:49 PM
On Sep 13, 7:48*pm, Orval Fairbairn >
wrote:
> In article >,
> *Robert Moore > wrote:
>
>
>
> > Orval Fairbairn wrote
> > > In the case of the firefighting plane, if it was flying level and
> > > dropped a large weight (slurry), the wings would have the same load,
> > > either with or without the dropped weight. Other airframe components,
> > > such as engine mounts, fixed equipment, crew, however, would
> > > experience a sudden increase in G loading. If the plane was flying at
> > > too high speed, sudden updrafts/gusts could overload the wings.
>
> > Say What!!
>
> > You may know what you are trying to say, but it sure didn't come out
> > making sense.
>
> > From Wikipedia:
> > The g-force experienced by an object is its acceleration relative to
> > free-fall. The term g-force is considered a misnomer, as g-force is not
> > a force but an acceleration.
>
> > You probably meant to say "wings would have the same LOAD FACTOR".
> > Clearly, the load supported by the wing of a loaded aircraft is more
> > than the wing loading of an empty aircraft even though both are
> > experiencing only 1g. If the pilot doesn't reduce the angle of attack
> > (amount of lift produced by the wing)as the load is dropped, the wing
> > root will experience an increase in g-force. G-force is equal to the
> > actual lift being produced by the wing (at that angle of attack and
> > airspeed) divided by the weight being lifted.
>
> > From a aerodynamic viewpoint, the smart thing to do would be to push-
> > over (reduce the angle of attack) just as the fire retardant is
> > released, thereby reducing the g-force on the wing root. This, however,
> > tends to prevent the retardant from exiting the aircraft. What the
> > pilots seem to be doing is pulling up AND turning at the point of drop
> > and thereby making a bad situation even worse.
>
> > Bob Moore
>
> It is a paradox here.
>
> Structurally, a wing really doesn't care WHAT the G-force is! All the
> structure is concerned about is the amount of stress on its components.
> A wing carrying, say 200K#, which suddenly drops 100K# will pull 2G, but
> the STRESS on the wings remains 200K#. Other components of the aircraft
> will experience 2G, but the wing's stresses remain the same.
>
> --
> Remove _'s *from email address to talk to me.

Wouldn't there also be a torsion moment caused by the pitch up, i.e.,
the configuration of trim for level flight would create a nose/pitch
up after release, increasing AOA rapidly and 'twisting' the wings off
at the roots? The wings would actually be trying to increase AOA
ahead of the more massive fuselage.

Or, if not torsion then indeed it was more like an updraft in that the
rapid pitchup increased the AOA at such a rate as to literally 'blow'
the wings off. Picture it as an effect similar to holding your hand
outside the car window then quickly rotating it for a positive AOA.
It can quickly get away from you and give you a wrenched shoulder...
Don't mind me, I'm just thinking out loud

Here's a good article:
http://findarticles.com/p/articles/mi_m0UBT/is_25_16/ai_87717208/?tag=untagged

Orval Fairbairn[_2_]
September 14th 09, 08:42 PM
In article
>,
a > wrote:

> On Sep 13, 9:54*pm, Mike Ash > wrote:
> > In article >,
> >
> >
> >
> >
> >
> > *"Morgans" > wrote:
> > > "Mike Ash" > wrote
> >
> > > > Makes sense to me. Seems like there are several potential explanations:
> > > > sudden flexing of the wings like I said, sudden pitch up like you said,
> > > > or simply CG changes causing increased load on the wing. Lots of ways
> > > > for this failure to occur given a weakened wing, but the idea of the
> > > > wings failing under constant load with more Gs due to less weight
> > > > doesn't seem to make sense.
> >
> > > *It does seem counter-intuitive. *I had problems with the concept when it
> > > came to explaining max maneuvering speed.
> >
> > > I had it explained to me, something like this: *You are cruising along at
> > > low weight, and hit a strong upward air column, suddenly. *With a light
> > > wing
> > > loading, the strength of the updraft will make the machine move upward
> > > rapidly, which will cause a G to register on your G meter.
> >
> > > Now, you take the same plane, loaded to max weight and going the same
> > > speed
> > > as before. *You hit the same updraft, but the plane has a higher wing
> > > loading, and higher mass, but the same wing area, so it will accelerate
> > > upwards more slowly. *That will register a lower G on your meter. *Same
> > > force applied to the higher mass is equal to less acceleration, as shown
> > > in
> > > F=MA.
> >
> > Up to this point I agree with you. The more heavily loaded plane,
> > hitting the same updraft, will accelerate more slowly due to F=ma.
> >
> > > In thinking about max maneuvering speed, the more gradually you move into
> > > an
> > > updraft, the less force will suddenly be applied, and I think another
> > > factor
> > > comes into play in this. *The same wing with a higher wing loading will
> > > not
> > > be as efficient at creating more lift. *It will slip, or "mush" through
> > > the
> > > air more at higher wing loading.
> >
> > Here I disagree. A more heavily loaded wing is just as efficient at
> > creating lift as a lightly loaded wing.
> >
> > The lift produced by a wing is dependent on factors like airspeed, angle
> > of attack, airfoil shape, density, etc., but it is not dependent on the
> > loading.
> >
> > No matter what the loading, the maximum amount of lift that can be
> > produced at a certain airspeed is the same. In the more lightly loaded
> > airplane this translates to more gees, and more force on structural
> > members holding fixed objects, but it does *not* translate into more
> > force on the wings themselves. The more lightly loaded aircraft may well
> > experience structural failure at lower speeds than the published Va for
> > max gross weight but that structural failure will be in something other
> > than the wing spars, which by definition can take the load they're
> > experiencing.
> >
> > > I believe the same factor took place in the fire fighting airplane that
> > > pulled the wing off. *With the lighter load, the wing slipped less, and
> > > created more lift at the lighter weight. *It changed direction much more
> > > quickly, which converts to higher G's, which broke it's wing.
> >
> > > I don't know. *I hope to always (usually?) explain things in the least
> > > technical way possible. *That is the teacher side of me trying to make
> > > things make sense to people who are not experts in the subject that I am
> > > attempting to explain. *It makes sense to me, but maybe I'm all wet.
> > > Something must make it true, because that is what people say who know how
> > > to
> > > make fancy math work as related to aeroplanes.
> >
> > An explanation that's compatible with what I'm saying and that's mostly
> > compatible with what you're saying is that the wings were caused to
> > generate more lift after the airplane released its load, either because
> > of changes in CG or because the pilots pulled back in order to start a
> > climb.
> >
> > --
> > Mike Ash
> > Radio Free Earth
> > Broadcasting from our climate-controlled studios deep inside the Moon- Hide
> > quoted text -
> >
> > - Show quoted text -
>
> Mike, dropping the load would have in itself resulted in a pitchup
> even if the pilots did not pull back on the yoke. It was trimmed for
> flight with one load, so the AoA would have been greater than needed
> without that load.When the load was dropped that trim would have
> suddenly cause a pitch up, and the sudden pitchup would have suddenly
> increased lift, and the bending moment on the wing spar since the
> airplane could not react immediately to new upward thrust.. That's my
> story and I'm sticking to it.

Nope. The lift remained the same, but the weight changed, resulting in a
pitch up. Inertial/torsional accelerations came into play upon the pitch
up -- whether or not they played a significant factor in the accident
remains to be seen. According to what I read on the accident, the wings
failed at the central wing box, which had not been inspected according
to standard schedules. Apparently there were cracks in that section that
propagated and failed the wing.

Sudden gust/updraft loadings are fairly common in the vicinity of such
major fires and probably were the significant factor in the accident.

--
Remove _'s from email address to talk to me.

Stealth Pilot[_3_]
September 15th 09, 01:33 PM
On Mon, 14 Sep 2009 15:42:06 -0400, Orval Fairbairn
> wrote:


>Nope. The lift remained the same, but the weight changed, resulting in a
>pitch up. Inertial/torsional accelerations came into play upon the pitch
>up -- whether or not they played a significant factor in the accident
>remains to be seen. According to what I read on the accident, the wings
>failed at the central wing box, which had not been inspected according
>to standard schedules. Apparently there were cracks in that section that
>propagated and failed the wing.
>
>Sudden gust/updraft loadings are fairly common in the vicinity of such
>major fires and probably were the significant factor in the accident.

now that last paragraph I could well believe.

to infer that the stresses increased because the weight being carried
reduced, I find implausible.

why are there no recordings of heavily overloaded ww2 bombers
collapsing on bomb release? didnt happen.
Stealth Pilot

a[_3_]
September 15th 09, 02:53 PM
On Sep 15, 8:33*am, Stealth Pilot > wrote:
> On Mon, 14 Sep 2009 15:42:06 -0400, Orval Fairbairn
>
> > wrote:
> >Nope. The lift remained the same, but the weight changed, resulting in a
> >pitch up. Inertial/torsional accelerations came into play upon the pitch
> >up -- whether or not they played a significant factor in the accident
> >remains to be seen. According to what I read on the accident, the wings
> >failed at the central wing box, which had not been inspected according
> >to standard schedules. Apparently there were cracks in that section that
> >propagated and failed the wing.
>
> >Sudden gust/updraft loadings are fairly common in the vicinity of such
> >major fires and probably were the significant factor in the accident.
>
> now that last paragraph I could well believe.
>
> to infer that the stresses increased because the weight being carried
> reduced, I find implausible.
>
> why are there no recordings of heavily overloaded ww2 bombers
> collapsing on bomb release? didnt happen.
> Stealth Pilot

WW2 bombers did not release all of their load at the same instant, and
they probably had better inspections than did this C130.

Look at the video, you will see the start of the pitch up when the
load is released, and in the next instant the wings are gone. You'll
have to slow down the video, but it's there.

Orval Fairbairn[_2_]
September 15th 09, 06:47 PM
In article >,
Stealth Pilot > wrote:

> On Mon, 14 Sep 2009 15:42:06 -0400, Orval Fairbairn
> > wrote:
>
>
> >Nope. The lift remained the same, but the weight changed, resulting in a
> >pitch up. Inertial/torsional accelerations came into play upon the pitch
> >up -- whether or not they played a significant factor in the accident
> >remains to be seen. According to what I read on the accident, the wings
> >failed at the central wing box, which had not been inspected according
> >to standard schedules. Apparently there were cracks in that section that
> >propagated and failed the wing.
> >
> >Sudden gust/updraft loadings are fairly common in the vicinity of such
> >major fires and probably were the significant factor in the accident.
>
> now that last paragraph I could well believe.
>
> to infer that the stresses increased because the weight being carried
> reduced, I find implausible.

Not in the wings themselves -- they carry the same aerodynamic load;
however, inertial/torsional loads from fixed equipment in the wings
(engines, propellers) transmit into the wing box structure with sudden
changes in G loading and attitude.
>
> why are there no recordings of heavily overloaded ww2 bombers
> collapsing on bomb release? didnt happen.
> Stealth Pilot

--
Remove _'s from email address to talk to me.

Morgans[_2_]
September 16th 09, 09:16 AM
"Stealth Pilot" > wrote

> why are there no recordings of heavily overloaded ww2 bombers
> collapsing on bomb release? didnt happen.

Plus the fact that the average life of a bomber before it was destroyed was
17 missions, on many types in many theatres.

They were all practically new aircraft, in comparison to the C-130 in the
video.
--
Jim in NC

Peter Dohm
September 16th 09, 12:10 PM
"Morgans" > wrote in message
...
>
> "Stealth Pilot" > wrote
>
>> why are there no recordings of heavily overloaded ww2 bombers
>> collapsing on bomb release? didnt happen.
>
> Plus the fact that the average life of a bomber before it was destroyed
> was 17 missions, on many types in many theatres.
>
> They were all practically new aircraft, in comparison to the C-130 in the
> video.
> --
> Jim in NC
That's very true.

Even the few that flew a full 100 missions should have had less that 2000
airframe hours when they retired and the engines should have run
considerably less. Something similar was presumably true of the fighters
and trainers as well. Those points have continued to be a source of endless
debate involving both "warbirds" and--although off topic for this
thread--some of the related engines.

BTW, at least to my eyes, that video that has been debated so long and hard
really looks like a considerable pull up was made before dropping the load
in essentially 1g climbing flight. It could be mostly the happenstance of
timing that the wing seperation became evident at the completion of the
drop--assuming that the load was completely dropped and that the outflow
didn't simply cease as the wings ceased providing lift. Obviously, that
last would/will be part of the final report.

Peter

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