Weight, Balance and Polar Moment of Inertia

[Colonel Sanders]

Time for a quick review (see Accidents forum).

As a pilot, you are taught to respect W&B. They
are tremendously important, and must be within
limits. Terrible things will happen if they are not,
you are told.

But then you hear about people flying over-gross
without incident (eg transatlantic ferry flights) and
the baby gets thrown out with the bathwater, and
people start disregarding W&B.

Despite what you have heard, weight is probably
the least important of the three topics of this thread.

You can increase the max gross of a 40-flap 172 by
150 lbs if you jigger the microswitch on the selector
so that you can't get more than 30 flaps. There's
even an STC for it.

I have flown airplanes that IMHO flew like they were
well over their max gross, they climbed so badly. A
fully loaded M20J on a hot day comes to mind, as does
a C150, which is frequently flown over max gross.

As I mentioned in the Accidents forum, an aircraft with
a forward C of G is going to be gentle and docile. It's
going to fly nicely and recover well from stalls and spins.

It is easy to develop contempt towards such an aircraft,
but beware that at the same weight, if you try to fly with
an aft C of G, like college girls on spring break, it's going
to behave badly. It's going to be sensitive in pitch, and
it's going to be a handful in stalls and spins. Might not be
recoverable at all.

tl;dr C of G is what matters, not weight. Weight just makes
your airplane climb like a pig, and a runway hog, which is
immediately obvious. A bad C of G, like your girlfriend off
drunk at Daytona on spring break, is not so obvious.


Now onto Polar Moment of Inertia (PMI) which is a measurement
of how the mass is located.

The best way to understand this is to take an egg carton with
four eggs in the center, close it, and spin it back and forth on
a shiny countertop, holding it from above in the center. See
how easy it is, to start and stop it spinning, with all the mass
in the center?

Now, open the egg carton and put two eggs in each end
and close it. Same weight, same balance, but a wildly
increased PMI. Now repeat the experiment, spinning it back
and forth, holding it at the center. See how much more torque
is required to start and stop it spinning?

This is why you want to centralize mass, which is actually
really hard on the wing spars, a subject for another time (ZFW).

We don't spin twin-engine airplanes, despite the fact that
they have lots of differential thrust to help with the recovery.

Why is that?

[photofly]

if anyone is interested in the kinds of things that go wrong when the CofG is in the wrong place, I have a pdf I can send them. Basic algebra only, is required, but that's too sophisticated to post here. It might "confuse" the unlearned, I'm told.

[Pop n Fresh]

I got in the habit of doing a second weight and balance with no fuel. Not because I intend on using it all, but because a flight instructor told me a tale about the center of gravity shifting too far back on a trip she took with a full load. She claimed it was scary, I figured I did not want to try it.

Paid off once, I switched my passengers front to back to keep that from happening on that flight.

[photofly]

told me a tale about the center of gravity shifting too far back on a trip....Paid off once, I switched my passengers front to back to keep that from happening on that flight.

Moving passengers front to back would move the C of G even further aft; if you're worried about a C of G too aft, then you would move passengers back to front.

[GyvAir]

From viewtopic.php?f=118&p=865198#p865198 :

CoG only takes into account the longitudinal distribution of the weight.
You could have 2 identical CoG, but in one aircraft, the cabin is lightly loaded and the wings full of fuel, and vice versa. The 2 aircraft will behave differently, expecially in a spin.

Thanks, trampbike.. I was trying to envision how a typical small aircraft could be loaded with significantly different polar moments of inertia, given the limited cabin length. I hadn't considered lateral weight distribution from fuel loading.

[lownslow]

told me a tale about the center of gravity shifting too far back on a trip....Paid off once, I switched my passengers front to back to keep that from happening on that flight.

Moving passengers front to back would move the C of G even further aft; if you're worried about a C of G too aft, then you would move passengers back to front.

I think he probably meant switching them around so the hottie has to ride in the back and his beer buddy gets to cram up front.

LnS.

[Pop n Fresh]

I did not word that very well did I. Unfortunately when photofly changed what I wrote it did not fix the statement.

LnS is nearly correct. But could get in the same situation in a plane that acts different from the one needing the lard up front.

I did not move them back to front, I switched them. One went to the backseat the other went to the front seat. My original seating plan was opposite of that.

I would have had a forward CofG once the fuel got low enough because it was a different plane than my usual C-172. Still we only had two rows, so my brother and his wife switched seats.

[Canoehead]

We don't spin twin-engine airplanes, despite the fact that
they have lots of differential thrust to help with the recovery.

Why is that?

Because they are placarded "INTENTIONAL SPINS PROHIBITED"... and I'm not a test pilot

Good post CS!

[photofly]

I imagine that's it's a design criterion for most aircraft that the CG doesn't move (very much) as the fuel is used; that means putting the fuel tanks somewhere close to the middle of the CG range. That's certainly true for the single-engined Cessnas with fuel in the wings.

I would have had a forward CofG once the fuel got low enough...

That's the opposite scenario of what your instructor was telling you about having a CG too aft as the fuel was used! In theory I guess either could be a problem: but what aircraft was she talking about, out of interest?

[photofly]

We don't spin twin-engine airplanes, despite the fact that
they have lots of differential thrust to help with the recovery.

To break out of a spin you need to get the nose down.

To get the nose down involves changing the axis of rotation of the aircraft - not in space, but the axis relative to the aircraft itself.

If the moment of inertia about a nose-down axis is smaller than the moment of inertia about the nose-flat axis of rotation, then the aircraft needs an input of extra energy to change it's spin to one with the nose down: the same angular momentum with a lower moment is a faster rotation and has higher energy.

To use the ice-skater example, when the skater pulls their arms in to reduce their moment and accelerate their spin their rotational kinetic energy increases, which is paid for by the muscular effort required to pull the arms inwards against the "centrifugal" force that tries to throw the arms outwards.

The flat-spinning aircraft may need extra energy to accelerate the spin as it goes nose down to recover, and it may not have a source of energy (no "arms") to provide it.

So it may be impossible to get the nose down.

[Docbrad]

So, you're saying that in a light twin, the extra mass on the wings means you need a lot more energy to push the nose forward, right? So much so, that the amount required is beyond the authority of the flight controls?

[trampbike]

Yes.
It can happen to many types of aircraft. The little Grumman AA1 fuel liked to pool towards the tips of the wings, flattening the spin, and making it so much harder to stop the yawing after a certain point.
https://www.youtube.com/watch?v=Kgzn9zILw6Q

[Colonel Sanders]

Twins also have their engines spaced out.

Note that the Extra 300 has a particularly stupid
fuel system, with wing tanks and tiny header tank
in the fuselage. POH says to not do acro with any
fuel in the wing tanks, but that's impossible unless
you are doing acro overhead your airport.

So, everyone breaks the rules and does acro with
fuel in the wing tanks of their Extra 300. Amazingly
no one has been killed yet, but it does smash the
tanks and the wings up pretty good.

Errata for the day: I remember being told by an F-14
driver that he could land on a carrier with his aux tanks
either completely empty or completely full. But not
inbetween.

[photofly]

So, you're saying that in a light twin, the extra mass on the wings means you need a lot more energy to push the nose forward, right? So much so, that the amount required is beyond the authority of the flight controls?

I don't have the expertise to say for certain that the amount "is" beyond the authority of the controls; but it should be clear that CS's point that the weight distribution of an aircraft can make a big difference to how and whether it recovers from a spin, even if the total weight is the same - is a good one.

[Docbrad]

I'm past the commercial stage in my training, and I have never heard discussion about PMI before. I'm glad I have an understanding of it now, as it seems to make a significant difference.

I almost got into a spin in a twin, and was criticized by the instructor for recovering so aggressively. Now I don't feel so bad about it better safe than sorry, right?

[photofly]

(Most of us just call it the "moment of inertia" but CS is quite a polarizing sort of person )

[trampbike]

I'm past the commercial stage in my training, and I have never heard discussion about PMI before. I'm glad I have an understanding of it now, as it seems to make a significant difference.

Check out this book:
http://www.amazon.ca/Light-Airplane-Pil ... 1879425432
It gives a pretty good overall understanding of the stall and spin principles.

[Big Pistons Forever]

An example where PMI matters is the Cessna 150 Aerobat. If it has been modified with a 150 hp Lycoming O 320 engine it is no longer certified for aerobatics or even spins. This is because the battery is moved to the tail cone. The combination of the heavier engine in front and the heavy bettery in the back so degrades the spin recovery that spins are now unsafe.

[PilotDAR]

Big Pistons speaks my mind about 150HP 150's. I used to fly one which had ten pounds of lead in the very back of the tail - if was not nice to fly, and completely turned my off the idea.

When I did spin testing in the Cessna Grand Caravan, the forward C of G spins when fine, but my loading for the aft C of G was a few hundred pound in the very back of the cabin. The C of G was right on, but it was a VERY different aircraft to recover a spin. It was find, just NOTHING like the forward C of G I had done the day before. Moment of inertia is certainly a factor in "long" cabin planes, or planes where fuel can slosh a long way in the wings.

But then, we're supposed to be spinning within limitations, aren't we.....

[Colonel Sanders]

PMI can either be a fuselage or a wing issue.

Decades ago, my father flew a Canadiar Northstar
around the world. Well, kind of. He rode along
until they reached the North Pole and then he flew
a circle around it. Took around 60 seconds.

Anyways, he found the Northstar (think DC-4
with V-12 Merlins) to be wild to fly, compared
to fighters. He would bank it, and it would
continue to roll past where you wanted it to
go. So he would roll it back, overshooting
again.

Back and forth in the slowest, stupidest PIO
you can imagine, caused by all that mass out
there on the wings, that fighters certainly don't
have.

Integral radius squared dm, IIRC.

[Pop n Fresh]

I imagine that's it's a design criterion for most aircraft that the CG doesn't move (very much) as the fuel is used; that means putting the fuel tanks somewhere close to the middle of the CG range. That's certainly true for the single-engined Cessnas with fuel in the wings.

I have not tried but I have never seen a 172 CofG calculated to be outside the normal category envelope so far. It is true that they do not seem to move much from full tanks to empty.

I would have had a forward CofG once the fuel got low enough...

That's the opposite scenario of what your instructor was telling you about having a CG too aft as the fuel was used! In theory I guess either could be a problem: but what aircraft was she talking about, out of interest?

Both events were a while ago. I don't remember for sure. I think her story was a about a Cherokee. The plane I was flying was a warrior II.

Now that we are writing about it, I think you're right. I'm not sure if my forward CofG was the "empty or the initial with fuel calculation." I am starting to think it might have been the full gross for take off weight. I don't have a POH for a warrior II. It seems unlikely that it would change backwards or opposite from the Cherokee.

I know I ended up putting my brother in the back and his wife up front instead of the way we originally planned to go, because the other way caused the CofG to be outside the limits on the chart on one of the sheets. I guess that is not relevant to my second calculation with no fuel if it was the take off weight and balance.

I think it's still a good plan to do the second weight and balance with the numbers for fuel tanks empty. I'm not going to empty the tanks intentionally but if the CofG is ok in that case, then I know it should be good for the entire flight.. If it's not I can see if moving something will fix it.

[FenderManDan]

Thank you for clarifying PMI. I haven't heard of it before and it seems a lot of PPLs, given the comic way they load gear in the single engine GA planes. I looked at my c172 POH and does not mention anything on that topic. I saw that Cessna add and driving the planes with a land-o-matic that CS had in the past and does not surprise. I tend to use my boating experience for loading purposes.

[Pop n Fresh]

My traditional load was headset, knee board, light duffle bag with socks, t-shirts and undies. Find out how much fuel we can take once everything is weighed.

[goldeneagle]

I imagine that's it's a design criterion for most aircraft that the CG doesn't move (very much) as the fuel is used;

While this is true of small / light aircraft, it is not true of larger more complex aircraft. Some larger aircraft have fuel tanks specifically located so they can be used for trimming balance. The most extreme example I can think of offhand, would be concorde. To reduce drag enroute, concorde was trimmed not by moving control surfaces, but by pumping fuel forward and aft. There was also a huge difference in required CG location for subsonic vs supersonic flight, also accomplished by pumping fuel between the trim tanks forward and rear, to the tune of many tons of fuel transferred, a 2 meter CG change was required. Roll trim was tuned by pumping fuel between port and starboard tanks.

Concorde was a rather complex aircraft in this respect, in that allowable CG range for takeoff / landing had no overlap with the allowable CG range for high supersonic cruise, so there was lots of pumping fuel between various tanks to keep it in balance, enroute.

[Big Pistons Forever]

I imagine that's it's a design criterion for most aircraft that the CG doesn't move (very much) as the fuel is used

Not necessarily. V tailed Bonanza's loaded to max weight and at the aft C of G limit at take off, will have a C of G significantly aft of the aft limit if most of the fuel is burned off prior to landing.