Thursday, November 27, 2008

The Balance Part of W&B

As well as not asking the airplane wings to lift more weight then the manufacturer intended, you need to make sure that the front-to-back balance of the weight is acceptable. To do this you take into consideration the weight of each component of the load, and the location at which it is loaded.

Weight, Arm and Reference Datum

Consider a teeter-totter (or a see-saw, whatever you call that spinal injury-inducing levered plaything in your neck of the woods). If you weren't denied the opportunity to have played on one, you know that if you put an adult (or two kids) at one end and one kid at the other, it's hard for those on the heavy end to push off the ground, and easy to launch the kid into orbit if the adult lets his end come down hard. For easy teeter-tottering, you have to pile up more kids at the light end, or have mom sit closer to the fulcrum (the hinge supporting the see-saw) where her weight has less leverage. The effect of someone sitting on the teeter-totter is directly proportional to how much she weighs, and to how far she is from the middle. If Mom weighs twice as much as Junior she has to sit halfway between the end and the middle to balance Junior at the other end. Mathematically put, the teeter-totter balances when Mom's weight times her distance from the middle equals Junior's weight times his distance from the middle.

The distance out is called arm and the combined effect of weight and distance, the weight times the arm, is called moment. As Mom's moment (say 60 kg x 2 m) tends to push the see-saw one way and Junior's (30 kg x 4 m) pushes it the other way, and they are equal, they balance. I can also call Mom's arm (distance to the right) positive and Junior's arm (to the left) negative, so that the total moment of the system becomes (60 x 2) + (30 x -4) = 0. A total moment of zero means no moment, so no tendency to move about the fulcrum and the see-saw balances. Whee. (I'm ignoring the weight of the teeter-totter itself, because presumably it's balanced). If Mom sits all the way out to her end, her arm increases to 4 (further to the right), so the total moment becomes (60 X 4) + (30 x -4) = 120. That 120 is a measure of the unbalancedness. [Physicists: please don't beat me up for using kilograms instead of newtons for weights here, and then ignoring dimensions. I know the difference. I just don't feel anything is added to the explanation by multiplying both sides of the equation by g.] The point I measure from is called the reference datum, and its position is irrelevant, as long as everything is measured from the same place.

Here is an example to explain that last italicized phrase. I postulated an eight metre teeter-totter, and the measurements I gave were from the middle. That wasn't too hard, as the middle is easy to find, but it did require the measurements in one direction to be negative, with some people find inconvenient. We could take the same teeter-totter and measure from one end, say Mom's end. So Mom's moment, when she is halfway to the middle in order to balance Junior, becomes (60 x 2 = 120) and Junior's becomes (30 x 8 = 240), for a total moment of 360. If we put Mom back at the end now, her arm would be zero, so the total moment would be (0 x 60) + (30 x 8) = 240. Notice that 360 (the ideal) minus 240 (the moment with Mom at the end) is 120. The unbalancedness is still 120. I could even measure from the swingset or from the edge of the playground and get the same result.

Centre of Gravity

When the reference datum is an arbitrary point, the moment, the measurement of unbalancedness, is an arbitrary number, and these numbers get quite large when you're dealing with long, heavy airplanes. So we put it all together to get a non-arbitrary number called the centre of gravity. Weight multiplied by arm equals moment not only for each component of the system, but also for the sum of all its components. And if weight times arm equals moment, then moment divided by weight equals arm. Back to the total weight and moment of the unbalanced teeter-totter.

Mom and Junior together make 90 kg, so that's the weight. And I already totalled the moments for each reference datum.

Measuring from the middle: 120 / 90 = 1.33

Measuring from the end: 240 / 90 = 2.67

Now, measuring 1.33 m from the middle towards Mom's end reaches the same point as measuring 2.67 m from Mom's end toward the middle. That is the point at which the teeter-totter would balance with Mom at one end and Junior at the other. That's still called the arm of the loaded airplane, but more commonly called the centre of gravity, often written CofG and pronounced see-uhv-gee.

Example

For the airplane, every manufacturer defines a reference datum, and publishes a list of the arms of every point at which weight will be loaded, such as each fuel tank, the pilot seat, each row of passenger seating, and each baggage area. The manufacturer also publishes a chart showing the acceptable range of moment and or centre of gravity (CofG). It's the pilot's job to calculate the total moment or CofG of the loaded airplane and ensure that it falls within the limits of the chart.

Here's an example, from a Piper Seneca. It's a fairly simple airplane, and I've simplified it further by not using one of the baggage areas. Note that this calculation applies to a particular Piper Seneca, one that once belonged to the Winnipeg Flying Club. A different Piper Seneca would have a different empty weight and arm.

We'll imagine a 200 lb pilot, 460 pounds of passengers, in two different rows, and 50 lbs of baggage in the back. The first line of the table I read off the individual airplane weight and balance document. The other weights I get from inspection. The other arms are from the Aircraft Flight Manual, based on a reference datum near the nose of the airplane. The moments are the product of the weights and arms. The total weight and total moment are simply a sum of the individual weights and moments. And the total CofG is the total moment, divided by the total weight.

WeightArmMoment
Airplane2944 85.82252674
Fuel 550 93.6 51480
Pilot 200 85.5 17100
Pax1 160 118.1 18896
Pax2 300 155.7 46710
Baggage 50 178.7 8935
------------------------------------
Total 4204 94.1 395795

The total weight of 4204 happens to be four pounds over the maximum weight allowed by the manufacturer, while the CofG is just forward of the maximum 94.6. In this case I would accept the load, because I know I will burn about eight pounds of fuel during taxi, runup and on the takeoff roll. By the time the airplane takes to the air, it will be within limits. The manufacturer tells me that the effect of fuel burn on CofG for this airplane is negligible.

It bears noting that this airplane has an additional baggage area in the nose, and two unoccupied seats. Three medium-weight passengers and their carry-ons is enough to load the airplane to its maximum. An airplane doesn't have to be stuffed to be full.

If the CofG numbers get unwieldy, there's another way to express them. I'll tell you about %MAC calculations some other time. And this isn't one of my best explanations, so ask away if I've said anything confusing.

9 comments:

Blake said...

I whipped up a worksheet to help me to W&B calculations for the C172.

http://fly.blakecrosby.com/files/worksheets.pdf

(page 2)

Richard said...

I've done the same for successive PDAs, and now re-factoring my Blackberry version. I'd be happy to post it when done if there is any interest.

curovalo said...

As one example of how the theory becomes practice:

I've calculated several typical loads for the aircraft that I fly as well as some extreme cases. I've 'set' red flags in my head that warm me when the balance might be an issue and when it'll be alright so long as the Max take-off weight is within limits.

Doesn't work all the time and not for everyone, but it keeps life simple for my private flying. But please NOTE. I still DO a weight and balance calculation in my head before each flight by checking how my pre-calculated examples apply (or fail to apply) and I mentally add up the weights to ensure that the total load is within limits.

zb said...

Oh dear. I just heard this funny story about a guy who obviously knew how to do his W&B on NPR's awesome Car Talk.

Sab said...

Here's a funny video showing that calculating weight and balance is also important in some more primitive forms of transport :)

http://www.youtube.com/watch?v=X7fkAF4nElA

Aprenta said...

How do you handle lateral center of gravity? Do you need to worry about the weight on each side of the airplane, or is the effect of imbalance too small to worry about?

I always wonder how much of an effect it has in a small plane like a C172 if there's just the pilot and he is heavy. Does the aircraft constantly try to roll to the left? Does anyone use ballast to compensate for this sort of thing?

Aviatrix said...

In a large airplane the momentary differential in power between the two engines far outweighs any lateral imbalance in loading. I've flown an airplane with a 1600 lb gross weight, probably less than 1400 lbs with just me and fuel by the end of the trip, and not noticed any lateral problems.

I have flown a side-by-side two-place ultralight that needed rudder trim adjustment with two on board (the adjustment needed to be done with a saw, so I didn't do it). The test pilot said it was in trim and he was a large guy flying single pilot, so I wondered if that might be an issue, but I didn't get a chance to test it.

If you hang an external load off one side, the drag has an effect that overwhelms any weight effects.

So no.

nec Timide said...

I was waiting for Aviatrix's comments on lateral balance. It is interesting how our experiences differ and are similar.

In flying Cessna 172s, which can take fuel from both tanks, and the tanks are relatively close to the fuselage I don't really notice any imbalance. In my Cherokee, which can take fuel from only one tank at a time, and the tanks are further out on the wings, I do tend to notice that too much asymmetric fuel burn can cause the need for constant significant aileron input. It is a bit annoying on long cross countries, but not a huge issue. It is usually enough to burn fuel off the heavy side during takeoff, climb and initial cruise. Then routine tank switching keeps things in trim.

My experience in a Piper Vagabond agrees with Aviatrix. The Vagabond had a centre main tank, and an aux tank in the left wing root. Even flying solo with my corpulent self and a full aux tank didn't result in any significant imbalance.

I've read reviews of single engine turboprops (Socata TBM 700, and Piper Meridian if memory serves) that have automation to actively prevent too much asymmetric fuel burn.

zb said...

Wow. Just as writing this helps you in knowing your stuff (as you often have mentioned), reading and commenting makes us in the audience wiser, too. This time, however, in my case, it's not so much the physics but the language of physics and their issues of translation.

Just as I was about to ask if that thing wouldn't have to be called torque instead of moment, I checked a mostly reliable source and was assured that I could trust the native speaker.

Note to self: When you think that there's a false friend in translations, it's actually time to relax.

Yes, there's more than the translation to torque of the word (Dreh-) Moment that they use in my neck of the woods; and the generalized physical quantitiy of moment (of a vector) makes it all so easy.