Definitions and Calculations — Page 33, Lesson 47

Right, let's get into the heart of mass and balance calculations. We're starting with the core definitions, and the very first one is the Balance Arm. The balance arm is the distance from the aircraft's datum to the CG position, or to the centroid of a body or mass. Now, let's unpack that. The datum is the reference point on the aircraft from which all horizontal distances are measured. The CG, or centre of gravity, is the point where the entire weight of the aircraft is considered to act. And the centroid is the geometric centre of a mass — the point its total weight acts through. For a simple shape, like a square or a rectangle, the centroid is simply the exact centre of that shape. So the balance arm would be the distance from the datum to that exact centre. But here's the practical problem: cargo bays are seldom exact squares or rectangles. So the manufacturer provides the centroid — the point the total weight acts through — for each bay. We don't have to calculate it; it's given to us. Now, the sign convention is critical. For the purposes of calculations, all balance arms ahead of the datum — that is, in front of it — are given a negative (-) prefix. Those behind the datum — aft of it — are given a positive (+) prefix. This is a classic trap. Inexperienced students mix up their signs and fail to arrive at the correct answer. So remember the arithmetic rules: a positive value multiplied by a negative value results in a negative answer. But two positives, or two negatives, multiplied together always produce a positive answer. Keep that straight and you'll avoid the classic error. Next, we have the Loading Index. Here's the problem it solves: when you multiply a force by a mass, the resulting answer — the moment — can be of such magnitude that it's too bulky and time-consuming to use. A loading index is simply a moment divided by a constant. Its effect is to reduce the magnitude of the moment to something much easier to handle. Let me show you with the example in front of us. Look at the table. We have items, their mass in kilograms, their arm in inches, and the moment in kilogram-inches. Take the BEM — that's the Basic Empty Mass — at 31,994 kg, with an arm of 691 inches. Multiply those together and you get a moment of 22,107,854 kg-in. That's a big, unwieldy number. So we divide it by a constant — here, 1,000,000 — and we get an index of 22. That's the loading index. Let's run through the rest. The flight crew — 180 kg at an arm of 183 inches — gives a moment of 32,940. Divided by 1,000,000, that's an index of 0. The cabin crew — 540 kg at 1,107 inches — gives 597,780, which divided by 1,000,000 gives 0.6. The special equipment — 12,000 kg at 701 inches — gives 8,412,000, which divided by 1,000,000 gives 8.4. Now, the DOM — that's the Dry Operating Mass — is the sum of all these. Its mass is 44,714 kg, and its total moment is 31,150,574 kg-in. Divide that by 1,000,000, and you get an index of 31. So the whole point is: instead of juggling millions of kilogram-inches, we work with small, manageable index numbers. That's the loading index in action.