Calculating the CG: Three Methods - ATPL 031 — Mass & Balance

Everything so far has gathered ingredients: masses, arms, moments, and the documents they live in. This lesson finally cooks. There are three working methods for finding where a loaded aircraft's CG sits, and they are one idea in three costumes:

The arithmetic method — computes the balance directly from moments.
The graphic method — lets printed graphs do the same multiplication with a pencil.
The index method — compresses the moments into small, addable numbers built for load sheets.

Every later lesson in this module uses one of the three.

The arithmetic method

The whole method is one formula, the same one the weighing lesson used to turn scale readings into a CG, now applied to a loading list: multiply every item's mass by its arm, add up the moments (signs included), add up the masses, and divide.

The result is an arm: the position, in the datum's own coordinate system, at which the whole loading acts. It can be compared against limits directly, converted to per cent MAC with the conversion formula from the per cent MAC lesson, or fed into the next stage of a load sheet. Two habits keep it honest: carry the moments' signs all the way through, and never divide until both sums are finished, because a CG computed from a partial list describes an aircraft that does not exist.

The graphic method

Light-aircraft manuals often replace the multiplication with printed loading graphs. Each loading station gets a line on a graph of mass against moment: enter with the mass on one axis, run across to the station's line, and read the moment off the other axis; the line's slope is doing the mass-times-arm multiplication for you. The station moments are then summed, exactly as in the arithmetic method, and the totals are checked on a second printed graph, the CG envelope, where a point plotted at the aircraft's total mass and total moment either falls inside the certified region or does not. Some sample envelopes are drawn against CG position instead of moment; there the division is done first and the point is plotted at mass against CG. The next lesson works a full example of that check; here the skill is reading the graphs the sample documents give.

The index method

Real moments are unwieldy: a 34,300 kg aircraft balanced around 650 in carries a moment in the tens of millions of kilogram-inches, a number no one wants to add in a paper column. The index fixes that. It is a scaled-down, shifted stand-in for the moment: the item's or aircraft's moment is measured about a chosen reference arm instead of the datum, divided by a large constant to shrink it, and often lifted by a fixed constant to keep everyday values positive. A generic index formula looks like this:

The dry operating index (DOI) is simply the index of the aircraft at its dry operating mass: the load sheet's starting value, published by the operator alongside the DOM. From the DOI, each loading entry adds or subtracts its own small index contribution, and the finished sum locates the loaded CG on the trim sheet's grid, which is the working method of the large-aeroplane lessons ahead.

The advantages of the index method are exactly what the example shows:

small numbers, quick and safe to add in a load-sheet column;
pre-printed contributions per seat zone and compartment, so loading becomes addition rather than multiplication;
direct plotting: the result lands on the trim sheet's mass-against-index grid, where the CG limits are drawn in the same currency.

The price is that an index only means something together with the formula and constants that made it, which is why every trim sheet states whose index it speaks.

Calculate the CG of a loading with CG = sum of moments ÷ total mass

Determine a CG using the loading graphs of sample documents

Explain the principle and the advantages of the index method, define index and DOI, and calculate a DOI from a given formula

The arithmetic method

Check you understood

The graphic method

The index method