How do I determine the center of gravity on aircraft?

Question

What are the parameters considered and what formulas are used to find the Center of Gravity on an aircraft?

Answer 1

You ask particularly about the Center of Gravity (CG), but not the weight. Any CG calculation will necessarily include the weight as a component of the calculation; therefore I refer to the calculations as weight and balance (or CG) calculations.

The two primary ways that weight and balance measurements are determined are by physically weighing the aircraft and by mathematically calculating based on additions or subtractions to the previous weight. The former is performed by maintenance personnel and forms the empty weight and CG for any given aircraft. The later may be performed by either flight operators or by maintenance and uses the predetermined empty weight and CG as the basis for the calculations; this may be needed due to the installation of new equipment, for example, or a pilot may do such a calculation to determine the weight and CG for a given flight with, for example, fuel and passengers.

For maintenance operations, most aircraft have a section in the maintenance manual or operating handbook that prescribes the process. For flight operations, most aircraft have an information manual or operating handbook that will prescribe the calculation.

One necessary parameter is an established reference datum.

The basic formula is:

$$ C_G=\frac{(W_1\cdot A_1)+(W_2\cdot A_2)}{W_1+W_2} $$

Where:

• $W_n$ is the weight of the aircraft scale's measurement, predetermined empty weight, fuel, or payload, etc.
• $A_n$ is the moment arm, in reference to the datum, of the of the aircraft scale's measurement, predetermined empty CG, fuel, or payload, etc.
• $C_G$ is the solution to the center of gravity for the aircraft.

The cancelled FAA Advisory Circular AC65-9A contains an entire chapter (Chapter 3) dedicated to the subject of weight and balance, including exactly what you ask, the parameters that must be considered and the formulae used to perform the calculations. Though cancelled, the AC is still available online (I have a print copy).

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EDIT: See the comment below from @voretaq7 referencing the excellent "Aircraft Weight & Balance Handbook" quoted here:

In addition to AC 65-9A the current Aircraft Weight & Balance Handbook (FAA-H-8083-1A - available here) contains more than you ever wanted to know about computing weight & balance.

This is currently the direct link to the PDF of that document.

Answer 2

It is not fully clear to me whether you are interested in determining the CG (Center of Gravity) of a new design or an existing one, I guess is the latter.

In this case the main parameters are:

• The Payload (i.e.: people, cargo, other (e.g.: ammunitions))
• The Fuel

The manufacturer provides you with tables to determine the CG due to the different loading condition (combination of the aircraft payload and fuel).

Answer 3

For light planes, you can get a good approximation with a bathroom scale. First you put it under the front/tailwheel, note the force F1, and then do the same for one of the main wheels. You note the force and take double that value, F2. The CG is in a point of a line along the fuselage, where the moments of F1 and F2 have the same value. It's elementary algebra for 10-year olds...

Answer 4

Very simple answer, but one familiar with modelers, is to lift it off the ground. Keep in mind your horizontal (front to back) CG is normally slightly forward of your Clift, and is "balanced" by the angle of incidence, or trim of elevator.

There is also a vertical CG (top to bottom) which should be compared with your center of frontal drag. Note most aircraft tail designs are assymetric, with more "up" than down, also adding to the up trim. The drag from the vertical stabilizer is therefor more efficiently used, saving some up trim on the horizontal.

For full scale, there are formulas in the POH based on the CG before loading fuel, people, luggage, etc. Very important pre-flight task to check this.