Calculating Weight and Balance in Large Model Aircraft

Mark Lipp
jflipp@aol.com

Once a model aircraft gets to a certain size, the traditional methods of calculating weight and balance in a model aircraft and confirming that the center of gravity (CG) is where it should be, such as suspending the model from two pivot points and balancing extra weights on the nose or tail, become difficult if not impossible.

This article describes a technique for not only determining weight and the actual CG, but also calculating how much weight should be added at a chosen location to establish the correct CG. This technique is the same as that used for many full scale general aviation aircraft.

For a copy of the Excel workbook described in this article, send an e-mail to Mark Lipp (jflipp@aol.com).

The figure below defines various points and measurements that are used in the calculations. The abbreviations in the figure are described in the following paragraphs. Even though a conventionally-geared aircraft (a tail dragger) is used in the figure and discussion, this technique can also be used for a tricycle geared aircraft.

Materials Needed

The materials necessary to use this technique are:

1. The aircraft in its final, ready-for-flight configuration but with no fuel.
2. The location of the desired CG. This is commonly found on the plans of the aircraft or described in the building instructions.
3. A set of accurate scales that will measure up to the full weight of the aircraft. I use a set that we also use in baking. My set is accurate to a tenth of an ounce.
4. Accurate measurements of several points on the aircraft (described in the figure above and in the detail below).
5. The Excel workbook that does the calculations. However, you do not need the workbook if you are willing to hand-calculate a few numbers. See the section titled “This is the Internal Calculations Part” to see these calculations.
6. Several 2x4s or other blocks to set the aircraft to the attitude of straight-and-level flight. These are usually required even if you have a tricycle geared aircraft.

The following paragraphs describe the process and some of the theory behind it. Skip to the section titled How To Do It if you want to skip the theory.

Definitions and Abbreviations

First, a few definitions:

Arm – The distance between a given point (the “datum”) and the point at which a weight measurement is taken. For example, this is the horizontal distance between the front of the firewall of an aircraft and the lowest point on the tailskid when the aircraft is in straight-and-level attitude. In this article, this is measured in inches. In the figure above, Am, Ap, At, and Ac are arms.

Datum – An arbitrary point on the longitudinal (nose to tail) axis of the aircraft from which all arms will be measured. The front of the firewall is commonly used, but any point (such as the nose of the spinner, or the axle of the main gear) can be used. This is “d” in the figure above.

Moment – The product of the arm and the weight measured at the end of that arm. In this article, this is measured in ounce-inches.

Weight – The weight measured at the end of the arm. In this article, this is measured in ounces. It is important that this weight is measured when the aircraft is in the attitude of straight-and-level flight.

Now for some abbreviations. These will be used to make the article and formulae a bit easier to read.

A – Stands for “arm” and is followed by a lowercase letter. For example, “Am” stands for the arm at the axle of the main gear (A for Arm and m for Main Gear).

W – Stands for “weight” and is followed by a lowercase letter. For example, “Wm” stands for the weight at the axle of the main gear (W for Weight and m for Main Gear).

m – Stands for “main gear”.

c – Stands for the desired CG. This is the CG shown on the plans or described in the building notes.

t – Stands for tailskid (or tailwheel) in a conventionally geared aircraft. This will stand for the nose gear if you are measuring a tricycle gear aircraft.

p – The point at which you wish to add weight to correct any deviations from the desired CG. Since most aircraft are initially tail heavy, this point is usually in front of the desired CG, sometimes at the front of the firewall or somewhere in the nose of the aircraft at which weight can be conveniently attached.

Using these abbreviations, you will see references below such as Wt (weight at the tailskid), Ac (arm of the desired CG), Ap (arm of the point at which you will add weight, if necessary). etc.

The formulae on which the calculations in the Excel workbook are based are at the end of this article. These formulae can relatively easily be used to hand calculate the results instead of using the Excel workbook. (SEE This is the Internal Calculations Part)

How To Do It

The example selections from the Excel workbook are for a BalsaUSA quarter scale Fokker D-VII with a Saito 180 engine.

1. Assemble the aircraft into its final ready-for-flight configuration. Do not add fuel because most model aircraft CG locations are set for when the aircraft is “dry”.
2. Determine the location of the datum. This can be anywhere (even in front of or behind the aircraft) on the longitudinal (nose to tail) axis of the aircraft. The front of the firewall, the main (or top) wing leading edge, or the main gear axle is commonly used. This is “d” in the figure above.
3. Determine the point at which you want to add weight, if necessary. This is usually at the front of the firewall or somewhere in the nose of the aircraft. This is “p” in the figure above.
4. Block up the aircraft into its straight-and-level attitude. I usually use 2x4s and strong cardboard boxes.
5. Accurately determine the following from the plans or by actually measuring the aircraft. These arms are measured from the datum. Use positive numbers for arms to the rear of the datum and negative numbers for arms forward of the datum. The opposite will work, so long as you are consistent.
   1. Arm, in inches, of the desired CG (Ac).
   1. Arm, in inches, of the main gear axle. This will be zero if you choose the main gear axle as the datum (Am).
   1. Arm, in inches, of the tailskid or nose gear. This will most likely be negative or very close to zero for an aircraft with a nose gear (At).
   1. Arm, in inches, of the point at which you will be adding weight (Ap).
6. Enter the arm values (in inches) into the Excel workbook.  The figure below shows the Excel cells into which these data are entered.

Desired CG Arm	6.75
Main Gear Arm	3.75
Tail Skid Arm	54.75
Added Weight Arm	0.80

• Accurately weight the aircraft at the left and right main gear wheels and at the tailskid or nose gear. Weights are in ounces.
• Enter the weights into the Excel workbook.  The figure below shows the Excel cells into which these data are entered. Notice the slight differences in weight between the left and right main gear. This is not uncommon.

Left Main Gear Weight	138.00
Right Main Gear Weight	135.00
Tail Skid Weight	18.00

• The Excel workbook will then calculate the results. The figure below shows how much added weight (if any) needs to be added at the point you choose. If the aircraft is already nose heavy, this weight will be negative. The distance by which the CG must be moved is informational and tells you how far off the actual CG of the aircraft is from the desired CG.

CG needs to be moved by 0.15 inches.
Add 7.5 oz. (0 lb./7 oz.) at the Added Weight Arm

1. If the weight to be added is negative, which means your aircraft is nose heavy, you can take either or both of the following actions:
   1. Move things around (such as moving the batteries towards the rear)
   1. Add weight to the tail of the aircraft (or some other place behind the desired CG)
2. If the weight to be added is larger than what you want, you can take any or all of the following actions:
   1. Move things around (such as moving batteries forward)
   1. Add weight as far forward in the nose as possible without compromising structural integrity
   1. Lighten the tail (I’m not sure how you would do this)
   1. Change the point at which you want to add weight by moving it forward. This changes the arm of the point at which weight is to be added (Ap), so you have to re-measure it.
3. If the weight to be added is within the range of what you want to add, then add the weight at point “p”.
4. If you had to take any action described in the above steps, reweigh the airplane, enter the new weights into the Excel workbook, and see what the new results are.

Comments

Straight-and-Level Flight Attitude: it is possible to do the measurements and calculations described in this article for any wings-level (regardless of pitch) attitude of the aircraft. However, this presents several difficulties, not the least of which is the CG shown on the plans of an aircraft. The CG marked on the plans is usually a point directly above (or below) the center of mass of the aircraft when the aircraft is in a straight-and-level attitude, and is usually shown (or described in the building documentation) as so many inches (or cm) behind the leading edge of the wing (or the top wing in a multi-plane aircraft). The actual CG is the same as the center of mass so the “plan” CG is a surrogate for the actual CG and is usually above or below the actual CG. If the aircraft weights and measurements are taken in a non-level pitch attitude, the arms must be calculated from the actual CG (which you probably will not be able to determine) to the points at which the weight measurements are taken.

This is the Internal Calculations Part

The following describes the formulae used by the Excel workbook to determine the weight to be added. This assumes you are familiar with algebra and can read formulae. These formulae can be used instead of the Excel workbook.

These formulae use simple arm/weight/moment calculations. That is, arm (length) multiplied by weight equals moment. Any moment or set of moments can be replaced by equivalent moments or moment. That is, the various weights at various arms of the aircraft (such as the main gear moment, the tailskid moment, etc.) can be replaced by an equivalent Desired CG moment.

As described above, all arms (and, therefore, the associated moments) are calculated as a distance from the single, arbitrarily selected datum.

The unknown is Wp, the weight to be added at the point at which the added weight is to mounted. This is arbitrarily selected and should be at a point at which weight can be added without compromising the structural integrity of the aircraft.

The moment at the desired CG is the arm at the desired CG times the total weight of the aircraft. The formula below includes the added weight.

Ac*(Wp + Wm + Wt) where Wm is the sum of the weights measured at the two main gear wheels.

This is equivalent to the sum of the other moments.

The main gear moment is Am*Wm where Wm is the sum of the weights measured at the two main gear wheels.

The tailskid moment is At*Wt

The added weight moment is Ap*Wp where Ap is known but Wp is unknown.

Therefore,

Ac*(Wp + Wm + Wt) = Am*Wm + At*Wt + Ap*Wp

Which is the same as

Ac*Wp + Ac*Wm + Ac*Wt = Am*Wm + At*Wt + Ap*Wp

Moving a few terms around so the unknowns are on the left side gives

Ac*Wp – Ap*Wp = Am*Wm + At*Wt – Ac*Wm – Ac*Wt

Factoring out Wp gives

Wp*(Ac – Ap) = Am*Wm + At*Wt – Ac*Wm – Ac*Wt

Dividing both sides by (Ac – Ap) gives

Wp = (Am*Wm + At*Wt – Ac*Wm – Ac*Wt) / (Ac – Ap)

All terms in the right of this equation are known so the calculation can be completed. This assigns a value to Wp, the amount of weight that must be added at the chosen point to cause the actual CG of the aircraft to be at the desired CG location.