Mathematical Influences on the Flight Checklist of Aircraft

The Basics

The key element of the pilots checklist that deals with mathematics and physics has to deal with the physical loading of the aircraft. The whole way that the pilot flies the plane is dependent upon how the plane is loaded, besides the amount that is loaded. The other components of the checklist will deal with many safety items and functionality of equipment and intrumentation. Additional notes concerning the intrumentation checks required may be covered in the instrumentation section.

In order to achieve the performance and flying characterisitics which are designed into an airplane, it must be flown with the weight and center of gravity position within an approved operating range (envelope). There is much flexibility in how an aircraft can be loaded, however, it cannot be flown with the macimum number of passengers, full fuel tanks and maximum baggage. The pilot must insure that the plane is loaded within the limits of the loading envelope before attempting a takeoff.

The affects of not loading the aircraft correctly can yield the following:

1. Inability to take off properly

2. Inability to climb or cruise properly

1. Weight of the load - The heavier the plane is loaded, the lower the climb performance will be.

2. Location of the load - The center of gravity (C.G.) is a determining factor in floght characteristics. If the C.G. is too far forward, it may be difficult to rotate for takeoff or landing. If the the C.G. is to far aft (to the rear of the plane), the plane may rotate prematurely on takeoff or tend to pitch up during a climb. Longitudinal stabiltiy may be reduced, which can lead to inadvertent stalls and possibly spins. Spin recovery may become more difficult, as the C.G. moves aft of the approved limits.

The weight and location of the C.G. are important from the manufacturer's perspective. Enough so that each aircraft is weighed empty before it is delivered and the empty weight C.G. is determined. (Note that this empty weight will include all the optional equipment to be delivered with the aircraft.) The pilot then can use this basic empty weight and C.G. to compute the the weight at C.G. position of the loaded airplane. This is where some mathematics and physics is involved.

C.G. Arm (torque/or moment) = (N (A) + (R + L) (B)) / T inches.

where T = N + R + L

The basic calculation for the moment is:

Moment = Weight * C.G. Arm, inches aft (the units are inch-lbs)

The method used to calculate this manually is as follows:

1. Add the weight of all items to be loaded to the basic empty weight.

2. Use a Loading Graph (sample below) to determine the moment of all items to be carried in the plane. Note that the graph has different lines for the different category of items. This is take into account their approximate loading location within the aircraft.

3. Add the moment of all the items to be loaded to the basic empty weight to determine the C.G. location.

4. Divide the total moment by the total weight to determine the C.G. location.

5. By using the figures of item 1., and item 4., locate a point on the C.G. range and weight graphs. (see below). If the point falls within the C.G. envelope, then the loading meets the weight and balance requirements.

An example is provided. Note that the assumptions made in this example are the same as shown in the example for the computer generated C.G. i.e. 2 adults in the front seats, weighing 180 and 200 lbs. 2 children in the back seats, weighing 80 and 100 lbs. 45 lbs. of baggage and 60 gallons of fuel.

 Description Weight (Lbs.) Arm Aft Datum (Inches) Moment (In-Lbs.) Basic Empty Weight 1,800 81.00 145,800 Pilot and Front Passenger 380 80.50 30,590 Passengers (Rear Seats) 180 118.10 21,258 Fuel 360 95.00 34,200 Baggage 45 142.80 6,426 Ramp Weight 2,765 86.18 238,274 Fuel Allowance for engine start, taxi and run-up -11 95.00 -1,045 Takeoff Weight 2,754 86.14 237,229

Now the manufacturers have noted that this is not as simple a process for many potential pilots. To simplify this process the manufacturers have created programs that allow for the calculation to be done and a plot created to show the location of the cargo and passengers, as well as the C.G. of the aircraft. The program determines the following:

1. The total weight and C.G. position

This is done in a graphical representation and a sample problem is presented in the owners manual as follows:

SAMPLE PROBLEM

Assume a basic weight and C.G. location of 1,800 lbs. at 81.00 inches. We also want to carry a pilot and 3 crew members. The two in the front seat will be men weighing 180 and 200 lbs. The two in the back will be children weighing 80 and 100 lbs. Two suitcases, weighing 25 lbs. and 20 lbs. are to be carried in the rear compartment. The plane will also carry 60 gallons of fuel. Will the plane be within the safe envelope?

a. Input the basic airplane information of 1,800 lbs. and 81.00 inches to represent the basic airplane.

b. Enter the data for the forward seats, 380 lbs. (180 lb. man and 200 lb. man in the front seats). The program notes the forward seats position and accommodates the C.G. for this location.

c. Account for the weight in the rear seats (80 lb. child and 100 lb. child). Again the program will determine the location of the rear seat and change C.G. accordingly.

d. Continue entering the data for the baggage compartment (20 lb. and 25 lb. suitcases) and the fuel tanks (360 lbs - 60 gallons @ 6 lbs. per gallon).

e. The program will end up showing the total weight of 2,765 lbs. to be at a C.G. of 86.18. The program will verify that this is within the envelope. From our plot of the envelope, we can see that this is well within the envelope, so there is actually room for more fuel.