Bank Angle and the Physics of Standard Rate Turns (continued)

IX - Other Useful Information that We Can Use From Our Study

As we constructed our mathematical model for standard rate turns we came across many formulas that are useful. In this section lets quickly highlight a few.

A - Radius of Turn Formula (as a function of speed and rate of turn)

where r is the radius of turn, v is the speed of the aircraft and is the rate of turn. Variables r, v and are in SI units: m, m/s and rad/s. We will apply the conversion multipliers so that r, v,and can be given in Nm, knots and ^o/s respectively. Applying the conversion multipliers on table 8-3:

Renaming r, v and to the more user friendly RadiusOfTurn, TAS (true airspeed) and RateOfTurn respectively, we have:

where RadiusOfTurn, TAS and RateOfTurn are given in Nm, knots and ^o/s respectively.

Evaluating and fractions to five significant digits:

For a standard rate turn (RateOfTurn = 3^o/s) ONLY, we have:

          RadiusOfTurn = 0.00053052 · TAS

where RadiusOfTurn and TAS are given in Nm and knots respectively. This equation is exact up to five significant digits.

B - Radius of Turn Formula (as a function of speed and bank angle)

Equation can be rearranged and solved for "r" as follows:

where r is the radius of turn, v is the speed of the aircraft, g is the acceleration of gravity and is the angle of bank. Variables r, v, g and are in SI units: m, m/s, m/s² and rad respectively. We are going to keep "g" in SI units and use the standard value of 9.80665 m/s² for it. We will apply the conversion multipliers so that r, v,and can be given in Nm, knots and ^o respectively. Applying the conversion multipliers on table 8-3:

Renaming r, v and to the more user friendly RadiusOfTurn, TAS (true airspeed) and BankAngle, we have:

where RadiusOfTurn, TAS and BankAngle are given in Nm, knots and ^o respectively.

Evaluating and fractions to five significant digits:

C - Load Factor Formula (as a function of bank angle)

Load factor is defined as the total lift to weight ratio:

(new equation)

Where L is the total lift and W is the weight of the aircraft. We now need to find an equation for L:

From our triangle that we got from analyzing the geometry of a standard rate turn in section III we can get the following equation that will give us total lift (L):

Where , L_V, and L are the bank angle, vertical lift and total lift respectively. Since vertical lift has to equal weight:

or from section III

we can substitute in to get:

we can substitute in to get:

Notice how weight (W) cancelled out. Note that is in SI unit: radians. LoadFactor has no unit. We will apply the conversion multiplier so that can be given in ^o. Applying the conversion multiplier on table 8-3:

b

Renaming to the more user friendly BankAngle, we have:

c

Evaluating and fractions to five significant digits:

d

where BankAngle is given in ^o. The charts below graph this formula.

Load Factor vs Bank Angle (at bank angles up to 80o)

Chart 9-1

Chart 9-1 graphs the load factor at bank angle up to 80^o. At 80^o the load factor is almost 6. This would mean that a pilot performing a level turn at this bank angle would experience almost 6 times the force of gravity (or 6g's).

Load Factor vs Bank Angle (at bank angles from 80o to 89o)

Chart 9-2

Chart 9-2 graphs the load factor at bank angles from 80^o to 89^o. Notice how abruptly the load factor increases. As the bank angle approaches 90^o, the load factor approaches infinity (not possible to show in the graph).