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

VI - Half Standard Rate Turns

Most pilots in heavier commercial aircraft normally don't exceed banks beyond 30^o. However notice how in chart 5-3, in the previous section, that at slightly above 200 knots, the bank angle for standard rate will exceed 30^o. To address this issue pilots will many times do a ½ standard rate turn. In other words instead of taking 2 minutes to complete a 360^o turn, it will take 4 minutes.

For a ½ standard rate turn, since:

          4 minutes = 4 · 60 seconds = 240 seconds

the aircraft will have to turn at a rate of:

          360^o / 240= 1.5^o per second

Now we will go back to equation ^b and plug-in 1.5^o/s instead of 3^o/s for the Rate of Turn:

b

Substituting: the Rate of Turn = 1.5^o/s for standard rate; using the conventional standard value for g, which is 9.80665 m/s²; and evaluating up to 5 significant digits we get:

          Bank Angle = 57.296 · atan (0.0013734 · TAS) ^d       (for ½ standard rate)

which we will call equation ^d. Applying our simplification of arctangent, equation ^d becomes:

          Bank Angle 57.296 ·0.0013734· TAS       (for ½ standard rate)

Now if we multiply the two numbers in our equation we will have:

          Bank Angle 0.078685 · TAS        (for ½ standard rate)

The equation above is for a ½ standard rate turn. It is basically the same as taking the approximate equation (below), from section IV, for the standard rate turn and dividing it by two (0.15737 / 2 = 0.078685).

          Bank Angle 0.15737 · TAS            (for standard rate)

Let's go ahead then and take the other equations that we used for standard rate turn ( and ) divide them by two and see how close we get to the bank bangle for ½ a standard rate turn given by the exact equation ^d.

Bank Angle Equations for ½ Standard Rate Turn
	Equation Number Equation Exact           d          Bank Angle = 57.296 · atan (0.0013734 · TAS) yes*                    Bank Angle 0.078685· TAS no                    Bank Angle (0.15 · TAS) /2 no                    Bank Angle (0.10 · TAS + 5) /2 no * Up to 5 significant digits

Table 6-1

The rule of thumb equations and that we used for standard rate turn are now divided by two and called and respectively.
 

Comparing the Approximations for ½ Standard Rate to the Exact Formula

At Low Speeds:

½ Standard Rate Turn Bank Angle (Low Speeds)

Chart 6-1

Chart 6-1 graphs each equation. The solid blue line is the exact equation ^d. At a quick glance it seems that equation and remain consistently close to the exact equation throughout the true airspeed (TAS) range.

½ Standard Rate Turn Bank Angle Error (Low Speeds)

Chart 6-2

In chart 6-2 we can take a closer look at the deviations from the exact equation (errors). Equation is the one with the least amount of error overall throughout the true airspeed range but this equation is not easily used for mental calculation. Equation is more precise than equation but the difference is less than half a degree. Equation is only more precise than the others throughout a very narrow range from about 80 to 100 knots.

At High Speeds:

½ Standard Rate Turn Bank Angle (High Speeds)

Chart 6-3

From chart 6-3 it looks like the bank angle passes 30^o at around 420 knots. The chart below will give a better picture of the errors.

½ Standard Rate Turn Bank Angle Error (High Speeds)

Chart 6-4

Chart 6-4 equation is better up to about 640 knots, however all equations are off by at least 7^o at that point. Equation has an error of less than 5^o up to about 580 knots and less than 2^o at or below 400 knots.

Conclusion:

If you used equation for estimating the bank angle for a regular standard rate turn, you can divide it by two (which yields equation ) and get a pretty good estimate of the bank angle for ½ standard rate turn for speeds up to 400 knots (less than 2^o error).