Expert: David Hemmer - 8/6/2007

Question
Dear Paul,

I have been working on a problem trying to solve the turn diameter of an aircraft in a steady state level turn.  I have been using the standard equations for turn radius and rate of turn.  The formulas I am using are:

1) Turn Radius = (Velocity*Velocity)/(11.26 *Tan (bank Angle)) , velocity in Knots, answer in feet.

2) Rate of Turn = (1092.95*Tan(bank angle))/ Velocity , velocity in Knots, answer in degrees/second

Here is my problem.  The Rate of Turn is an angular velocity, ù.  Angular velocity where the radius and the velocity vector are perpendicular is given by ù = velocity/radius.  Using this equation to solve for radius does not give me the same answer I get using equation number 1.

Last, I tried to use the Rate of Turn value given using equation number 2 in a vector solution to find turn diameter.  I made the assumption that initial heading was 0 degrees, 150 knots, bank angle of 45 degrees.  I then set up an excel spreadsheet to solve for the vector values of X and Y using the following formulas,

X= (150knots *Cos (heading angle after 1 second)) *(1.15*5280/3600), answer in feet

Y= (150knots *Sin (heading angle after 1 second)) *(1.15*5280/3600), answer in feet

Since I changed my heading angle by the Rate of Turn every second, I would have a new heading angle for each second that went by.  Rate of Turn in degrees per second.  I would them sum all my X and Y distances.  It should make sense that the total distance in the X direction should be zero for 180 degrees of turn.  Then the Y distance should sum up to the turn diameter.

Would you please consider my methods and explain what I am doing wrong if possible.

Thanks You.


Answer
I'm not sure how much help I can be to you on this one. I'm not even sure how your equations (1) and two can be correct. The tangent of an angle will have no dimensions, since it's feet/feet. Thus the units on the right hand side of your first equation will be velocity^2, how can this be a radius? I have similar problems with your second equation. I can also warn you that these  type of equations usually require all your angle be in radians not in degrees.