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I want to deduce the rate of turn of targets that move in a uniform circular motion. The radius of the different circles are known in advance. I have searched for my question and found the following formulas to help me: $$R=\frac{V^2}{11.26\tan\theta}$$

$$\omega=\frac{1,091\tan\theta}{V}$$

The variables used are:

  • $V$ = true airspeed in knots
  • $R$ = turning radius in feet
  • $\theta$ = bank angle in degrees
  • $\omega$ = rate of turn in degrees per second

As we see from the previous equations that we can get the velocity from the first equation and then substitute in the second one to get the rate of turn. But the problem in the bank angle, how can one put a reasonable values for this angle on different radius. Of course the targets that rotates in circles with small radius will different from that moves in a large radius each one will have its own reasonable bank angle and hence its own speed that it will fly with and hence its own rate of rotation.

Note:

all that because i am trying to sample the plane movement over different circles each T (secs). So i want the rate of rotation to be reasonable to be able to get enough samples to represent that movement. Also i have read that the civilian planes is different from the military ones, so it seems to be complicated topic. Any help is appreciated, thanks in advance.

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    $\begingroup$ The required bank angle depends on both the speed and the radius of turn. You need to know any two of these to compute the third. $\endgroup$
    – Dan Hulme
    Jul 26 '18 at 21:17
  • $\begingroup$ A full description of these formulas are in the FAA Pilot's Handbook on page 4-34: $\endgroup$
    – jwzumwalt
    Jul 28 '18 at 18:46
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I hope you also appreciate help that uses the metric system. You should. It makes the calculation much simpler.

So your question is: What bank angle $\varphi$ do I get when flying a circle of a known radius $R$?

All by itself, the question cannot be answered. You need to specify a speed, either the flight speed $v$ or the angular velocity $\omega$, in order to find the proper bank angle. Then it is easy: $$tan\varphi = \frac{v^2}{R\cdot g}\;\; \text{or}\;\;\varphi = arctan\left(\frac{v^2}{R\cdot g}\right)$$ $$tan\varphi = \frac{\omega\cdot v}{g}\;\; \text{or}\;\;\varphi = arctan\left(\frac{\omega\cdot v}{g}\right)$$ where $g$ is the gravitational acceleration and all parameters should be inserted in their SI units. $\varphi$ will be in radians!

If you don't know either speed, try some values until you get a reasonable angle, which would be not more than 30° for civilian and 75° and more for military aircraft. You can find an upper bound for that angle if you know the maximum sustained load factor of the respective plane because the roll angle and the load factor $n_z$ in a turn are directly related: $$tan\varphi = \sqrt{n_z^2 - 1}\;\; \text{or}\;\;\varphi = arctan\left(\sqrt{n_z^2 - 1}\right)$$

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  • $\begingroup$ to be sure that i received it well, the units of v is m/s for r is meter for g is m/(s*s) and for bank angle is degree and for w is degree/sec. am i right? $\endgroup$
    – AAEM
    Jul 27 '18 at 9:38
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    $\begingroup$ @Ahmed: Yes for length and time, no for angles: Please use radians for $\varphi$ and rad/s for $\omega$. $\omega$ = 1 means 2$\pi$ (6.283) seconds per full circle and a bank angle of 30° is 0.5236 rad. $\endgroup$ Jul 27 '18 at 11:51

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