My question: Is rudder angle equal to bank angle? Or, does rudder angle has some relationship with ROT formula?

Below is my steps trying to solve this question.

Step 1: Found below graph from this article showing that vessel's rudder angle (guess it is a close case study) should be in a quadratic equation to the radius like radius = a * rudder_angle^2 + b * rudder_angle + c

enter image description here

Step 2: I check back the ROT equation, ROT (°/sec) = 1091 * tan(bank angle) / speed in knots. Step 1 is somehow make sense to me because it used tan for the angle.

Then, assume all factors are fixed, any suggestion for the next step to prove the relationship or coefficient between rudder angle and bank angle? Thanks.

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  • 4
    $\begingroup$ An aircraft rudder has very little in common with a ship’s rudder. $\endgroup$ Jan 14 at 13:43
  • $\begingroup$ Once established in a coordinated turn the rudder angle should be zero. $\endgroup$ Jan 14 at 17:33
  • $\begingroup$ @MichaelHall not really, you need some $\endgroup$
    – Federico
    Jan 15 at 11:18
  • $\begingroup$ @Federico, I’ve never found it necessary to hold any rudder pressure once the adverse yaw from the initial roll has been compensated for. Heck, half the time I don’t even bother with rudder for that. (laziness from years of flying a jet with yaw damper I guess…) $\endgroup$ Jan 15 at 17:52
  • $\begingroup$ Bank angle, as in the angle of roll of the vehicle. $\endgroup$
    – Koyovis
    Jan 16 at 4:51

Then, assume all factors are fixed, any suggestion for the next step to prove the relationship or coefficient between rudder angle and bank angle? Thanks.

There is absolutely no relationship or coefficient between rudder angle and bank angle in an aircraft turning. Your understanding of aerodynamics is incorrect and your theory is flawed.

An aircraft rudder is only there to control yaw. The effectiveness of the rudder also can’t be quantified by the rudder angle alone because there are too many variables.

A few of the variables include density of the air, speed of the air, size of the rudder, the moment of the rudder relative to Center of Gravity, the airfoil shape of the rudder, size and shape of trim tabs and aerodynamic horns, etc. I am sure there are many more variables.

  • $\begingroup$ Thanks for the direct answer "no relationship or coefficient between rudder angle and bank angle", so that I don't need to stuggle in those angles. $\endgroup$ Jan 14 at 18:07

You are mixing two completely different effects here. With a boat, the rudder is the primary steering control, while with an airplane, the bank angle is the primary driver of a turn, and bank angle is controlled by varying the roll rate with the ailerons. And naturally, a roll rate of zero is still compatible with banked flight, so we can still be turning even with ailerons and rudder centered. (The subtle yaw and roll trim effects that comprise the basis of lateral "stability" will be left beyond the scope of this brief answer.)

I won't try to assess the accuracy of the article about boats, but I can tell you that, depending on the airspeed, the bank angle, the shape of the aircraft itself (e.g. high aspect ratio sailplane versus medium aspect ratio light plane versus low aspect ratio jet fighter), and whether or not any thrust asymmetry exists (e.g. p-factor), in a constant-bank angle turn, for optimal "coordination" (yaw centered or slip-skid ball centered), the rudder may be need to be kept significantly deflected toward the inside of the turn, or no significant deflection may be needed. (Deflection toward the outside of the turn would be unusual, except to compensate for p-factor.) The rudder required for optimal coordination while actually changing the bank angle (rolling) is another matter; generally the rudder must be deflected toward the descending wingtip, especially with high aspect-ratio aircraft.

But to a very rough first approximation, airplane turning dynamics are all about the bank angle, and the rudder can be considered optional. Many small, fast, low aspect-ratio radio-controlled model airplanes have no rudder at all, and still are very maneuverable and aerobatic, apart from snap rolls, spins, knife-edge flight, and other such sideslip-based maneuvers.

So no, you can't just combine a formula for boats with a formula for airplanes like this.

  • $\begingroup$ Any stuff I should take a look if I want to explore more on the relationship between turning and rudder angle $\endgroup$ Jan 14 at 13:54
  • $\begingroup$ @PakHoCheung -- you could start by looking at any ASE answers dealing with turn "coordination", "yaw string", "sideforce" etc. I've written more than a few myself. I highly recommend John S Denker's "See How It Flies" website too. Aircraft yaw dynamics are complex, it's not an easy thing to really understand. Google an article "Circling the Holighaus way". But start by understanding the basic relationship between banking and turning. Many small, fast, low aspect-ratio radio-controlled model airplanes have no rudder at all, and still are very maneuverable and aerobatic. $\endgroup$ Jan 14 at 13:57
  • $\begingroup$ Thanks for your suggestion. I'm trying to learn more about the numbers used in aircraft, so that I can apply more real case for my work as I'm a programmer to design a simple software. Will look at those books and see if having some numbers I can apply:) $\endgroup$ Jan 14 at 18:06
  • $\begingroup$ @PakHoCheung -- basically I think unless you have a rather sophisticated computer model of the aerodynamics, you should ignore the rudder, i.e. assume no rudder deflection is needed, just start by figuring out the relationship between bank angle and turn rate, which depends on airspeed. $\endgroup$ Jan 14 at 20:35

The dynamics of a turn involve bank angle AND the weathervaning effect of the vertical fin (the rudder has a secondary function) (plus pitch, but we'll leave that part out to keep it simple).

When you bank, the wing's lift vector is tilted, and the lateral force introduced by the tilt drives the aircraft sideways (it sideslips). Without a weathervaning effect, it would move laterally while remaining pointed in its original direction; just continue sideslipping. This sideslip happens initially (briefly) when bank is first applied, and is critical for dihedral effect to work for roll stability.

The result of the sideslip is a lateral angle of attack acting on the airplane's vertical profile, resulting from the sideways movement created by the bank, mostly acting on the fin, or more correctly acting on the vertical aerodynamic center, the vertical Neutral Point, of the entire aircraft. As long as the Neutral Point is aft of the airplane's C of G (what the fin is there to do), there is a positive weathervaning tendency, or positive static stability in yaw.

So the bank makes the plane slew sideways, the fin develops lateral lift in response (after a slight lag to allow dihedral effect to do its thing in the event you didn't actually want to bank), and rotates the body about the yaw axis to reduce the angle of attack on the fin to near zero.

So the airplane banks, starts to slew sideways, but immediately afterward the weathervaning effect of the fin rotates the body in yaw to keep the body aligned into the airstream and the sideways motion is accompanied by a rotation that makes the airplane follow the arc of the turn.

What the rudder does is allow the camber of the vertical fin to be varied by the pilot (being more or less a wing flap that works in both directions), to apply forces beyond the basic weathervaning tendency, when the weathervaning tendency is insufficient. What makes the weathervaning tendency insufficient is aileron adverse yaw.

So rudder application is used to adjust the camber of the vertical tail, to apply force beyond the basic weathervaning effect, to cancel out the yawing moment created by adverse yaw from the ailerons. If the ailerons are neutral while banked, little to no rudder is required because there is no adverse yaw.

So for a turn:

  • Bank creates a lateral thrust component to move the plane sideways.
  • The vertical fin responds to the sideways movement by continuously weathervaning the body into the lateral airflow created by the bank. The fin needs to be small enough to allow a slight lag in this reaction to allow dihedral effect to work, where the bank was induced by a bump and the objective is to make the plane return to level flight on its own, but large enough to give a positive weathervaning tendency beyond that when the bank is deliberate.
  • The rudder allows the camber of the vertical tail to be varied to increase or decrease the weathervaning force, mostly to cancel out the yaw forces induced by the aileron displacement. The rudder can also be used to take out the small lag in weathervaning effect you normally get when you bank as mentioned previously. Rudder application is normally roughly in proportion to the up aileron displacement. Airplanes with rudder interconnect systems, like the Piper Tripacer, mechanically directly gear rudder movement to aileron movement (using bungee springs) and the pilot doesn't need to move the pedals to maintain a coordinated turn - left aileron gives left rudder and right aileron gives right rudder and neutral aileron gives neutral rudder.

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