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Thread: Help- mathematical results for a banked turn

  1. #1
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    Help- mathematical results for a banked turn

    Any Mathematicians out there

    I am fitting a G meter to the plane for testing and would like to predict mathematically if possible the expected results in a steady state balanced banked turn.

    I have no idea on the theory behind the formula but using 1/cos(banked angle) seems to give incorrect results for the steeper angles which you may expect as it then becomes more of a sideways climb but subject to 1G side load.

    So we move on to centrifugal load using G=F/mass*Velocity(squared) / turn radius ( thanks Adrian )

    As I understand it , and please correct me , From level flight when going into a banked turn is the first formula but when you reach a 90 degree angle it is not banked anymore so all down to the centrifugal force plus the downward G load.

    This suggests to me that the two values are interacting in different proportions throughout the angle changes.

    Believe me if you explain the mathematical solutions there is a very good chance of completely losing me or giving me a mental breakdown so the ideal question is can I find a spread sheet were I could input the bank angle, static mass, speed and radius of turn and get the G loads expected between the bank angles of 0-90 degrees?

    Somebody must have worked all this out !!!!
    Hopefully what the predictions will give me is confidence in the G meter readout and also maybe tell me if it really is a balanced turn

    Thanks

  2. #2
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    Help- mathematical results for a banked turn

    You can't do a level banked turn at 90 degrees because there would be nothing to oppose the gravity acting on the mass of the aircraft.

    Draw a diagram of forces to see what I mean. I would do the figures up to 60 degrees for a level turn.

    The 1/cos is correct...

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    Help- mathematical results for a banked turn

    Hi Steve,

    Thanks for the input , if I may ask further

    I agree you cannot do a banked turn at 90 so does that mean for that the 1/cos does not work at that or approaching that angle.
    The banked turn becomes a climb or at least a change in direction in the vertical sense seen from the plane and its not limited as in a normal climb because it does not need to deal with gravity ( except 1G sideways )
    The arbitrary 60 degree bank is only of interest for regulated planes and comes from Section S. I need to go to 90 degrees ( worse case)
    From a loading on the plane if I have it correct the gravitational pull giving the balanced turn is replaced by the centrifugal loads due to the change in direction as the dominant force as the turn rate and angle increases.
    What I don't know is does this mean that the figures produced by 1/cos being dominated by gravity are good till 45 degrees then it drifts off as the centrifugal loads replace gravity and presumably at 90 degrees to the first.

    Or have I got it completely wrong?

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    Help- mathematical results for a banked turn

    Yes that's quite correct Mick. You have got it completely wrong.

    If you are talking about a fully balanced turn, maintaining the exact same height, then it is simply 1 / cos (bank angle). This is true for all bank angles but it will approach infinity at 90deg. 2g at 60deg, 3g at 70.5deg, 4g at 75.5deg, 5g at 78.5deg, 6g at 80.4deg, 7g at 81.8deg, 8g at 82.8deg. You can see that as you approach 90deg the rate of change increases rapidly.

    If you are not worried about maintaining a balanced turn then you could bank at 90deg and all of the lift would go into turning. Using the formula that you have quoted above, if you know the g you are pulling and the speed you are flying then you can work out what the turn radius would be. Obvious while you are at 90deg bank angle you would be accelerating towards the ground at 1g. If you go into the bank with an element of climb then you won't loose too much altitude during the turn.

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    Help- mathematical results for a banked turn

    Hi Adrian,

    Now this is were I have the problem .

    You state that as you approach the 90 degree bank angle the G values go through the roof which if correct would lead to the structural failure of the plane or the thing stalling when this is obviously not the case.

    More help required

    british team practice

  6. #6

    Help- mathematical results for a banked turn

    Only if you attempt to maintain a level turn using wing lift.

    Basically as soon as you are not maintaining a level turn using wing lift, the bank angle to g relationship is no longer valid.

    You can pull whatever g you like at 90 degrees, assuming you ignore plummeting downwards or generate some lift from yawing the fuselage.

    It becomes a g to turn-radius and speed relationship, which is going to require some deft in-flight tape-measure use from you Mick!

    Cheers,
    Paul.

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    Help- mathematical results for a banked turn

    That makes sense

    I think we are all saying the same thing here so do we plot out the G turn values for the banked turn and the G value due to a G turn and take the first value up to 45 degrees bank then the next set up to 90 degrees.

    Or is there a workable formula which combines both

    New toy Paul ( not working yet )

    GPS Logger

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    Help- mathematical results for a banked turn

    You need to think very carefully about this if it involves a flex wing.

    Recently the DHV investigated a flex wing accident where the wing was banked to close to 90 degrees.

    The wing did not recover and spiralled into the ground. The pilot survived but only just.

    If you are banked up close to 90 degrees with a flex wing, and pulling huge amounts of G, do you think you will ever get your weight high enough to exit?

    This video is not for the faint hearted. Think about placards with care.

    https://vimeo.com/106099298

    The manoeuvre was repeated by a DHV test pilot and he threw his chute at high level.

    If you think there isn't a similar problem with microlights then think again. We won't name the affected.

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    Help- mathematical results for a banked turn

    Hi Steve,

    Interesting Video, as my German is non existent what was the conclusion on the cause?

    The thing is I believe the G loads are not that high because its in a G turn and a radius speed issue.

    The wings do show an indication of the higher loads and you can see the increase in speed with either wing behaving differently but I am not clever enough to know what its telling me.

    At the end of the day the wings were still on and so how accurate is our G load prediction. The ratio of available weight to effect control in an accelerated G condition is something which requires more thought.

    Thanks for the post , I am learning all the time

  10. #10
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    Help- mathematical results for a banked turn

    Mick Broom wrote:
    Interesting Video, as my German is non existent what was the conclusion on the cause?
    Here is the DHV report in English:

    http://www.dhv.de/db1/source/technic...ng=en&item=233

    The conclusion after the technical ones was:

    Irrespective of this, in general it counts that by exceeding the operating limits, stated in the owners manual by the manufacturer, uncontrollable flight attitudes/situations can occur.

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