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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
    Mick Broom
    Member 909
    Shadow G-MWTN

  • #2
    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...

    Comment


    • #3
      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?
      Mick Broom
      Member 909
      Shadow G-MWTN

      Comment


      • #4
        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.

        Comment


        • #5
          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
          Mick Broom
          Member 909
          Shadow G-MWTN

          Comment


          • #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.

            Comment


            • #7
              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
              Mick Broom
              Member 909
              Shadow G-MWTN

              Comment


              • #8
                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.

                Comment


                • #9
                  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
                  Mick Broom
                  Member 909
                  Shadow G-MWTN

                  Comment


                  • #10
                    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.

                    Comment


                    • #11
                      Help- mathematical results for a banked turn

                      Thanks again Steve,

                      I agree with you that manufacturers limitations should not be exceeded and it obviously initiated the problem but with the inside wing unloading the batten clips I believe it became an aerodynamic failure, not a lot to do with G loads or is it?

                      The question I have been asking is to understand level turns as I thought it best to understand them before thinking about descending and climbing turns

                      Thank you for your caution

                      Remember you are on the section of the forum which deals with SSDR so any knowledge which helps understand the dangers is needed so the right decisions can be made.
                      At least in this section I believe we need to go beyond the limitation of just accepting a 60 degrees bank angle and understand the true picture so we can see the reasons behind it.

                      Sometimes a plane is very happy at 60 degrees and others I have flown I would get nervous after 30 degrees.

                      Its a bit like a speed limit through a village there is the legal limit which at school times is way too fast and then there is Sunday morning on the way to the field when above the legal limit is no problem.

                      Well we have dereg so someone's thrown the legal limit out and made us think for ourselves, its a big responsibility and challenge but is a lot more interesting, at least to me as I discover I know nothing but believe it will lead to better pilots.
                      Mick Broom
                      Member 909
                      Shadow G-MWTN

                      Comment


                      • #12
                        Help- mathematical results for a banked turn

                        Are we saying that the G loads are so great the pilot is incapable of pulling himself up the bar? I must say that I don't see the hang strap move much out of the vertical (perpendicular) throughout the maneuver. At one point he seems to be doing the classic HG beginner error of moving across the bar, but allowing his body to twist, losing all weightshift effect. Although that is close to the point where he gives up and goes for his chest pack.
                        Dave

                        Comment


                        • #13
                          Help- mathematical results for a banked turn

                          Hi Dave,

                          That is the message coming from a few but it plainly is not happening.

                          If as stated with the wing at about 90 degrees using the formula as now gives say 8G , well the pilot would have lost vision and certainly not be in a position to do anything but hang on without success plus the real possibility of the plane /wing failure.

                          I believe when in a steep turn the G loads are speed and turn dependant so a bit of speed but more important a tighter radius will produce the G load but this is a lot less than 1/cos( angle) and is probably half the value.

                          Now all I need is a mathematical case which supports this or someone to say its not !

                          Apart from any structural or pilot limitations without an accurate prediction of the G we cannot establish where the stall would occur.
                          Mick Broom
                          Member 909
                          Shadow G-MWTN

                          Comment


                          • #14
                            Help- mathematical results for a banked turn

                            Dave Smith wrote: Are we saying that the G loads are so great the pilot is incapable of pulling himself up the bar? I must say that I don't see the hang strap move much out of the vertical (perpendicular) throughout the maneuver. At one point he seems to be doing the classic HG beginner error of moving across the bar, but allowing his body to twist, losing all weightshift effect. Although that is close to the point where he gives up and goes for his chest pack.
                            Dave
                            It's worth drawing a triangle of forces.



                            As you get closer to the 90 degree point, the G forces build up, forcing you more and more to the lowest part of the arc through which you swing, plus the effect of gravity is always downwards. In extremis you have massive G away from the direction of turn plus nothing to balance gravity (any force will have a zero element at 90 degrees to it).

                            If you look to the end of the video, you see an earlier successful exit from a spiral. Part of the twisting is looking for the reserve handle.

                            Mick Broom wrote:
                            I agree with you that manufacturers limitations should not be exceeded and it obviously initiated the problem but with the inside wing unloading the batten clips I believe it became an aerodynamic failure, not a lot to do with G loads or is it?
                            It is to do with G loads. If you read the report, you will see that the extra loading on the sail caused the batten clips to break, and therefore to disengage from the sail. In the report minimum failure loads for the batten clips were specified.

                            I have seen a fair few aerobatic manoeuvres on HG, and I've seen a few go wrong and parachutes get deployed.

                            When you go beyond 60 degrees angle of bank (or 45 degrees with a Flash 2), you are headed into unknown territory where hospital food and wooden boxes wait for the unwary.

                            From my spreadsheet, at 75 degrees angle of bank you are pulling 4G. 80 degrees is 6G. Beyond this lies AAIB report land. Your flex wing may shed load by washing out, or it may break. Not many wings go much over 7G on the test rig.

                            Beyond 4G you need to have access to a reserve.

                            I'm fairly sure you know what you are doing but just in case I'll have a clear conscience in stating that once you get near 80 degrees there is a fair chance you'll be packing the aircraft into a carrier bag using a dust pan and brush.

                            Here's a quick example of a brief excursion outside the placards:

                            https://www.youtube.com/watch?v=b0gu8UWDqL0

                            PM me an e-mail address and I'll send you the reverse engineered G loading to bank angle.

                            Comment


                            • #15
                              Help- mathematical results for a banked turn

                              Of course you can be at 90 degrees or greater with any g load. It's only if you start in level flight and increase bank whilst maintaining level flight that G will rise according to the hard relationship law.

                              With aerodynamic pitch control you can be at 90 degrees and have negative G if you wish.

                              In a flexwing doing a pylon turn ( which I think is Micks interest as the WAG is coming up and son is flying in it?) the G is governed by the pilot and how much push forward on the bar is made. The machine may briefly get to 90 degrees or greater at the turn apex, but no hard relationship with G can be assumed - the machine can be considered to maintain height 'ballistically' for the very small amount of time it is at that bank angle.

                              An accelerometer is a very useful instrument so the pilot can get a feel for what load they are subjecting the machine to. I think last time at the WAG, with accelerometer fitted they found it very difficult to exceed 4g. As they were flying the Quikr solo this was equivalent to around 3/4 the limit load values.

                              Paul

                              Paul

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