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#1
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A properly performed barrel roll is a 1G manuever. The aircraft's
flight path describes a helix, as David described below. An aileron roll is a variable-G operation, since you feel -1G while inverted. |
#2
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Chris W wrote:
Do we have any who is a math whiz here? I want to find a formula to calculate the position of an airplane throughout a 1G roll. The reason I'm doing this is so I can build a "roll track" for a remote control car so the car will alway have a positive g force on it to keep it on the track. Anyone have any ideas? So far my attempts have have all come up short. They don't pass what my college calculus instructor called the "warm and fuzzy" test. I think it has been too long since I took those classes. Chris, I suggest that you forget about trying to model the path of an airplane in a 1 G roll and, instead, make your car track a simple helix. With a simple helix you should be able to keep your car's front wheels straight as the car goes through the helix. Now for the details... Envision a helix laid out on the inside surface of a cylinder. The cylinder will have a radius and a length. Let's assume for this discussion that the helix makes one revolution in that length. Now all we have to do is find a radius and a length for the cylinder that, for a given car speed, will keep your car on the track throughout. For your car to remain on the track at as it goes inverted, the centripetal acceleration due to the car's rotation about the cylinder axis will have to exceed the acceleration of gravity. We'll specify the target centripetal acceleration by defining a multiplicative factor which we will call the "G factor". A G factor of 1.5, for example, would mean that the target centripetal acceleration is 1.5 G, where G is the acceleration of gravity. With a G factor of 1.5, then, at the top of the helix the net acceleration would be the centripetal acceleration minus the acceleration of gravity or 1.5 G -1 G, or 0.5 G. The force of the car pushing on the track at that point would be 0.5mG where m is the mass of the car. We don't have to use 1.5 G for the G factor. We could use, for example, 1.2 or 2.0. At the end of this post, I'll give you a link to a couple of spreadsheets. In those spreadsheets, "G factor" will be one of the user inputs. A cylindrical helix is nothing but a straight line on a cylindrical surface. "Unroll" that cylinder onto a plane surface and the helix becomes a straight line. Knowing this, it becomes quite straightforward to relate the path length of the helix to the cylinder radius and cylinder length. Remember that we're talking about just one turn of the helix for the cylinder length. Once the relationship of helix path length to cylinder radius and cylinder length is formulated, it is again straightforward to split the car speed (which we shall assume is known), into two components, one along the cylinder axis and the other around the cylinder circumference. With the velocity around the cylinder circumference now formulated, and specifying the cylinder radius as a known, the car speed as a known, the acceleration of gravity as a known, and choosing a G factor, we have all that is necessary to compute the cylinder length necessary to achieve the target centripetal acceleration. I'll not write the formula here because it would be too cumbersome to write in text form. Instead, I will give you a link to an Excel (5.0) spreadsheet in which you can inspect the formula if you wish. http://www.airplanezone.com/PubDir/Helix01.xls In the spreadsheet, I used 9.8 meters per second squared for G, the acceleration of gravity, so all distances are in meters and all velocities are in meters per second. If you changed G to 32.2 then all distances would be in feet and all velocities would be in ft/sec. The numbers in green are user inputs and the numbers in burnt red are the calculated results. Note that I've not locked any cells. Of course, you are free to alter the user inputs as you wish but let's talk about the spreadsheet with numbers that I put in. Note that I specified a car speed of 4 m/s and a G factor of 1.5. The results table shows the cylinder radii and the resulting cylinder lengths to achieve the specified G factor. Note the interesting result that there are clearly two usable radii for most of the cylinder lengths within the solution range. The smaller radius results in a long narrow corkscrew while the larger radius result in a short wide corkscrew. Also note that at the extreme, with a cylinder radius of 1.0884 (for the inputs I used), the cylinder length becomes quite small. At this extreme, the solution is quickly approaching a loop instead of a corkscrew. As an aside, it should be noted that the formula I used in the spreadsheet was not derived to solve for a loop (i.e. for a cylinder length of zero) and it is ill suited for that purpose. In the argot of numerical analysts, the calculation is "ill conditioned" for that purpose. For completeness, then, for the data given, the radius for a loop is 1.088435374... . Of course, the spreadsheet results will change as you change Vcar or G factor, or whatever. You will also note as you play around that some speeds just won't work. I don't know your model scale or your model speeds so you will have to play with the data yourself to find a good solution for your needs. Once you choose a cylinder radius and cylinder length for your helix, you can use the following spreadsheet to see how the centripetal acceleration varies with your model car speed. Of course, you'll not want to let your car's centripetal acceleration fall below 1 G at the inverted point. http://www.airplanezone.com/PubDir/Helix02.xls Now let's talk about the approach and exit from the helix. Let's call the tracks leading to and away from the helix the "approach tracks". You'll probably not want to have to turn your car as you enter the helix so the approach tracks should be straight for some distance before reaching the helix and should be tangent to the helix at the helix entry and exit points. This means that the helix cylinder axis will be at an angle to the approach tracks and that the approach tracks will be parallel to each other but will be offset laterally. Lastly, I'm fairly sure of my physics and math but I'll leave it to others to vet. Good thing you posted your query on a Sunday. Cheers, David O -- (David at AirplaneZone dot com) |
#3
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David O wrote:
Chris W wrote: Do we have any who is a math whiz here? I want to find a formula to calculate the position of an airplane throughout a 1G roll. The reason I'm doing this is so I can build a "roll track" for a remote control car so the car will alway have a positive g force on it to keep it on the track. Anyone have any ideas? So far my attempts have have all come up short. They don't pass what my college calculus instructor called the "warm and fuzzy" test. I think it has been too long since I took those classes. Chris, I suggest that you forget about trying to model the path of an airplane in a 1 G roll and, instead, make your car track a simple helix. With a simple helix you should be able to keep your car's front wheels straight as the car goes through the helix. Now for the details... Why didn't I think of that. That is a much simpler solution. I can even do those calculations but thanks for doing them for me. If I get a 3d model going I will send you an image. -- Chris W Gift Giving Made Easy Get the gifts you want & give the gifts they want http://thewishzone.com |
#4
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Chris W wrote:
If I get a 3d model going I will send you an image. Yes, please do. David O -- email: David at AirplaneZone dot com |
#5
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On Sun, 19 Jun 2005 11:04:39 -0500, Chris W wrote:
4 Groups? Do we have any who is a math whiz here? I want to find a formula to calculate the position of an airplane throughout a 1G roll. The reason At any rate, do you want to maintain 1G, or just positive G? it's a big difference. You can do a barrel roll and maintain positive G all the way around. It's a very simple maneuver and very easy to do. It's also probably one of the easiest to screw up. I'm doing this is so I can build a "roll track" for a remote control car Remember that in straight and level flight you are pulling 1G. If you start a roll you will have to start adding nose up stick to maintain 1G to the point of 1G when inverted. " "Theoretically" as you rolled past inverted you would start reducing back pressure until you were back wings level. At this point it takes some one much more versed in aeronautic theory (and practice) than I, but... A barrel roll comes the closest to what you are asking. It, however starts out at more than 1G. Typically 2Gs and it can be more. With a 2G pull up at the start, you will be pulling 1G when passing inverted. Remember you started out in a nose high attitude to get to this point. So in the theoretically description you would most likely be way nose low at the 180 degree inverted position and I think you will probably get well past 2 Gs getting back to the wings level position. so the car will alway have a positive g force on it to keep it on the But, if it's just the positive Gs you need, shape the track like the path a plane would take through a barrel roll. It would go up and curve to the right forming a corkscrew shape with the end right back at the same level as the beginning. You can add turns as well "as long as the car is changing direction in relation to *its" own vertical axis. For example if the car is on its right side the track needs to be curving right, if on its left then the track needs to be curving left. If the car is inverted the track needs to be curving down. Remember too that the car has to be going fast enough to maintain the desired G forces and traction. Slow down and it'll just fall off. Roger Halstead (K8RI & ARRL life member) (N833R, S# CD-2 Worlds oldest Debonair) www.rogerhalstead.com track. Anyone have any ideas? So far my attempts have have all come up short. They don't pass what my college calculus instructor called the "warm and fuzzy" test. I think it has been too long since I took those classes. |
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