Equations for determining MOI's?

[Guest Ionizeyou]

Hello all. Am looking for assistance in determining MOI characteristics for the .cfg file. What formulas or equations are used to help determine these numbers. The formulas in the SDK don't appear to be correct, especially when running the numbers on the stock aircraft, so I'm wondering if they're correct. I've done well in algebra in the past, so I'm certain that I'm using the equations correctly. Can anyone shed some new light on the subject for me?KevinBoise, Idaho

[Guest Ron Freimuth]

>Hello all. Am looking for assistance in determining MOI >characteristics for the .cfg file. What formulas or >equations are used to help determine these numbers...>>Kevin >Boise, Idaho Normally, the best way to set MoI's is from a list that gives them for different aircraft. However, such lists are often for typically loaded AC, while FS/CFS need the empty weight MoI's set. I have also found discrepencies in such data. So, use some logic and try different values if you get conflicting results. Checking the values for good, similar AC is a good idea. MS appears to get at least the Moments of Inertia, Weights, and distances correct in aircraft.cfg of default AC. Here are the equations I use. Based on Dr. Roskam's info on Radii of Gyration for various AC. First, 'Radius of Gyration' is the RADIUS of a rod, etc. with all the mass at the end point(s) that has the same Moment of Inertia as the actual structure. It is somewhat intuitive, since this RoG is a distance to some point on the structure. Which might be a wing. It is not the distance to the Center of Gravity of the structure, since Moments of Intertia involve d^2, not simply d. For a point mass, MoI = m*d^2. Where m is mass. In English units, m=weight/32.2. 32.2 lb has a mass of 1 slug. One measures distances in the English system in feet. A mass of 32.2 lb (1 sl) at a distance of 2 ft gives a MoI of 2^2 * 1 = 4 sl-ft^2. Moving it to 4 ft makes the MoI 4^2 * 1 = 16 sl. Note mass components further from the point one is figuring MoI for are much more significant in the end result. In the formulas below, b, d, and e are the normalized "Radii of Gyration" for various AC configurations. A twin has a relatively high Radius of Gryation about the Roll and Yaw axis since it has heavy engines on the wing. Intuition suggests that Roll MoI may be higher than Pitch MoI for such AC, since rotation in pitch doesn't move the engines much, but rotation in roll and yaw does. --- Estimate Aircraft Moments of Inertia - from Dr. Roskam ---Roll Ix=(W/g)*(Rx*b/2)^2 Pitch Iy=(W/g)*(Ry*d/2)^2 Yaw Iz=(W/g)*(Rz*e/2)^2 Roll-Yaw Ixz <= 0.02 Iz, often 0.0 (added, Ron Freimuth)g=32.2 ft/s^2W=gross wt. lbb= span ftd= length fte=(b+d)/2 [just average of span and length) (formatting messed up by Forum software)TYPE OF AIRCRAFT - ROLLRx PITCHRy YAWRz Single Low Wing .248 .338 .393 Single High Wing(182R) .242 .397 .393 Light Twin .373 .269 .461 Biz Jet, Light .293 .312 .420 Biz Jet, Heavy .370 .356 .503 Twin Turbo Prop .235 .363 .416 Jet Airliner 4 eng .322 .339 .464 Jet Airliner 3 aft eng .249 .375 .452 Jet Airliner 2 eng wing .246 .382 .456 Prop Airliner 4 eng .322 .324 .456 Prop Airliner 2 eng .308 .345 .497 Jet Fighter .266 .346 .400 Prop Fighter 1 eng .268 .360 .420 Prop Fighter 2 eng .330 .299 .447Prop Bomber 2 eng .270 .320 .410 Prop Bomber 4 eng .316 .320 .376 Concorde Delta Wing .255 .38 .39 (added, Ron Freimuth) Note Delta Wing adds more to Pitch MoI due to Long Root Chord. NOTE: Yaw MoI ~ Pitch MoI + Roll MoI or a bit less. Check this! I assumed the MoI's in the FS2K2 aircraft.cfg for the Concorde were correct. They appear to be reasonable. Then, figured out the three values to add to Roskam's table. I use them for delta wing AC similar to the Concorde. Then, I made some calculations for the Concorde. This illustrates use of the formulas. One complication is I think Dr. Roskam's values apply to AC with typical payload and fuel. While one needs the empty MoI values for FS settings. FS accounts for the station loads and fuel independently. As fuel load changes so do the AC's MoI's. Adding station loads along the fuselage increases pitch MoI above the "empty" values set. Fuel in wing tanks increases Roll and Yaw MoI's automatically. Now the forumlas require a weight, so using the empty weight for the FS AC will at least account for the difference from 'loaded weight'. However, the Radii of gyration may also change with loading. Sometimes I modify the values the formulas give a bit for that. Regardless, these formulas give at least approximate values for the three MoI's required for FS/CFS AC. They may not be critical in well damped AC, but jets I set up with more realistic (lower) dampings are more critical. One can experiment to see how stable the AC is with different fuel loading, station loads, etc. and modify the MoI values in aircraft.cfg some to see if it helps. Ixy, the 'cross_moment_of_inertia' in aircraft.cfg is often zero. I found a value there can make an AC less stable on HDG hold with the FS2K2 autopilot. AC such as the 747-400, which store fuel in the tail, including vertical fin have a significant Ixy due to the assymetry in the up/down plane. Due to the extra mass above the AC's CG. However, if one sets up such a tank, FS accounts for Ixy automatically. So, I reduced the published value of Ixy I had set in my 747-400 settings.-------- EXAMPLE ----------- Concorde:Max TO - 408,000 lbNo Fuel - 197,768 lbLength - 204.5 ft (.38*102)^2 = 1502Span - 84.25 ft (.253*42)^2 = 113(Span+Length)/2 - 144.8 ft (.39*72.4)^2 = 797Width - 9.5 ftEmpty Mass - 197,768/32.2 = 6,142 slugsMax Mass - 408,000/32.2 = 12,670 slugs ---- Empty Pitch: 1502 * 6142 = 9,227,000, Max: 19,034,000 slug ft^2Roll: 113 * 6142 = 694,000, Max: 1,432,000Yaw: 797 * 6142 = 4,897,000, Max: 10,100,000 FS2K Concorde: Pitch is about 9,300,000,Yaw is 4,900,000, Roll is 696,000. ----------------------- That was calculated to check that the d,b,e I'd calculated gave results consistent with the values set by MS for the Concorde. I just change the weight for empty and loaded to see the end effect (where d,b,and e were assumed to stay the same). ------------------- This can all be done automatically with a program Greg Pearson wrote "CFSCalculator.exe" some time ago. It was somewhere at but I can't find it now. If one has the appropriate DLL's in his system (probably most do) I can email "CFSCalculator.exe" as an attachment. {CFSCalculator also calculates required Flaps Lift. However, since it was written we found one has to adjust the increase in Lift due to Flaps by dividing by 'max flaps deflection, radians'. This is in TBL 320 of the AIR file, typically it is 0.698 [radians] for 40 deg flaps deflection. FS2K2/CFS2 have the flaps deflection(s) set in aircraft.cfg, so that has to be divided by 57.3 to convert to radians. That value for 'Lift - Flaps' is set in the AIR file.}Ron

[Guest Ionizeyou]

Hi Ron, Thanks for your detailed answer. It's much appreciated.In plugging in some of the numbers tonight, I found that my 757 (Mike Stone's model) collapsed its landing gear upon loading into FS.Here are the numbers I came up with using the formulas above:Weight of the aircraft was determined by adding weight of, empty weight, 128730 + fuel weight, 77187 + payload weight (10,00 less than maximum) 39083 = 245,000 lbs. total weight. Maximum gross weight is 255,000 lbs.To obtain my numbers for MOI, I did the math as follows:For Roll; (245000/32.2^2)*(.246*124.833/2)^2=55708.746For Pitch; (245000/32.2^2)*(.382*155.25/2)^2=207770.695For Yaw; (245000/32.2^2)*(.456*140.0415/2)^2=240900.412This is what I came up with and obviously I'm doing something wrong here. Perhaps you can help me. Thanks again for your help...Kevin

[Guest Ron Freimuth]

>Hi Ron, >>Thanks for your detailed answer. It's much appreciated. >>In plugging in some of the numbers tonight, I found that my >757 (Mike Stone's model) collapsed its landing gear upon >loading into FS. Hmm. Unless the MoI's make an AC unstable on the ground, the LG shouldn't collapse. >Here are the numbers I came up with using the formulas >above: >>Weight of the aircraft was determined by adding weight of, >empty weight, 128730 + fuel weight, 77187 + payload weight >(10,00 less than maximum) 39083 = 245,000 lbs. total weight. > Maximum gross weight is 255,000 lbs. >>To obtain my numbers for MOI, I did the math as follows: >>For Roll; (245000/32.2^2)*(.246*124.833/2)^2=55708.746 >For Pitch; (245000/32.2^2)*(.382*155.25/2)^2=207770.695 >For Yaw; (245000/32.2^2)*(.456*140.0415/2)^2=240900.412 >>This is what I came up with and obviously I'm doing >something wrong here. Perhaps you can help me. >Kevin I showed g = 32.2 ft/sec^2. (ft/sec^2) is the units for the conversion. Just use (245000/32.2) above. I brought up CFSCalc and for a 'Jet Airliner 2 eng wing' it shows '6,690,216' sl-ft^2 for Pitch MoI. When I multiply your Pitch value above by 32.2 to take out the extra division by 32.2 I get '6,690,194'. Virtually the same as CFSCalc gives. HOWEVER, use "128730", your empty AC weight, in the equations. The fuel and payload are added automatically in flight. Incidently, if the form of an AC stays constant, Weight increases with l^3. While MoI's incrase with l^5. Where 'l' is a length measurement, such as span. Double wing span, fuselage length and diameter and weight increases by 2^3 = 8X. While MoI's increase by 32X. Especially useful when one is modifying the total size of an AC by some small factor. At least as a check.Ron

[Guest Ionizeyou]

Thanks again Ron for your help. The corrections worked and have provided numbers which I feel help the realism in flight for the 757.Kevin