Estimation of Moments of Inertia

SPIDERMANN:

What are you trying to do? The moment of inertia is for planar areas only. What you possibly want is the mass moment of inertia which is for 3 dimensional volumes. If you have access to dimensioned drawings you can esitmate the MOI by adding up the individual MOI's in the plane. If you have an actual section you can find the MOI experimentally.

Regards
Dave

Hello Dave

These are calculations for a simulator I've been working with that is dependent on the MoI's for proper flight handling characteristics. I require an MOI for roll, pitch and yaw referenced to the three standard aircraft axes. I think you are correct in your assumption that I need the mass moment of inertia, but I'm not entirely sure of how to achieve the desired results. I understand that I need the radii of gyration to be able to calculate the MoI's.

Normally I would use the formulae above to calculate MoI's, but it is dependant on Radii of Gyration coeffiecents given in a table of known values, but the plane in question is not listed, and has no characteristics in common with any that are.

----EXAMPLE----
Using:
Roll Ix = (W/g)*(Rx*b/2)^2
Pitch Iy = (W/g)*(Ry*d/2)^2
Yaw Iz = (W/g)*(Rz*e/2)^2

where:
g = 32.174 ft/s^2
W = wt. lb
b = span ft
d = length ft
e = (b+d)/2

TYPE OF AIRCRAFT - ROLL,Rx PITCH,Ry YAW,Rz
Single Low Wing .248 .338 .393
Single High Wing(C182R) .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 .447
Prop Bomber 2 eng .270 .320 .410
Prop Bomber 4 eng .316 .320 .376
Concorde Delta Wing .253 .380 .390

-----TO CALCULATE-----
Concorde:
W = 197,768 lb
d = 204.5 ft (.380*102)^2 = 1502
b = 84.25 ft (.253*42)^2 = 113
e = (Span+Length)/2 - 144.8 ft (.390*72.4)^2 = 797
Empty Mass - 197,768/32.174 = 6,142 slugs

Pitch MoI: 1502 * 6142 = 9,227,000,
Roll MoI: 113 * 6142 = 694,000,
Yaw MoI: 797 * 6142 = 4,897,000,

How can I calculate the missing coefficent for the aircraft I'm currently working with?

Thanks for your assistance, and tell me if I'm traveling down the wrong path here.

-Spider

SPIDERMANN: There is something that I do not understand.

In the Roll Ix = (W/g)*(Rx*b/2)^2 equation if we calculate the units we find that Ix(moment of inertia?) has the units of
#-sec^2. The units for moment of inertia are "distance^4" such as "ft^4"

Also working out the units in the equation I=mk^2 we find the units of I(mass moment of inertia) are #-sec^2-ft These units differ fron the Ix units by "ft" term.

It appears there is something wrong in the equations for Ix, Iy, Iz or in the assumptions.

This is about as far as I go, I believe there is more information needed

Regards
Dave

Dave,

You are absolutely right - the example calculation is severely flawed (sorry about that).

Lets try this again -

This is the formula I originally started with:

MOI = EmptyWeight * (D^2 / K)
Where:
Pitch Roll Yaw
D = Length (feet) Wingspan (feet) 0.5* (Length+Wingspan)
K = 810 1870 770

But this proved to not be accurate enough due to "K" being a general term.

So I thought I would try to get more accurate results by the formulas above.

If I use the data from the plane I'm working on now,
Weight(34000 lbs), Length(62.1 ft), Wingspan(44.5 ft), how can I work out the needed "K" value above?

Any ideas?

Thaks again for your help, and sorry about the flawed example!
-Spider

---EDIT---

Sorry, the units are slug - ft^2.

Thanks.

I suggest that you will be able to use plug values for pitch based on the overall length. Yaw and roll are more difficult. The yaw values don't seem to vary much, so use 0.42. Roll will be a bit less than yaw.

An alternative is to make a solid model of the a/c out of uniform density material and establish your own values. If your engines are somewhere odd you'll need to take account of that.



Cheers

Greg Locock

Please see FAQ731-376 for tips on how to make the best use of Eng-Tips.

SPIDERMANN: Here goes more information. The moment of inertia, Im=the integral over the volume of (r^2*p*dV). The second moment of the area is Ia=intergral over the area of ((y^2)*dA).

It looks to me like what has to be done is to take incremental volumes (area * distance) from a fixed point, say teh wing root, and add them up to get the Im for say, the wing. You would take a section of the wing with the skin, the spar and stringers and compute the volume using some fixed spanwise distance "dx" maybe 6" or 1 foot. Then add them up to get Im and from that you can find the radius of gyration. The origin of "r" could be from the root. A lengthy way to do it, but you could closer to the exact answer. There must be a better way or doing this, but I am at a loss.
Sorry

Regards
Dave

Great, thanks guys!

That's a lot more info than I had, so that's cool.

I've got a call in to my old Physics Prof. so if that yields any more useful information, I'll post here first.

Thanks again,
-Spider

And what happens as the mass fraction of the fuel decreases during flight? Fuel mass can account for almost 1/3 of GTOW on a typical aircraft, right?

I'm no expert on the subject, so please correct me if I'm wrong. But on large aircraft, with fuel in the wings, isn't fuel transferred during flight from the wings to a central tank in order to maintain c-of-g? And wouldn't such a transfer of fuel decrease the aircraft's mass MOI, primarily in roll?

Interesting discussion though. Reverse engineering is always a challenge! Trying to figure out why something is designed the way it is......

Don't lose sight of how accurate you want to get. Other things you could consider, but haven't been mentioned are gyroscopic effects from propellers and increased drag as the attitude becomes more assymetric.

Individual components also have their own mass-moments of inertia, not just the mass-moment due to their distance from the center of gravity, so the equation you're writing could include that, too.


Steven Fahey, CET