There's something awry with that data; it's not internally consistent.
Ignore for the moment the pitch attitude and the fuel (I'm scratching my head over that; is this special fuel that has no weight or something?). Then the info reduces to:
Originally Posted by
nobbyknownowt
An aircraft on wheels weighing
Aircraft main wheels are at 12.871M
Nose wheel 4.089M
Main jack points at 12.763
Nose jack 4.426
Though I think thats irrellevant (I agree, since it's specified as 'on wheels)
LEMAC 11.425
MAC 2.285
r = reaction on nose landing gear tyre
W = total wt
The H-arm M calc given is
12.908 - (8.832 r / W)
The XG(%MAC) given is
64.9 - (386.5 r/ W)
The H-arm equation is stating that the distance between the two weighing points - in this case the wheels - is 8.832m, and that the mainwheel weighing point is 12.908m aft of the reference. The data states that the values are (12.871m - 4.089m =) 8.782m and 12.871m respectively. That's errors of 5cm and 3.7cm respectively.
If you assume 10% of weight on nosewheels (a reasonable enough value) then the equation gives:
H-arm=12.908-0.1*8.832=12.0248m (excuse the excess decimals!)
while first principles gives:
H-arm=12.871-0.1*8.782=11.9928m
- a 3.2cm difference. For the quoted MAC that's almost a 1.5% discrepancy.
I'm already somewhat sceptical of the equations, but let's press on, assuming the one given is correct somehow.
Given that a=12.908 and b=8.832, previous posts have given:
c = 100/MAC*(a - LEMAC)
d = 100b/MAC
For LEMAC=11.425 and MAC=2.285
We should have:
c=64.9
d=386.5
At least the two equations given are consistent. Even if I'm not convinced either is correct.
Next, let's add the pitch attitude terms:
Originally Posted by
nobbyknownowt
(The fuselage datum lime is at 3M if thats any use)
a = Aircraft pitch attitude (+nose up attitude)
The H-arm M calc given is
12.908 - 0.686 sin a - (8.832 r / W)
The XG(%MAC) given is
64.9 - 30 sin a - (386.5 r/ W)
As JT stated, the reason for the pitch attitude correction is the vertical position of the cg/centre of mass. If the mass all lay on one horizontal plane then you'd not need this correction, even at a pitch angle other than zero. Consider a ridiculous example of an aircraft with a huge AWACS-type mounting on its back, full of lead, such that the vertical position of the cg is as high above the reference as it is forward of the mainwheels. If we pitched the aircraft up to 45 degrees and looked at reactions we'd see there was NO reaction on the nosewheel, and with no correction we'd end up assuming that the cg is at the mainwheels - which it is, but only at this silly pitch attitude.
The pitch terms are making a similar correction, but for a more realistic case. They still heavily depend on ASSUMING the vertical cg position - and you may wish to consider how its possible to know the vertical cg if you don't know the weight or the longitudinal cg when you start the weighing .... (hint: it isn't - its an eductaed guess)
Looking at the correction term: - 0.686 sin a; we can see that it provides a smaller value for H-arm (i.e. moves the calculated cg position forward) if the aircraft is pitched up. This means that the raw, uncorrected, value is too far aft, which means that the vertical centre of mass is ABOVE the reference. The 0.686 tells us it's 0.686m above the reference.
In the MAC equation, the 0.686m becaomes 100*0.686/2.285=30.0, which it indeed does...
Since the fuel term is a correction to the 0.686m term, it follows that its an attempt to correct for vertical cg movement with fuel. It appears to be stating that the vertical cg moves by 0.00016m per kg - or, rather less annoyingly, 0.16m per tonne. Interestingly, this term is additive, which indicates that fuel is moving the cg upwards - which would seem to say it's either a high wing type with integral fuel, or fuel in the fuselage. Whatever it is, its another source of error, and best eliminated if you want a real answer.
PS thinking a bit, I wonder if the 5cm and 3.7cm apparent errors are due to either gear travel under load (trailing link type?) or some kind of wheel contact area offset.