Mathematically find forward CG limit on the sloping envelope?

[Tinstaafl]

I need to be able to find the forward CG limit at a known weight where the weight lies on the sloping forward limit. I used to know how to do this but have forgotten after years of no use.

I remember the formula was *something* like:

X = CG position
Y = weight for the X CG position
X1 = arm for the least weight point on the slope
X2 = arm for the max weight point on the slope
Y1 = lowest weight for the slope, corresponds to X1
Y2 = higher weight for the slope, corresponds to X2

X = Y x (Y2-Y1 / X2-X1) + X1

It annoys me that I can't remember it. Even more so that I need to use the formula. Can anyone help?

regards
T.

[Genghis the Engineer]

It can be done mathematically, but frankly it's far easier just to draw the diagram and do it graphically.

G

[john_tullamarine]

Genghis is being a little tough on you Tinny...

In general, if you are looking at a graph of weight against CG, the upper forward limit usually is a sloping straight line (which appears to be your object of interest).

Recall a long ways back to secondary school and the equation for a straight line of known slope m, and vertical intercept b, is

y = m.x + b

A bit of algebraic juggling (and it was rather difficult on this occasion as I have had a number of very nice glasses of wine this evening so do check that I have come up with the correct oranges and lemons ..) gives, for such a straight line passing through known points (x1,y1) and (x2,y2),

m = (y2-y1)/(x2-x1)

b = y1 - m.x1

So, for instance, the straight line through (-6,-33) and (6,3) is

y = 3.x - 15

In the case of the sloping upper forward limit one usually sees in lighties, one can magically call

x = weight and

y = CG

and do the same thing with two known points on the line, albeit this time of the form (w, CG) rather than (x,y).

Keep in mind that, if you are looking at the typical ICAO/GAMA POH chart presentation, the axes are weight and moment (IU, if you prefer) rather than weight and arm. In this case the straight line becomes a parabola and the equation a bit more complicated.

So, good Tinnie .. go forth and put your POH data into a spreadsheet and calculate CG loading things to your heart's desire.

If you check my things above and they don't work out, do let me know and I shall revisit the exercise in the morning ....

[Tinstaafl]

Thanks gentlemen. Finding a the forward limit on the sloping forward face for a given weight is exactly what I'm trying to do. It's for a basic W&B spreadsheet my new employer has done. At the moment he has an image of the graph inserted into the spreadsheet. I'd like to replace that with plotted & computed points. By testing for the forward limit at the computed weight I can use conditional formatting in the data entry cells to warn of an exceedance.

The forward limit is a straight line because it's an arm:weight graph so equations for straight lines are appropriate here. Mathematical rebalancing used to be part of the CPL syllabus & I used to teach it so it's a annoying not being able to remember how! My notes are half a world away so I can't look it up.

I'm having trouble working out what the '+ b' part should be in the formula X=Y m + b, where X is the arm to be found, Y the calculated weight, m the slope & b a constant: Is it the distance up the Y axis where the line would intercept at x=0?

[I.B. Pilot]

I have a better solution for you yet. Choose a computerised W&B system from delete
Check their web site delete

Their system includes both, tabulated data and graphical data (display of your CG envelop and the respective CGs (Zero Feul, Takeoff and Landing). Their system warns you AND displays out of envelop CG and at the same time does not let you print or message an out of limit condition. Same with weight exceedence.

The name of the system is delete. It is the best product available, proven and reliable too. Take a look at it.

It may be the best product on the market .. I haven't the time to check their website. However, overt free advertising is a no no on PPRuNe.

We'll give you the benefit of the doubt in that you presumably use the commercial product and are just ecstatic about its qualities .. we surely wouldn't suspect that you are tied up with it and are just giving it a free plug now, would we ?

Generating a spreadsheet model to do this sort of work with a few bells and whistles is child's play, by the way

[Tinstaafl]

Thank I.B., but that's rather more than we need. AFAIK we're not approved for a load control system such as that. Our Ops. Specs. certainly make reference to not approved for something along those lines.

[Genghis the Engineer]

I'm having trouble working out what the '+ b' part should be in the formula X=Y m + b, where X is the arm to be found, Y the calculated weight, m the slope & b a constant: Is it the distance up the Y axis where the line would intercept at x=0?

If X=Ym+b, then b=X-Ym.

If you know X and Y for any point, and m (the slope) then it's easy. If you don't know m, then you need two points - we'll call them 1 and 2, so..

(#1) X1=Y1m + b
(#2) X2=Y2m + b

Subtract #2 from #1, and you get...

(#3) X1-X2 = Y1m - Y2m = (Y1-Y2)m

Divide through by (Y1-Y2) and you get...

(#4) m = (X1-X2) / (Y1-Y2)

Plug this back into your original equation and you get...

b = X1 - (Y1*(X1-X2)/(Y1-Y2))

G

[Tinstaafl]

Thanks Ghenghis. I'll get busy on the white board next time I'm at work. I want to make sure that my logic/reasoning & equation set up is correct if I have to derive the technique. It's been 25 years since I last had to play silly buggers with slopes of a line. As I said, I used to have the (correct) formula for just this problem ie find a point on a line given slope & y-coordinate (plus offset if it didn't pass through the origin), memorised & noted down.

[john_tullamarine]

To tie my and Genghis' posts together on the m point, you'll note that the equations for m and b are the same form.

Re the original post, the steps to calculate b are

(i) do the calculate m thing

(ii) pick one of the points, plug in the calculated value for m and the answer is b

B is, indeed, the line's intercept with the vertical axis. In my earlier post, this is the y axis, in Genghis's post, the x axis, and in Tinny's end required equation, the CG axis .. ie the dependant variable axis.

More simply, if

y = m.x + b

put x = 0 and

y = m. 0 + b

ie y = b for x = 0

In the spreadsheet simple conditional tests for forward and aft limits are all that is needed. Be aware that quite a few aircraft have varying aft limits .. indeed some are not straight lines as well ... in which case you may need to run a regression to figure a polynomial for the spreadsheet design.

However, the spreadsheet exercise is no different to any other non-trivial sheet calculation .. plan the logic carefully but do a heap of error testing of the beta before showing it to the world.

[Tinstaafl]

Ta muchly. The CG envelope consists of straight lines so the testing can't be too awkward. The most cumbersome part for me is the reason for this thread.

Appreciate your help.

[Taildragger]

I generally punch the data into a programme on our mainfarme, and insert a new type and it is all worked out beautifully after comparison with a Balance Chart.!! Voyla as they say in Frogland.
Now... The Red Writing ..... do you poor Mods have to read every word on these forums to edit Commercial/Non-Commercial,
Offensive/Non Offensive etc etc etc etc etc.
You poor so and so's..... My sincere sympathies.

[john_tullamarine]

Taildragger,

Life's not quite as bad for us as you might imagine. Danny assigns us each a 500 percent pay increase each and every year ... wonderful, heady stuff ... makes us feel loved but doesn't cost PPRuNe all that much ... any calculation with a zero in the multiplication has that sort of characteristic.

However, the rules of the site (which eveyone agreed to) require paid, rather than free, advertising. I have no idea how much the site costs to maintain but I know I am glad it doesn't come out of my pocket money .. paid advertising keeps it available for the enjoyment of us all.

As you would realise from your time on the site, we don't get terribly introverted about things but, recalling the original post which I amended ... it was a bit like reading advertising copy. Fair go, guys and gals ....

I prefer to think that the poster is a convert to the particular commercial package of his reference (I have no knowledge of the package myself) and sees his/her duty to be one of converting all us do-it-yourself weight and balance heathens to the true path.

All it takes is the simple technique of the commercial package owners approaching the top levels of the totem pole and do a commercial advertising deal .. all quite simple, really.


Tinny,

You can go down Taildragger's path and run up a full blown song and dance routine spreadsheet .. been there, done that .. good fun at the time. Suggest that you keep it all as simple as you need to achieve the task at hand. If you come across any big hurdles, give me a call ... one of my bread and butter skillsets.

[Tinstaafl]

Very kind of you J_T.

At the moment I only intend modifying the basic spreadsheet the boss has done by replacing a scanned image of the envelope with a plotted graph. Not sure how successful I'll be but it's something to do while waiting for the passengers.

He's not hugely PC or spreadsheet literate & I'm no huge expert either. He's already incorporated some simple conditional formatting changes I suggested are possible to red flag out-of-limits conditions - hence the need for the sloping forward face calculation to complete the test.

Eventually I think he plans to use PDAs for this sort of stuff so a simple spreadsheet is all it will be.

[john_tullamarine]

Suggestion for that problem -

(a) do the loading sums and determine a CG and weight value for the required load configuration - same as a longhand calc

(b) determine the forward and aft limits at the weight in (a) from your table/equations for the envelope

(c) do a simple comparison test to make sure that the calculated CG in (a) is aft of the forward limit and forward of the aft limit as determined in (b).

Plotting a picture is fun and makes the screen output pretty but doesn't necessarily achieve much beyond what a basic calculation will do for you. The aim is to keep the CG and weight inside the approved envelope .. how you do that is up to your imagination.

[DFC]

I have used a spreadsheet for a number of years to calculate weight and balance and then plot both the ZFW, TOW and LDW on a graph in the spreadsheet. I don't use any serious math.

What I do is place the weight and arm limits from the POH into the spreadsheet and using that info, plot a line graph which (provded that the first weight/arm is the same as the last weight/arm), produces an envelope just like the one often produced in the POH.

Now simply plot the appropriate weight and arm on the same graph as a point and there you have the C of G.....as one varies weight or loading position one can watch the result move about.

Regards,

DFC

[john_tullamarine]

.. only one caveat, DFC.

I read your post to indicate that you plot only a final loaded point for specific loading cases .. no problem with that.

If you are looking to see the variation of CG as load varies, eg with fuel burn, that is most conveniently done on a graph of weight against moment or IU with a CG overlay.

Not so easily done on a graph of weight against CG unless you continually redo the sums and replot the one point for each exercise.

[Tinstaafl]

That's what I had in mind, J_T. The current spreadsheet already tests for some items & then changes the cells red or green if out or in limits, respectively.

It can't yet test around the sloping face, hence the envelope image. I'd like to keep the graphical depiction in addition to the standard table layout for the easy comprehension of what is where.

[john_tullamarine]

.. sounds good ..

[DFC]

You can plot the empty weight values, the ZFW weight values, the LDW values for destination and alternates and the TOW values..........all using figures from the loading details and planned fuel burns

Join them all with a line and provided it remains inside the envelope there is nothing to worry about.

Regards,

DFC

[john_tullamarine]

DFC (and the VERY great majority of pilots),

aahhh .... yeeeesss ... and/but no !!!

This post addresses some important things which are very poorly understood amongst the aviating fraternity. Hopefully it will generate some follow-on posts to make sure that those watching the thread end up understanding a bit more about these concerns.

Beware of being too cavalier and relaxed with plots as there are a few little traps lurking in the background for young players who are insufficiently cautious due to technical ignorance (and I use the latter term in a strict definition sense .. not at all wanting to be impolite here).

First up, I apologise to the purists for whom "mass" is preferable to "weight" .. and they are correct ... but I'm a "weight" dinosaur ... so please make allowances for me ...


Now, there are two main forms in which we might plot loading data

(a) weight vs CG

(b) weight vs moment

Both of these are shown in the typical POH. Such manuals nowadays usually follow the ICAO ops manual format which, subsequently, was adopted by GAMA. This all dates back some years ago ...

Now, if one does the normal longhand calculation sets and plots the RESULTS of the calculations on either format, there is no problem .. one ends up with a few points representing, eg,

(a) empty (or operating .. whatever particular starting configuration definition you might like to use) weight,

(b) zero fuel weight,

(c) landing weight (for as many fuel states as you might find convenient, useful, or otherwise interesting) and

(d) takeoff (or ramp) weight.

The problems arise when people start drawing lines between these points without understanding what the lines might represent and whether such lines have any relationship with real world loading cases.

We need to look at two sets of graphing problems .. and I need to make the point that the scales and datum positions chosen can vary the visual appearance quite dramatically ... but that is another concern altogether ...


First, simple loading cases where we might PRESUME that the loading arm remains (reasonably) constant (and this is the simple case to which I think DFC refers).

Looking at the two plot cases ..

(a) if the plot is drawn on a weight vs CG chart, then one MUST NOT join points with a straight line and presume that the line represents the variation of CG with weight. The actual plot is a complicated curve. Test it .. do a series of calculations and plot the incrementing (weight, CG) points ... not a straight line ....

(b) if the plot is drawn on a weight vs moment (or IU) chart, then the straight line DOES represent the actual CG variation.


Second, consider the notion of constant arm for particular loading cases .. and this is where the main problems can arise

(a) approximately correct for loads such as personnel sitting in seats and we would not normally worry about any minor residual errors

(b) a problem with baggage and freight compartments as the Manual-quoted arm will be for a notional centroidal loading. This is rarely achieved in practice so there will near always be a residual error to consider

(c) fuel loads .. and here is the BIG problem.

(i) as an aside, consider that the "standard" fuel SG values often quoted in pilot training and other documents are not so standard. Fuel density varies with a bunch of parameters .. mainly oil field origin, refining, and fuel temperature. If you want to know the value today at your fuel bowser while you are gassing up the bird, you need either to take a sample and measure the SG in an hydrometer (a simple float gadget to measure liquid density) or check with the fuel farm people who do just that for each incoming load of bulk fuel. Now I can do either as I have half a dozen calibrated hydrometers .. but most pilots can only get the information by ringing the fuel farm office folk .. and they will give you the data without any worries ...

(ii) if we are looking at a small prismatic (eg a nice box shape) tank then the presumption of constant arm usually is OK

(iii) most aircraft tanks are not quite prismatic and there will be a variation of resultant CG as the tank fuel level varies. Be wary that, sometimes, documents will quote a CG for the full fuel case ... which may not be accurate for a part load case. Mostly, the POH will give information if this is a problem.

(iv) many tanks are wildly non-prismatic, eg, swept wing jet fuel tanks .... to use a single arm figure is nonsensical. (Having said that, sometimes we might do just that for particular reasons and take out the error in envelope adjustments on a trim sheet .. but let's not go there at the moment ... )

(v) in the case of multi-tank installations the overall fuel line will comprise several separate lines representing the various tanks. Often a multi-tank installation will have a combination of prismatic and non-prismatic tanks and the overall fuel line can be quite strange-looking

(vi) and, one also needs to be aware of the POH-mandated fuel usage sequence as that will constrain how the fuel is managed and how the loading case for fuel varies.


Another thing one needs to consider is the shape of the CG envelope.

(a) for very small lighties, the envelope might be a simple single forward and aft limit.

(b) for larger lighties, it might include an upper forward (and sometimes aft) sloping limit where the forward limit moves aft at higher weights and the aft limit moves forward.

The problem here is that the weight-CG variation typically is linear (straight line) - although there are aircraft around where (usually aft) limits are complex curves. When plotted on a weight-moment chart, such a straight line becomes a fairly simple curve. Often the curve is drawn as a straight line and, depending on the datum and scaling chosen by the drafter, this difference may be obvious to a lesser or greater extent. However, for the normal lightie chart, using a straight line will be conservative (as it will cut off a bit of the available envelope) so we are not concerned from a safety point of view. We might, however, have a loading (read "dollar") penalty ...

(c) for large aircraft, the envelopes can be quite complicated and there may be some additional problem areas. Two which come to mind involve

(i) sloping limits which go in the reverse sense to what one sees in lighties .. ie the limit becomes less restrictive with increasing weight. In this case, using a straight line on the moment chart is non-conservative and ought not to be done.

(ii) aircraft which check CG for the zero fuel case and stab setting for the takeoff case.


I can recall a number of aircraft where inappropriate use of straight lines can either suggest you are inside (when you are outside) or outside (when you are inside) the envelope ..... as I suggested at the start in respect of drawing straight lines on charts .... not necessarily as simple as people might think.


On a related subject .. years ago I was commissioned to run up a trim sheet for a quite complicated aircraft .. and that was an interesting job as I had to figure out some way of actually doing the task.

However, the point of the tale is that the operator, on the very first local asy training exercise after introducing the new loading system .. rang me up to query how come their standard training ballast load put them outside the envelope ? .. they weren't at all impressed when, on checking it out, I suggested that, for the past however many years, they had been operating outside the envelope on asy sessions ... this was a simple consequence of an inappropriate and ill-conditioned envelope datum/scale in the superseded system.

The plot said it was OK .. but the longhand calculation clearly showed it was not ... The chap who had developed the earlier system was a very experienced consulting engineer for whom I had a high regard .. he just got caught out by a simple design consideration error.

Beware of traps out there, good folk ..


Cup of coffee now finished so I shall stop. There probably are a few other things I should mention but enough is enough for now.