the following is partly old, and then somewhat new. So please bear with me.
In order to speed up the W&B and performance considerations - which are very similar in most cases anyway - I have found a new workflow that I would like to introduce here. Of course, the specific values are from my aircraft (Mooney M20J MSE), also with the specific basic empty weight and CG, but the approach is adaptable to any other airplane.
1. The graphic approach to W&B: My loading scenarios are 99% of the time between a few "extremes", one extreme for example (i) only me in front left, relatively little fuel, or another (ii) myself front left, my wife front right, my son and the heavy mother-in-law in the back, and more fuel. And so on. All of these "extreme" loading scenarios are contained in a single image (the first attached image), and that can be printed on the back of a checklist, for example. There's a lot of information in there. For loading scenarios that are between two extremes, one can simply interpolate. Bonus: the optimal approach speed for the remaining fuel can be read off the right side (in my case the "short field" column, because at my home base with a 600m runway it is important to use the correct, weight-adjusted approach speed). The numbers on the lines within the chart from full to empty fuel are placed in intervals of about an hour's flight.
For loading situations that are "exotic" and cannot be interpolated between these extremes, I have (like many others) a spreadsheet with which I can calculate and print out the same line (full fuel to empty fuel) for this situation.
Attached is the spreadsheet.
So far so good. Now let's make it even simpler than that.
2. As you can see from the picture, it is almost impossible (in the case of the M20J) to miss the allowed region for the CG for any common loading scenarios (unless you put lead weights in the aft baggage area :-). The fuel tank that empties during the flight also does not shift the CG outside of the allowed region (something that Bonanzas sometimes do, I've been told).
Therefore, it is in practice sufficient to calculate the weight and check it is below MTOW. For that, I have designed a "rotating slide rule", using Excel, making fittings to the curves in the POH and a 3D printer.
The calculator works from the inside out: Align the number of adults (80kg each) with the number of children or suitcases (30kg each), then read at the fuel level -> If you get into the red area, the plane is overloaded, otherwise OK. Example in the picture: 3 adults, 2 "children" (one of them is the baggage of 30kg), 46gal of fuel -> outside of red, so it's ok.
Bonus (i): at the weight, you can also read the final approach speed (in the example, 70kt).
Bonus (ii): going further out, If you follow the guide lines on the static middle ring with your eye (these lines are a nonlinear function, namely a logarithm, because from here outwards the game is no longer "adding" weights but "multiplying" influence factors). Then using outside temperature, pressure altitude and headwind component, you can read off the take-off run on asphalt (black) and grass (green) in meters. In the picture, taking the above mentioned 3 adults, "2 children" (= 1 child and a suitcase of 30kg), 46 gal fuel, and using 20 °C, 1000ft PA and 10kt headwind you arrive at 500m ground roll.
Needless to say, this is an approximation (as are the curves in the POH on which this is based), and the results tend to be conservative, still if conditions are marginal, please do use the tables of the official POH, think about reducing weight or wait for the next morning when it's cooler. Also, check full RPM, MP and FF is achieved at the beginning of the takeoff, be disciplined with takeoff abortion etc..
I have also attached the spreadsheet that has been used to calculate the "rotating slide rule". Unfortunately, it is not exactly self-explanatory and well structured (very different from how I work otherwise :-).
In the following, I describe the approach and hope that with these explanations, anyone who wants to go through this can either adjust the spreadsheet or completely rebuild it.
(i) First the inner part of the arithmetic wheel: this is computationally trivial (just add up the weight). You can use this part to understand the graphic creation in Excel. In the block from cell J8 you can find: in column J the number of adults (1,2,3,4), in column K the angle of the rotary knob (20 degrees corresponds to 80kg), in column L and M the X and Y Coordinates of the labels in the rotary wheel, and in the graphic (from line 79) you can see the arrangement of the numbers (1,2,3,4). The yellow input cell K9 is used to be able to turn the rotary wheel in the graphic (for testing in the Excel graphic). Columns N, O, P are the data for the first circle along which it is cut. And so on; the columns to the right give the data for the circles and labels that are always further out.
(ii) It gets more complicated with the performance part. The spreadsheet contains a calculation block at the top left from cell B9 with which you can try out the calculation. Yellow cells are input cells, orange cells are model parameters. The basic assumption of the fit is that the ground roll can be calculated as the product of a constant times four influencing factors (I have "stolen" this idea from https://www.eaa393.org/Cleco/Cleco03/to-m20e_3.htm, where you can also find the reasons for the somewhat odd numerical constants in the formulas; they contain unit conversions and the properties of the standard atmosphere). At MTOW, sea level and 15 °C without wind, the ground roll of the M20J is according to POH
G_standard = 500m
If weight, PA, temperature or wind are different, the ground roll is approximated as a product
Ground Roll = G_standard * f_PA * f_T * f_M * f_W with the correction factors
f_PA = (1 - PA[ft] / 145442) ^ (- 5,255876 * k))
f_T = ((273.15 + Temp[° C]) / 288.15) ^ k
f_M = (actual weight / MTOW) ^ m
f_W = (1 + headwind component / rotation speed) ^ (- w)
(MTOW is 1315kg, rotation speed is 59kt in the case of the M20J MSE).
The parameters k, m and w as well as G_standard were determined from the curves in the pilot's POH using the Excel "solver" as well as trial and error. You can find this in the block from line 56, separately for the three topics PA + Temp, Weight and Wind. There, also the errors of the approximation are estimated and can be minimized using the Excel "solver". It makes sense to use a higher "penalty" for values undershoot than overshoot, to be conservative.
Of course, these fits have to be re-done for another aircraft, and their quality assessed accordingly.
In my case (M20J) I found k = 2.46889, m = 2.7268673 and w = 0.4047607 as a good fit.
(iii) Now for the graphics on the mechanical calculator.
Addition is simple. As described above, on the inside of the stator (above table in the spreadsheet to the left of column AC), 20 degrees correspond to exactly 80 kg, i.e. an additive weight.
But for the performance part, we have to multiply, therefore we have to change from an additive thinking scheme to a multiplicative (logarithmic scale). On the outside of the stator (above table in the spreadsheet to the right of column AM), we are switching from the additive (20 degrees = 80kg) to a logarithmic scale: here, 100 degrees correspond to a factor e in the ground roll, or an addition of 1 in the logarithm of the ground roll. (This means that 69.3 degrees corresponds to a factor of 2, i.e. a doubling of the ground roll).
Columns AC to AM are the labels of the stator with the approach speeds. There are also the guiding lines that translate the additive weight logic (20 degrees = 80 kg) into the multiplicative ground roll logic (69.3 degrees = doubling), i.e. draw the logarithm.
The above formula
Ground Roll = G_standard * f_PA * f_T * f_M * f_W
is equivalent to
ln (Ground Roll) = ln (G_standard) + ln (f_PA) + ln (f_T) + ln (f_M) + ln (f_W)
where all summation terms are reflected on the slide rule according to the relation 1 degree = addition of 0.01. (--> 69.3 degrees is an addition of ln(2) and thus a factor of 2 in the ground roll).
The calculations of the angles for the calculator labels can be found in the spreadsheet from column AS (temperature) to the right. Here, for example, are the temperatures and next to them (derived from the above formulas) the appropriate angles and the corresponding graphic coordinates for the labels.
(iv) For the physical production of the sticky labels on the computing wheel, I copied the graphic of the spreadsheet to PowerPoint and embellished it, for example, rotating the numbers at the correct reading angle depending on their position. Then the whole thing printed out on a sticker, sprayed with clear lacquer (to protect the printer color) and then cut out and glued on.
(v) The mechanical calculator / rotational slide rule itself is a 3D print. Attached are the STL files. All have to be scaled to 10%. The rotors (Rotoren) are inserted into the base plate (Grundplatte), and then the stators (Statoren) are glued to the ground plate for fastening. If someone follows my approach and builds an similar slide rule, I would be happy about a comment here. Have fun.
Weight and Balance MULTI.xlsx Performancerechner Spreadsheet.xlsx Takeoff Grundplatte.stl Takeoff Rotoren.stl Takeoff Statoren.stl