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Maths formulas used in flight Sim

Pro Member First Officer
lionlicker First Officer

In this thread all are welcome to post their maths formulas used in sim flight and add your comments. 😀

I have several posts yet to be added to this thread covering additional formulas - but will start here with a recently discussed one.

My favourite and most used equation relates to keeping the aircraft on a desired feet per Nautical Mile glideslope. No matter about the variations in indicated airspeed and changing wind conditions - you can easily stick to the nominated glideslope angle using the following equation.

FINDING FPM GIVEN FT/NM

A = height above the threshold in feet
B = distance to threshold in nautical miles
C = ground speed (shown on GPS or DME instrument)

fpm = [(A/B)/60]*C

Also the same concept equation just expressed differently:-

D = designated ft/Nm (300 on most finals)
E = resultant constant = D/60 (5 on most finals)

fpm = E * GS

Because your ground speed is constantly changing, you have to keep re-evaluating and re-setting fpm as you descend.

Compensation will also be required. Blindly following this formula will not precisely get you exactly where you want to be.

FINDING FT/NM GIVEN FPM and ground speed(GS)

This is the same above equation just backwards:

ft/Nm = (fpm * 60)/ GS

There are times when you would like to know the angle of your glide slope. Example: Engine failure,.... you are established in a glide and you quickly need to know what is your range. Then you would use this formula.

CONVERSIONS often used in sim:

1 Nautical Mile = 1.151 Miles
1 Nautical Mile = 1.852 Kilometers
1 Nautical Mile = 6076 feet

Aviation Fuel:-
1 US gallons = 6 lbs
1 litre = 1.567 lbs

Temperature:-
Farenheit = 9/5 * Celcius + 32
Celcius = 5/9 * (Farenheit - 32)

7 Responses

Pro Member First Officer
lionlicker First Officer

HOW TO FIND DISTANCE AND DIRECTION BETWEEN depart' AND dest' POINTS.

This following method is 99.7% accurate most of the time. Computing resolutions near or over the poles are not accurate. Computing resolutions where depart' and dest' points are in opposite hemispheres renders about a 97- 99.5% accuracy.

The following seems at first glance to be long and complicated. Once you understand and use it a few times, it become suprisingly simple and quick.

Example:

Departure___LPLA__Lajas Portugal______N38 45.72____W27 05.45
Destination_LSMP__Payerne Switzerland_N46 50.60____E06 54.90

1) Define your variables according to this format:
Departure___Latitude A B Longtitude C D
Destination_Latitude E F Longtitude G H
** signage of variables is important **
North positive...South Negative
East positive....West Negative
....therefore
A = 38______B = 45.72______C = -27____D = -05.45
E = 46______F = 50.60______G = 06_____H = 54.90

2) I = Latitudinal Displacement
I = (E*60)+F-(A*60)-B
.....therefore.... I = 484.88 Nm

3) J = Equatorial Longtitudinal Displacement
J = (G*60)+H-(C*60)-D
.....therefore.... J = 2040.35
***Important Clause****
If J>10800 then J=J-21600
If J<-10800 then J=J+21600
(so you don't compute the long way around the globe between points)

4) K = Mean Latitude
In this example depart' and dest' are in the same hemisphere; - thus
K = [(A*60)+B+(E*60)+F]/120
.....therefore..... K = 42.8
Please Note:___More complicated if depart' and dest' points are not in same hemisphere. Say depart' latitude was 12 South and dest' latitude was 21 North:- you would have to construct a "SUM OF" sequence and get the average by dividing "THE SUM OF" by the number of entries in the sequence. As follows (at 3 degree steps) ;- {12+9+6+3+0+3+6+9+12+15+18+21} = 114. Divided by 12 entries = 9.5. Therefore K would equal 9.5 . End of Note.
.....therefore.... K = 42.8

5) L = Horizontal Displacement
L = cos(K)*J
.....therefore.... L = 1497.06 Nm

6) M = Distance between depart' and dest'
M = sqrt(I^2 + L^2)
..... therefore..... M = 1573.64 Nm

FINDING THE DIRECTION

7) N = Tangent into the appropriate Quadrant
If I>L then N = INVtan(L/I)
If L>I then N = INVtan(I/L)
In this case L>I so N = INVtan(I/L)
.....therefore.... N = 17.94 degrees

😎 O = Direction True
O = N either plus or minus of North, South, East, or West
...in this case it is minus of East (90 degrees)
......therefore.... O = 72.05 degrees True

9) P = Direction Magnetic
If you don't already have one, you could download an A4 sized printable chart showing magnetic deviations over the whole globe. Search engine "Main Field Declination".

Pro Member First Officer
lionlicker First Officer

Finding TrueAirSpeed (TAS)
using GPS and Shift[Z] (red text info line at top of screen - shows direction wind from and wind speed)

A = ground track on GPS
B = ground speed on GPS
C = direction wind from
D = wind speed

E = TAS
E = sqrt[(sin{A}*B + sin{C}*D)^2 + (cos{A}*B + cos{C}*D)^2]

You can use this formula when performing trial runs to construct your own IAS:TAS ratio charts.

The ratio between IAS:TAS varies at different speeds:-
ie; If by testing you find IAS:TAS = 100:120, . . . does NOT mean that IAS:TAS = 50:60
The ratio between IAS:TAS varies with OutsideAirTemperature(OAT)
The ratio between IAS:TAS varies with Altitude

Therefore construct a chart at TYPICAL Alt/Temp/Cruise speeds: -

Example the Grumman "Goose"

__lbs___RPM__Throttle__4000ft,52F___6000ft,49F___8000ft,41F___10000ft,34F__12000ft,27F__14000ft,20F
7200___1900____25"______131:140______130:144______129:147______128:150______127:154______127:159
8500___2000____27"______136:145______135:149______134:152______132:155______131:159______123:154
9900___2100____29"______140:150______140:154______139:158______137:161______129:156______103:130

It is real handy to have access to your TAS:IAS ratios. I use my charts all the time.
Need to know TAS to detirmine wind influence.
Need to know TAS when calculating fuel requirements for a given journey.
🙄

Pro Member First Officer
lionlicker First Officer

More comprehensive IAS:TAS charting.

These days I do not use the formula in the previous post any more when operating performance trials for IAS:TAS and other things. Instead, I edit the weather in Alt{W}{W}>>[User Weather], to be wind free -

and therefore just read the TAS straight off as the GPS(ground speed).

With no wind . . TAS = GPS ground speed.

When charting IAS:TAS, I would do the same test flight in three separate scenarios - COLD , MID TEMP, and HOT.

When setting up your User Weather you might want to choose an appropriate location and date to be consistent with your desired conditions. Choose airports with near Sea Level elevations. :Example : my choices are -
COLD : NZNV Invercargill New Zealand, Middle of Winter
MID TEMP: YPPH Perth W. Australia , Spring or Autumn
HOT: YCIN Curtin W.Australia, Middle of Summer

1) Spawn the weather model with [Weather Schemes]>>>[Clear Skies. (Clears all weather)]. Then click

[OK].
2) Re-enter the weather modeller and go into [User defined weather]>>[Customize weather]>>[All weather

stations]>>[Advanced Weather]>>[Temp/Pressure]
3) Set up Temp' layers as from appropriate column in the following chart.

(Temps given in Farenheit . . format as temp:dewpoint )

___ALT_______COLD_______MID_TEMP_________HOT___
40000______-60:-61_______-57:-58________-52:-54
35000______-58:-59_______-52:-57________-40:-46
30000______-55:-57_______-35:-51________-20:-30
25000______-50:-54_______-19:-37__________0:-14
20000______-34:-42_________0:-20_________18:0
15000______-17:-27________17:-5__________34:12
10000________0:-13________34:10__________52:26
6000________20:2__________49:23__________70:40
3000________24:5__________54:26__________75:45
Ground______32:10_________66:35_________100:60

4) Fly the test flight and gather the IAS:TAS figures.

Example: - Here are a set of figures I gathered for the King Air 350 with RPM about 1575, throttle

about N1=70. . . format temp_IAS:TAS

__ALT_______COLD__________MID_TEMP___________HOT___
_4000____24_212:218______54_210:223_______75_209:226
_8000____11_206:225______42_203:229_______63_200:231
12000____-5_200:231______29_195:235_______47_194:238
16000___-18_192:236______15_187:240_______32_186:244
20000___-32_184:243_______2_184:251_______19_180:252
24000___-45_182:256_____-13_186:272________5_183:274
28000___-52_182:276_____-27_180:281______-10_179:285
32000___-55_174:288_____-40_173:292______-26_174:298
35000___-56_169:299_____-50_175:311______-39_172:310

I then plotted these three lines onto a graph to make it easier to make visual interpolation when seeking a specific detirmination.

😉

Pro Member First Officer
lionlicker First Officer

How to calculate WIND DIRECTION FROM AND SPEED

Sure, you can just Shift[Z] and it will tell you -- . . . but that's cheating! Banned

Seeking more realism in flight sim suggests that you do not use shift[Z] except in test flights.

Here is a method to calculate the wind vector whilst in flight.

It's accuracy is dependent upon your ability to come up with a reasonable TAS figure.

You will need a TAS... so have a IAS:TAS chart on hand. (See previous posts - how to compile one.)

This proceedure looks long and complicated - but it's not! It's very simple! ROFL

A = Ground Track (GPS)
B = Ground speed (GPS)
C = Aircraft Bearing (cockpit instruments)
D = Aircraft TrueAirSpeed (you will have to approximate one from your chart)
(If your IAS is given in mph, divide by 1.151 to convert to knots - then approximate TAS)

FINDING THE WIND SPEED (Knots)

E = Horizontal component
E = sin(A)*B - sin(C)*D

F = Vertical component
F = cos(A)*B - cos(C)*D

G = Wind Speed
G = sqrt(E^2 + F^2)

***Important*** maintain signage of variables - variables will often be minus signed.

FINDING THE WIND DIRECTION FROM

H = the first sine returned
H = INVsin(E/G)

I = the second sine deduced
I = 180 - H
if I>180 then I=I-360

J = the first cosine returned
J = INVcos(F/G)

K = the second cosine deduced
K = 0 - J

L = the common sine and cosine
if H=J then L=H
if H=K then L=H
if I=J then L=I
if I=K then L=I

M = wind direction from
M = L+180

Pro Member First Officer
lionlicker First Officer

Performing test flights to detirmine key characteristics and Vrefs.

When trialing FuelUsage(next post), stall speeds at 2000ft, practical ceilings, elevator trim settings for take off, rotate speeds, minimum cruise settings, emergency descent settings, engines fail glide settings, final approach configs (how far do you want to go??), . . . . it is more significant to vary the weight of the aircraft than to vary the temperature, thus you can perform all your testflights in the same User-defined weather setup if you wish.

If you are really worried about the temperature affect on performance just remember -
- in HOTTER Conditions slightly INCREASE throttle and Vrefs -
- in COLDER Conditions slightly DECREASE throttle and Vrefs.

But again I say - you might as well perform all test trails in MID TEMP conditions. (except IAS:TAS trials)
My choice would be the following: - no wind with these temperature layers (see earlier post for more details on setting up your weather.) format: tempF:dewpointF

___ALT____MID_TEMP
40000____-57:-58
35000____-52:-57
30000____-35:-51
25000____-19:-37
20000______0:-20
15000_____17:-5
10000_____34:10
6000______49:23
3000______54:26
Ground____66:35

Conduct the same testflight in these three scenarios - LIGHTEST, MIDWEIGHT, and HEAVIEST.

To me, LIGHTEST means just you (170lbs) and just enough fuel to ascend to a practical cruise level - turn around descend and land. This varies from aircraft to aircraft. Example: the Mooney Bravo = 85 lbs fuel or 15%. The Boeing747 = 19150 lbs fuel or 5%.
To me, HEAVIEST means full fuel and enough payload to bring you up to Max Gross weight.
MIDWEIGHT would be right in between.

The natural progression of testing and detirmining would go something like this:-

1) THE PRACTICAL CEILING
Jump into a fully loaded aircraft and climb until it becomes ridiculous to climb much higher. It is your personal preference. I would say the Practical Ceiling for the Douglas DC3 is 16000 ft, - but if you want spend another half an hour to climb to 17000 , - that's your choice. Ermm...
(** remember when pushing altitude limits you should increase RPM higher than normal climb or cruise RPM **)

2) Find the LIGHTEST config'. Put about 15% fuel in aircraft (with 170lbs pilot) and fly up to practical ceiling and back to land. Work out how much fuel you used. Define LIGHTEST, MIDWEIGHT, and HEAVIEST weights.

3) Fly up to 2000ft and test for stall speeds (clean config')

4) Decide and trial rotate and Take Off configs and speeds.(elevator trim also)
example: Douglas DC3

__________T.O.lbs___Stall___T.O.Flaps___T.O.Elev___T.O.RPM_____Throttle"____Rotate
LIGHTEST___16750_____70_____none_______4.2%____2400(77%)_____40_________90
MIDWEIGHT_21475_____75_____none_______4.7%____2500(85%)_____42_________95
HEAVIEST___26200_____80_____none_______5.2%____2600(92%)_____44________100

5) Trial and chart CLIMB SETTINGS and PERFORMANCE
example: The Douglas DC3

LIGHTEST 37"throttle
_ALT____IAS____RPM_____fpm___Fuel(lbs)_Dist
_2000___110___2250____1100_____37_______2.3
_4000___108___2250____1000_____60_______6.1
_6000___106___2250_____950_____86_______9.5
_8000___104___2250_____850____112______13.8
10000___102___2250_____800____141______19.1
12000___100___2250_____750____172______24.2
14000____98___2275_____600____211______30.0
16000____96___2325_____600____247______37.0

MIDWEIGHT 39"throttle
_ALT____IAS____RPM_____fpm___Fuel(lbs)_Dist
_2000___115___2350_____800_____47_______3.0
_4000___113___2350_____750_____81_______7.7
_6000___111___2350_____700____117______13.2
_8000___109___2350_____650____158______18.5
10000___107___2350_____600____202______25.1
12000___105___2375_____550____251______32.5
14000___103___2425_____500____303______41.9
16000___101___2475_____450____357______51.4

HEAVIEST 41"throttle
_ALT____IAS____RPM_____fpm___Fuel(lbs)_Dist
_2000___120___2450_____650_____70_______4.1
_4000___118___2450_____650____114______10.4
_6000___116___2450_____550____159______16.8
_8000___114___2450_____500____219______25.1
10000___112___2450_____425____283______33.7
12000___110___2500_____400____355______44.4
14000___108___2550_____350____436______56.7
16000___106___2600_____250____524______73.2

6) For some aircraft you could compile a list of approximate mixture settings
NOTE: In COLDER conditions set mixture slightly RICHER - - - - HOTTER = LEANER

example : - Douglas DC3
_ALT____Mixture%
_2000_____64
_4000_____55
_6000_____47
_8000_____40
10000_____34
12000_____29
14000_____25
16000_____22
18000_____20

Geek

Pro Member First Officer
lionlicker First Officer

Charting Fuel Usage

Set User-defined weather as No Wind/MID TEMP (see earlier posts). Fly at TYPICAL Cruise speeds at different levels and calculate for fuel usage as follows:-

When the aircraft has reached equilibrium at the desired cruise speed, use the cockpit panel clock to time yourself - at the first stroke - hit the pause [P] button. Go into Alt[A][F] and record the exact amount of fuel in tanks. A = first reading of fuel (lbs).
Un-pause and fly for exactly 60 seconds - then hit pause and B = second reading of fuel (lbs)
C = the amount of fuel used in 60 seconds
C = A - B
D = TAS = GPS(Ground Speed)with no wind
E = lbs/Nm
E = (C*60)/D

(hint: with a bit of practice you can save time by timimg in 2X time compression mode) Secret

*** remember E represents a "no wind" scenario - lbs used to fly 1 Nm ***

Example: Boeing747 Cruise Data Chart

format : N1__lbs/Nm

_ALT___Mach__IAS__TAS__LIGHTEST 45000lbs__MIDWEIGHT 58000lbs__HEAVIEST 71000lbs
_4000________230__245_____37__39.59___________43__57.31___________49__83.90
_8000___.43__250__282_____40__36.30___________45__50.18___________50__68.94
12000___.52__284__336_____44__34.03___________47__42.86___________51__55.36
16000___.61__311__388_____48__33.33___________50__40.05___________53__48.71
20000___.69__328__432_____54__40.37___________56__45.14___________60__50.56
24000___.77__341__474_____63__45.41___________66__49.98___________69__54.94
28000___.85__351__515_____85__62.10___________86__64.78___________91__67.92
32000___.85__320__507_____84__52.17___________87__55.85___________94__61.78
36000___.85__291__500_____83__45.41___________90__50.04___________97__55.33
40000___.85__265__498_____85__38.99___________94__44.34___________98__48.49

(hint: you could start up at the highest level (practical ceiling) and descend progressively. Top up the fuel as you burn it up so as to keep the weight approximately the same for the subsequent levels. )

Pro Member First Officer
lionlicker First Officer

Calculating Fuel usage for a given flight.

This post is concerned with using your tailor made charts to calculate fuel requirements when planning and setting up for a flight.

This explanation asumes you have downloaded Real Weather.

1) Check for wind conditions en-route. You can do this by entering [User-defined Weather] to check out the weather stations along the path of your flight. Your aim is to come up with an approximate over-all wind effect (at cruise level) for the entire flight path, so you need to average out the individual instances of wind occurrance reported along the flight path. Just use "fuzzy logic" to come up with a final figure - you could express it as "20 knots on the nose" for example or "45 knots on the tail" for example.

(remember *** keep clicking on [Cancel] to back your way out of User defined Weather)

2) Express this as a mathmatical figure
A = Wind influence
if "35 knots on the nose" ..therefore A = -35, because the wind is pushing you back
if "10 knots on the tail" ..therefore A = 10, because you have a 10 knot advantage

3) Estimate ground speed
B = TAS . refer to charts for given cruise altitude TAS
C = estimated ground speed
C = A + B

4) Estimate Distance through Air
D = ground distance between departure and destination airfields (see earlier post)
E = Air Distance
E = (B/C)*D

5) Distance involved in climb, and fuel used for climb
F = Distance involved in climb (refer to your charts)
G = Fuel used for climb (refer to your charts)

6) Distance involved in descent, and fuel needed for descent and landing
H = Distance involved in descent (you calculate the distance out from dest' airport that you would begin your descent)
I = Fuel allocated for descent and pattern, approach, missed approach etc ( you designate an amount - *** a good rule of thumb is that I = G at least ***)

7) Distance involved in cruise, and fuel involved in cruise
J = distance involved in cruise
J = E-F-H ,{ or J = E-(F+H) same thing }
K = fuel used per one Nautical mile
K = lbs/Nm ,(refer to your charts)
L = fuel needed for cruise
L = J*K

😎 Total fuel needed for journey
M = Total fuel
M = G+I+L

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