See my other question here for the background context.
Consider this simple plane. I have applied the 4 basic forces: lift, drag, weight, and thrust. My question is, where, precisely, do I apply the lift force?
I've read through this which says it's at a quarter of the chord length. But where does that sit in 3 dimensional space? Most airliners have slightly swept wings. The chord length also changes along the wing. How can I model this mathematically?
The Centre of Lift (CoL) of the whole aeroplane is a summation of the lift forces of the wing, the fuselage and the horizontal stabiliser. The wing has the largest contribution to the lift of course.
Above figure is from Torenbeek, Synthesis of Subsonic Airplane Design, and shows how to construct the Mean Aerodynamic Chord of a swept wing. The CoL of a subsonic wing is taken at 25% of the MAC in the pre-design phase - this answer provides some more detailed information.
Above pic from this answer shows the relation between CoL, Centre of Gravity and tailplane lift/downforce: it is an equation of moments about the CoG.
The arrow at dC$_{NW}$ is the wing lift, dC$_N$ the composed aeroplane lift, and at dC$_{Nh}$ the tailplane lift. Which can be positive in cruise, however at take-off the stabiliser will need to generate a downforce, shifting the aeroplane lift balance point forwards (the wing lift can still be assumed to be at 25% MAC in this stage of your design).
If CoL of the complete plane is in front of the CoG, the nose of the aeroplane will tilt up.
My question is, where, precisely, do I apply the lift force?
I've read through this[3] which says it's at a quarter of the chord length.
The idea of the lift force acting at the quarter-chord point 1) is only a convention and 2) is only valid if you also specify an aerodynamic pitching moment, which compensates for where the lift force is actuallyeffectively acting.
An exception is the case of fully symmetrical airfoil (i.e. an airfoil with no camber), in which case the lift force does act very near the quarter-chord point, with no additional aerodynamic pitching moment to consider.
If the plane you are modelling has a cambered airfoil, then you can't pick a single point where the lift force effectively acts. The Center of Lift moves forward as angle-of-attack increases, and moves aft as angle-of-attack decreases. This destabilizing effect is the reason that most aircraft need tails.
In the air, in steady state flight at a given airspeed, the 4 forces of flight are in balance. Therefor, it might be easier to work this backwards.
for zero pitching moment the center of all lifting forces are directly at the center of gravity
This includes lift contributions from all parts of the airplane and the thrust vector. From your modeling, start with the wing and tail. The clue is "elevator force ON".
The key is to find the Cm plot of the wing you are using. Starting with your straight "candy bar" wing design, pick an airfoil to determine elevator force required for a given center of gravity location. Cm tells you about the stability of the wing itself at various angles of attack by comparing center of pressure to the aerodynamic center, and is stable in negative values. It may vary with AoA.
pitching moments of the wing center of pressure relative to the center of gravity must be balanced with the elevator.
Location of center of gravity will become important later on as you model pitch stability. In other words, do you want your aircraft to require computer control like a fighter jet or to be rock stable like a cargo plane.
