Newton has worked out for us that every movement is the result of a force, or a balance of forces. The minimum requirement is to have a force model of lift - weight - drag - thrust.
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With a fixed wing aircraft, the forces are a function of:
- Thrust
- Aircraft speed
Aircraft configuration (high lift devices etc.)
- Aircraft attitude in 6 Degrees of Freedom: fwd/aft, left/right, up/down, pitch, roll, yaw. Both relative to gravity and to free airstream.
- Flight path
Once all the forces are known, work out the equations and direction of motion as a result of the forces, as a function of time.
All in all not a simple model. You would be able to find the force equations and the flight dynamics in open source software such as FlightGear. One has to adhere to the open source license of course, but one could read the source code to understand what the equations are. These are computed every so often per second, and parameters updated for the next clock tick.
Another possibility, a bit more straightforward, is to consider the energy balance of the aircraft. Engine power or thrust setting is what moves the aircraft, and flight starts with thrust as a function of speed. This energy is converted into increase/decrease of kinetic energy, or increase/decrease in potential energy.
You would still have to know weight, attitude relative to free stream etc, but would now compute vertical lift power (weight * climb speed) and horizontal aerodynamic drag power ( a constant * speed$^3$).
More info in
- this question which is about excess lift and power for climbing,
- this answer on how to compute aerodynamic constants, then flight characteristics per time step
- this answer for torque vs thrust for a helicopter rotor.
EDIT
The above is of course a bit complicated due to the many forces acting, and the fluid dynamic nature of the forces. The second method briefly mentioned above is much simpler: use the aircraft performance curve.
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- Power required is a curve, determined by aerodynamic factors and weight.
- Power available can be modelled as a straight line, for simplicity's sake.
- Increase throttle, and power available goes up/down.
- The intersection point between Power available and Required is the airspeed at which the aircraft settles in trimmed flight.
- Vertical distance between available and required can be used for climb. Airspeed will then reduce.
- The right hand side of Power Required, after Minimum Power Airspeed, is roughly proportional to $V^3$. You could only model this bit for simplicity sake.
- Climbing will use up power roughly according $P_{climb} = m \cdot g \cdot c$, with c = climbing speed in [m/s], m = mass in [kg]
- Aircraft nose-up pitch is a linear function of climbing power available (simplicity!)
You can see that the airspeed will need to be high enough for the aircraft not to stall.
Inputs change all the time (your power setting and stick pitch), resulting in a new Power Available line and climbing power demand. The response to the new requirements is not instantaneous of course, but slowly creeps to the new equilibrium. How fast or slow can be regulated with a software gain factor.
So with $\delta_{throw}$ the throttle deflection, $\delta_{pitch}$ the stick pitch, and $c$ the climb speed:
$$ P_a = k \cdot \delta_{throt} \tag{1}$$
$$ c = \delta_{pitch} \tag{2}$$
$$ P_r = f(m, V^3) + m \cdot g \cdot c \tag{3}$$
On the runway, throttle setting = power = $m \cdot a \cdot V$
