# A transfer function relating jet engine thrust to pitch angle during a phugoid motion

I am doing a school project that requires me to find a transfer function that relates the thrust of a jet engine (which could change with time in one way or another) and the aircraft's pitch angle during a phugoid motion. Phugoid motion is when an aircraft pitches and flies up along this pitch angle while maintaining constant thrust. The drag and the weight would slow the aircraft down, causing the pitch angle to decrease as lift--which depends on aircraft velocity along the flight path. Due to the lift being no longer able to counter the weight, the aircraft droops down, accelerated by gravity and its own thrust while speeding up. This increases lift and allows it to climb back up again. Then the cycle is repeated because all of the flight controls are down. My project's goal is to use fuzzy logic to increase and decrease the thrust so the pitch angle and rate of pitch angle change would be damped to 0.

The $$\theta$$ is the pitch angle: https://www.math.stonybrook.edu/~scott/Book331/Phugoid_model.html
The way I see it, the acceleration/deceleration of the aircraft along the flight path can be found through: $$a_{fp}(t)=T(t)-D(t)-mg\sin(\theta(t))$$ m is the mass of the aircraft,g is the gravity, T is thrust, and D is drag.
The velocity along the flight path can be found through $$v_{fp}=v_{fp_0}-\int_{t_0}^{t_1}a_{fp}(t)dt$$ $$v_{fp_0}$$ is the initial aircraft speed along flight path (a given value). The thrust is a time-varied input, the drag can be found through $$D(t)=C_d\frac{\rho(v_{fp}(t))^2}{2}S_{wet}$$ Everything except vfp is known. The lift of the aircraft is just like the drag: $$L(t)=C_L\frac{\rho(v_{fp}(t))^2}{2}S_{wet}$$ The lift and drag equation here are all with respect to the inertia reference frame instead of with respect to the flight path. The pitch angle should be $$\theta(t)=\theta_0-\int_{t_0}^{t_1}\frac{\frac{L(t)-W}{m}}{\int_{t_0}^{t_1}v_{fp}(t)dt}dt$$ Where $$\frac{L(t)-W}{m}$$ is the vertical acceleration of the aircraft, $$\int_{t_0}^{t_1}v_{fp}(t)dt$$ is the displacement of the aircraft from $$t_0$$ to $$t_1$$ So this $$\frac{\frac{L(t)-W}{m}}{\int_{t_0}^{t_1}v_{fp}(t)dt}$$ is actually an angular acceleration. So we integrate this angular acceleration and find the change in pitch angle from initial $$t_0$$ to "final" $$t_1$$.