Angle of attack from pitch angle acceleration

[Ballistic]

I am analyzing the landing of an aircraft with the following assumptions:

- I consider the main and the nose landing gear wheels as skids, in order to ignore the tyre deflection;

- I have a 3 DOF aircraft model, with X and Y-axis indicating the forces acting respectively horizontally and vertically and the theta angle positive clockwise. This frame of reference is referred to the center of gravity;

- I assume that the plane has already touched the ground, so I don't consider the gliding phase towards the runway.

Now the summation of moments aroung the c.g. gives:

ƩM_cg = I_yy$\ddot{\vartheta}$= R_NLG * L_n - R_MLG * L_m - h_cg * $\mu$F_vert

Where:
$\ddot{\vartheta}$ is the angular acceleration of the pitch moment;
R_NLG and R_MLG are the vertical reactions of the nose and of the main landing gear shock absorber;
L_n and L_m are the distances respectively of the nose landing gear and of the main landing gear from the c.g.;
$\mu$ is the friction coefficient;
h_cg is the vertical distance between the runway and the c.g.;
F_vert is the resultant of the vertical forces.


Knowing the value of I_yy, which is the airplane pitch moment of intertia I can then compute $\ddot{\vartheta}$.

From this angular acceleration of the pitch angle I want to find the angle of attack, which is the difference from the pitch angle and the flight path angle:
A.o.A. -> α = θ - γ

My questions are:

1) Can I assume γ = 0 (flight path angle), since the plane has already touched the ground when I start my computations?

2) If so, how do I integrate $\ddot{\vartheta}$ in order to get the picth angle? What are my intitial conditions?


I hope that everything is clear and sorry for the long post.

 

[Navin A S]

1) Yes, you can assume γ = 0 since the plane has already touched the ground. 2) To integrate \ddot{\vartheta} to obtain the pitch angle, you need to use the following equation: θ = θ_0 + \int_0^t \ddot{\vartheta}dt Where θ_0 is the initial pitch angle and t is the time. The initial conditions for the integration will be θ_0, which can be assumed to be 0, and the initial angular velocity, which can be assumed to be 0.