Introduction

In the tutorial Equations of Motion, we had derived the 6 DOF (degree of freedom) equations of motion. Now what we are going to do is learn a more simplified version of the 6 DOF AC model. Simplified models are preferred because they are much easier to work with and are faster, and hence cheaper, computation-wise!

This new model we will be looking at is the point mass dynamical model. With this model, we will be working in the wind axes frame (from the Traditional Aircraft Axes tutorial) because all aerodynamic forces and moments are given relative to the free airstream velocity.

Point Mass Model

Let's remember back to the Traditional Aircraft Axes tutorial when we talked about the wind axes frame.

Fig. 1 - Depiction of traditional AC axes.

The wind axes are found by rotating the stability axes' z axis (adjusting its yaw) by a degree $ \beta $. The wind axes represent the flow velocity vector of the wind.

We can also write a relation between pitch angle and the angle of attack and flightpath angle using Figure 2 below.

Fig. 2 - Angle of Attack, Pitch Angle, and Flightpath Angle Depiction

We can see from Figure 2, the summation of $ \alpha $ and $ \gamma $ equal the pitch angle, $ \theta $. This is an important relation to note for later use in other tutorials.

Let's now define Euler angles in the wind axes; we can let

• $ \psi_w $ - velocity heading
• $ \gamma_w $ - flight path angle
• $ \phi_w $ - roll angle

Now, using directional cosine matrices, we can write the relation between the wind and local frame.

In the inertial frame, gravity can be written as

$$ g_{x_I} = 0 $$
$$ g_{y_I} = 0 $$
$$ g_{z_I} = g $$

We can use the expanded version of Eq. (1) (by multiplying everything out) to get gravity in the wind axes is

$$ g_{x_w} = -g \sin \gamma_w $$
$$ g_{y_w} = g \sin \phi_w \cos \gamma_w $$
$$ g_{z_w} = g \cos \phi_w \cos \gamma_w $$

Now the velocity vector in the wind frame, by definition, is

$$ \vec{v} = v \hat{i_w} + 0 \hat{j_w} + 0 \hat{k_w} $$

Now since aerodynamic forces are given relative to the wind axes frame, we can write $ F_{aero} $ in the wind axes.

$$ F_{aero} = -D \hat{i_w} + S \hat{j_w} - L \hat{k_w} $$

Drag is going to be negative along the velocity direction as lift is along the negative $ \hat{k_w} $ direction. The other term $ S $ stands for sideforce and that is in the direction along $ \hat{j_w} $. We can also write vector for the force from propulsion in the wind axes frame.

$$ F_{prop} = F'_{p_x} \hat{i_w} + F'_{p_y} \hat{j_w} + F'_{p_z} \hat{k_w} $$

We can then break up each component of the propulsion force vector to relate thrust, the angle of attack, and the sideslip angle.

$$ F'_{p_x} = T \cos \alpha \cos \beta $$
$$ F'_{p_y} = T \cos \alpha \sin (-\beta) $$
$$ F'_{p_z} = -T \sin \alpha $$

And similar to the body frame analysis, we can also express the wind rates.

$$ \omega_{w,I} = \dot{\phi_w} \hat{i_w} + \dot{\gamma_w} \hat{j_2} + \dot{\psi_w} \hat{k_I} $$

where,

$$ \hat{j_2} = \cos \phi_w \hat{j_w} + \sin \phi_w \hat{k_w} $$
$$ \hat{k_I} = -\sin \gamma \hat{i_w} + \sin \phi_w \cos \gamma \hat{j_w} + \cos \phi_w \cos \gamma \hat{k_w} $$

Equating the terms in each direction, we get

$$ P_w = \dot{\phi_w} - \dot{\psi_w} \sin \gamma $$
$$ Q_w = \dot{\gamma} \cos \phi_w + \dot{\psi_w} \cos \gamma \sin \phi_w $$
$$ R_w = \dot{\psi_w} \cos \gamma \cos \phi_w - \dot{\gamma} \sin \phi_w $$

Lastly, we can use Newton's 2nd Law and Chasle's Theorem, where

$$ \text{Newton's 2nd Law: } m\frac{d}{dt} \vec{v} = \Sigma \vec{F} $$
$$ \text{Chasle's Thm: } \frac{d}{dt} \vec{v} = (\frac{d}{dt} \vec{v})_{w} + \vec{\omega} \times \vec{v} $$

We can then write the following equations of motion as:

If Eqs. (2) are put into ODE format, $ \dot{\psi} $ and $ \dot{\gamma} $ can be solved for. We can also set $ \psi = \psi_w $ and $ \phi = \phi_w $ to reduce the point mass model down to

The equations in Eqs. (3) are used mainly for performance analysis and optimization. It saves time and money to use simpler models!