EoMs for Translation

With the expression for the absolute acceleration now obtained, we can apply Newton’s second law,

(27)\[\frac{d\left(m\vec{V}\right)}{dt}_{abs}=\sum\vec{F}_{ext}\]

Where the \(()_{abs}\) refers to absolute acceleration, defined in an inertially-fixed reference frame - which, for our purposes, we may treat Earth axes as being. As we have shown, we have defined forces in body axes, which is moving with respect to Earth axes.

When we have accelerations and translations defined in a non-inertial reference frame, we must use Eqn (25) to determine absolute accelerations.

\[\vec{a}_{b_{abs}} = \left.\frac{\text{d}\vec{V}_b}{\text{d}t}\right|_{Oxyz} + \vec{\omega_b}\times\vec{V}_b\]

where

\[\begin{split}\begin{align}\vec{V}_b&=\begin{bmatrix}U\\V\\W\end{bmatrix}\\ \vec{\omega}_b&=\begin{bmatrix}P\\Q\\R\end{bmatrix}\end{align}\end{split}\]

so we have

\[\begin{split}\begin{aligned} \vec{a}_{b_{abs}} &=\begin{bmatrix}\dot{U}\\\dot{V}\\\dot{W}\end{bmatrix} + \left|\begin{matrix}i & j & k \\ P & Q & R \\U & V & W\end{matrix}\right|\\ &= \begin{bmatrix} \dot{U} + Q\,W - R\,V \\ \dot{V} + R\,U-P\,W\\\dot{W}+P\,V-Q\,U\end{bmatrix}\end{aligned}\end{split}\]

Thus we have defined our absolute acceleration terms in body axes, which means we can define the LHS of Equation (27).

\[\begin{split}\frac{d\left(m\vec{V}\right)}{dt}_{abs}=m\begin{bmatrix} \dot{U} + Q\,W - R\,V \\ \dot{V} + R\,U-P\,W\\\dot{W}+P\,V-Q\,U\end{bmatrix}_b\end{split}\]

The RHS of Equation (27) are the sum of gravitational/weight, aerodynamic, and propulsive forces defined in body axes.

\[\sum{\vec{F}_b} = \vec{F}_{G_b} + \vec{F}_{A_b} + \vec{F}_{P_b}\]

We have already expressed the aerodynamic and gravitational forces in body axes, Equations (16) and (15), respectively. We now define the propulsive forces.

We presume the aircraft has one or more propulsors providing thrust, \(T\), along a vector defined in the \(x/z\) plane, at angle \(\theta_T\) to \(x\). We include an additional term for sidewash effects due to propulsion, \(F_{Ty}\):

\[\begin{split}\vec{F}_{T_b}=\begin{bmatrix} F_{Tx} \\F_{Ty} \\F_{Tz}\end{bmatrix}=\begin{bmatrix} T\cdot\cos\theta_T \\F_{Ty} \\-T\cdot\sin\theta_T\end{bmatrix}\end{split}\]

Thus we can write out equations of motion for translation:

(28)\[\begin{split}m\begin{bmatrix} \dot{U} + Q\,W - R\,V \\ \dot{V} + R\,U-P\,W\\\dot{W}+P\,V-Q\,U\end{bmatrix}=\begin{matrix} -mg\sin\theta - D\cos\alpha + L\sin\alpha + T\cos\theta_T\\mg\sin\phi\cos\theta + F_{A_Y} + F_{T_Y}\\mg\cos\phi\cos\theta - D\sin\alpha - L\cos\alpha - T\sin\theta_T\end{matrix}\end{split}\]

Which is a pretty big deal to have derived. Equations (28) describe the response of the aircraft translational acceleration to the aerodynamic forces and vice versa.

This isn’t enough to describe flight, though - for that, we require the ability to describe rotation and attitude.