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Source of that document : http://www.math.sunysb.edu/~scott/Book331/Phugoid_model.html

The Phugoid model

The Phugoid model is a system of two nonlinear differential equations in a frame of reference relative to the plane. Let v(t) be the speed the plane is moving forward at time t, and $ \theta$(t) be the angle the nose makes with the horizontal. As is common, we will suppress the functional notation and just write v when we mean v(t), but it is important to remember that v and $ \theta$ are functions of time.

\begin{mfigure}\centerline{\psfig{figure=plane.eps,height=1.75in} \qquad
\psfig{figure=planeforce.eps,height=1.75in}}\end{mfigure}

If we apply Newton's second law of motion (force = mass × acceleration) and examine the major forces acting on the plane, we see easily the force acting in the forward direction of the plane is

m$\displaystyle {\frac{dv}{dt}}$ = - mg sin$\displaystyle \theta$ - drag.

This matches with our intuition: When $ \theta$ is negative, the nose is pointing down and the plane will accelerate due to gravity. When $ \theta$ > 0, the plane must fight against gravity.

In the normal direction, we have centripetal force, which is often expressed as mv2/r, where r is the instantaneous radius of curvature. After noticing that that $ {\frac{d\theta}{dt}}$ = v/r, this can be expressed as v$ {\frac{d\theta}{dt}}$, giving

mv$\displaystyle {\frac{d\theta}{dt}}$ = - mg cos$\displaystyle \theta$ + lift.

Experiments show that both drag and lift are proportional to v2, and we can choose our units to absorb most of the constants. Thus, the equations simplify to the system

$\displaystyle {\frac{dv}{dt}}$ = - sin$\displaystyle \theta$ - Rv2                $\displaystyle {\frac{d\theta}{dt}}$ = $\displaystyle {\frac{v^2 - \cos\theta}{v}}$

which is what we will use henceforth. Note that we must always have v > 0.

It is also common to use the notation $ \dot{v}$ for $ {\frac{dv}{dt}}$ and $ \dot{\theta}$ for $ {\frac{d\theta}{dt}}$. We will use these notations interchangeably.


Translated from LaTeX by Scott Sutherland
2002-08-29
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