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 (*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 are functions of time.

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

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

In the normal direction, we have centripetal force, which is often
expressed as *mv*^{2}/*r*, where *r* is the instantaneous radius of
curvature. After noticing that that
= *v*/*r*,
this can be expressed as
*v*, giving

Experiments show that both drag and lift are proportional to *v*^{2},
and we can choose our units to absorb most of the constants. Thus, the
equations simplify to the system

= - sin - *Rv*^{2} =

which is what we will use henceforth. Note that we must always have It is also common to use the notation for and for . We will use these notations interchangeably.

2002-08-29