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Next: Dealing with the Singularity Up: The Art of Phugoid Previous: Fixed Point Analysis

Qualitative Classification of Solutions

This analysis of the previous section allows us to completely classify what happens in the Phugoid model for all R $ \in$ [0,$ \infty$). In all cases, the fixed point in ($ \theta$, v) coordinates corresponds to the solution

$\displaystyle \theta$(t) = - arctan$\displaystyle \left(\vphantom{ R \sqrt{\frac{1}{1 + R^2}} }\right.$R$\displaystyle \sqrt{\frac{1}{1 + R^2}}$$\displaystyle \left.\vphantom{ R \sqrt{\frac{1}{1 + R^2}} }\right)$    v(t) = $\displaystyle \sqrt[4]{\frac{1}{1 + R^2}}$    x(t) = $\displaystyle {\frac{t}{\left(1 + R^2\right)^{\frac{3}{4}}}}$    y(t) = $\displaystyle {\frac{-R t}{\left(1 + R^2\right)^{\frac{3}{4}}}}$

R = 0:
In this case, there is no drag. The fixed point at $ \theta$ = 0, v = 1 corresponds to a glider flying level with a constant speed (labeled A in the figures below). The fixed point is a center in $ \theta$, v coordinates: nearby solutions are closed curves and correspond to a glider with an oscillatory path, alternately diving and climbing (labeled B). For initial conditions further away from the fixed point, v(t) oscillates, but $ \theta$(t) constantly increases (see D). Such solutions correspond to a glider endlessly looping, and a pilot with a severe case of nausea. Between these two is a solution which cannot be continued beyond a certain time (C), because v(t) becomes zero and our equations are no longer defined. This corresponds to a glider which stalls.

\begin{mfigure}\centerline{ \psfig{figure=glidersum-0a.eps,height=1.5in} \qquad\qquad

0 < R < 2$ \sqrt{2}$:
Here the fixed point (A) corresponds to a glider which dives with a constant velocity and not too steep an angle (the steepest angle is slightly less than $ \pi$/4 below horizontal). The fixed point is a spiral sink: nearby initial conditions spiral into it (B). Such a spirals corresponds to a glider for which the angle and velocity constantly oscillate, but these oscillations get smaller and smaller as time passes, limiting on the same angle and velocity as the fixed point. Solutions with initial conditions further away (D) from the fixed point do some number of loops before settling into a pattern of oscillations which limit on the diving solution. Between the solutions which do k loops and those which do k + 1 loops lies a solution which stalls out after k loops (C) -- at some time the solution gets to $ \theta$ = 2k$ \pi$ + $ \pi$/2 (i.e. going straight up) and velocity v = 0.

\begin{mfigure}\centerline{ \psfig{figure=glidersum-02a.eps,height=1.5in} \qquad\qquad

R$ \ge$2$ \sqrt{2}$:
The fixed point is a sink corresponding to a steeply diving solution.3.6Causing the glider to loop becomes increasingly difficult as R increases. For example when R = 3, if the initial angle $ \theta$ is zero, an initial velocity of larger than 86.3 is required to get the glider to do one loop. If the initial angle differs from that of the fixed point, the glider fairly quickly turns towards that angle. No oscillation occurs; the angle can pass the limiting angle at most once.

\begin{mfigure}\centerline{ \psfig{figure=glidersum-4a.eps,height=1.4in} \qquad\qquad

We have shown that for R < 2$ \sqrt{2}$, the fixed point is a spiral sink: nearby solutions oscillate toward it. We should, however, point out that while the oscillations are always present mathematically, they can be very hard to discern. To emphasize this, we will compare a solution in the $ \theta$, v-plane for R = 2 (a spiral sink) and R = 3 (a sink with two real eigenvalues).

\begin{mfigure}\centerline{ \psfig{figure=glidersum-2a.eps,height=1.5in} \qquad\qquad

The two plots look rather similar. However, if we look closely at the solutions near the fixed point (we can use zoom to do this without recomputing the picture), we see a significant difference.

\begin{mfigure}\centerline{ \psfig{figure=glidersum-2az.eps,height=1.5in} \qquad\qquad

For R = 2, there is a ``hook'' at the end, but for R = 3, the solution goes straight in to the fixed point. Further magnifications show the same pattern, as you may want to verify for yourself. The spiral for R = 2 is always present, but very tight and hard to detect. Similarly, the oscillations in the x, y-plane when R is just slightly less than 2$ \sqrt{2}$ are nearly indiscernable, but present.


... solution.3.6
For R = 2$ \sqrt{2}$, the linearization corresponds to a degenerate case with a double eigenvalue, rather than a regular sink. However, what we say here still applies.

next up previous
Next: Dealing with the Singularity Up: The Art of Phugoid Previous: Fixed Point Analysis

Translated from LaTeX by Scott Sutherland