Approximate Solutions for the Phugoid Mode

The observation that the phugoid involves little change in angle of attack allows its analysis to be made by eliminating the Z equation out of the set of the three longitudinal equations of motion. Retaining only the speed and pitch equations gives:

and the phugoid’s characteristic equation may be written as:

/ /„. dX длЛ 2 1 dX dM

yG. W./g di: + dq J + G. W./g dx dq

For the example helicopter with the three different-sized horizontal stabilizers, the root loci of this mode are given in the second portion of Figure 9.16. It may be seen that although the phugoid period is approximately correct, the simplification has sacrificed reasonableness for the phugoid damping.

An even simpler way of calculating the phugoid frequency an be obtained by applying the same assumption used in hover—that since the pitching motion is occurring about a virtual center far away from the flight path, the aircraft’s moment of inertia about its center of gravity can be ignored. This reduces the

which is the same equation as that derived for hover and gives results almost identical to those. produced either by the full three-degree-of-freedom system or the two-degree-of-freedom system in which the moment of inertia is retained. For illustration, using the 54 square foot horizontal stabilizer on the example helicopter:

Since the phugoid motion is primarily an interchange between kinetic and potential energy (speed and altitude), anything that dissipates energy in the process will add damping. A natural energy dissipator is parasite drag. Thus the aerodynamicist is faced with a dilemma because anything he does to clean up the aircraft will result in the phugoid having less damping.