Phugoid (Long-Period Mode)

The phugoid represents motion at a nearly constant angle of attack. Although the pitch angle is varying periodically, the altitude is also changing so as to maintain a nearly constant a. This is depicted in Figure 9.5. The displacements are exaggerated for clarity. Beginning at the top of one cycle, the airplane has slowed down to its minimum airspeed, and its attitude is nearly level. It then begins to lose altitude. As it does so, its speed increases, followed by a nose-down attitude. At the bottom of the cycle its airspeed is a maximum and its attitude is again nearly level. It then begins to climb. The

O’ as constant Figure 9.5 The phugoid mode; oscillating, longitudinal motion at a constant angle of attack.

airspeed begins to drop off, followed by a nose-up attitude, and the cycle is repeated.

Since a, and hence CL, is approximately constant for the phugoid, an approximate value for the period of the mode can be obtained as follows. At any instant of time, the net force on the airplane in the vertical direction can be written approximately as

Fz=W-p{U02 + 2Uau)C^S But

w = Pu02cus

Thus, the unsteady force accelerating the airplane vertically downward is

Fz = — pUoCi^Su

The approximate equation of motion in this direction is

Fz = mZ

Integrating this equation, assuming и to be of the form

и = мтах sin wt

gives

_ P 1-А) Sll max _

Z =——– 5—- Sin bit

ты

Thus the maximum change in the potential energy (PE) of the airplane is

2gpU0CLoSu

max

APE =———- 5—-

a>

This must equal the maximum change in the kinetic energy (KE) of the airplane, given by

AKE = 2mU0umax

It follows that ы is given approximately by

– V2-^rrad/sec

C’o

This approximate result is seen to be independent of airplane geometry. It can be found in other references, but is frequently written in a somewhat disguised form as

rad/air sec

As written here, the units of ы are radians per air seconds. Expressed in real time, this equation reduces to Equation 9.79. For the example case of the Cherokee 180 at 50m/s, Equation 9.79 predicts an ы of 0.278 rad/sec, or a

period of 22.6 sec. This period is 2.7 sec, or approximately 10%, shorter than the exact value calculated previously.

Short-Period Mode

A very close approximation to the short-period mode is obtained by assuming that и is constant for this mode. With this approximation, Equation 9.33 for the control-fixed case reduces to

(2ijl – CzJd – Cza – (2fi + С2.)в + Сц tan в0в = 0 (9.80a)

— Смаос — Cmjh + іуд — См .в = 0 (9.80 b)

Both CZa and Cz. are normally small compared to 2ft. These two terms will therefore be neglected. If we let в0 = 0, the following quadratic is obtained for the characteristic equation.

2/i/ycr2 — [CzJy + 2(і(См^ + См.)](г + СгСмц ~ 2и-Сма = 0 (9.81)

or

Cz CM’ + CM<i T /Cz См’ + СмЛ2 /2цСма-СгСмЛЛт

a 4n 2iy ~ l 4/i 2iy / 2fiiy /J

For the Cherokee 180 example, this approximation to <r becomes

<t = -0.0391 ± 0.0544/

Thus, with this approximation, the damping is predicted within 1% and the frequency within 4% of the previously calculated values.