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             S2: 4.9612 0.2296
          S1: 0~0763
              SO: 0.2296
We observe that all the elements of the first column of Routh's array are positive
or have the same sign; hence the sufficiency condition is also satisfied. Hence,
the characteristic polynomial has no positive real root or a complex root with
positive real part. Therefore, the given system is stable as we have noted from
direct determination of eigenvalues of matrix A.
    The standard form of the characteristic equation of a second-order system is
A2 +2<WnA +Wr~ - O
(6.34)

544           PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
Let
                                     A. -. -r i: js                                            (6.35)
                                           = -CWn + jcon./                                      (6.36)
where g is the damping ratio and tOn iS the natural frequency of the system. Then,
                                       r
                               <- - -                        .       (6.37)
                                                               '  ,/77~
                                   tOn :,                                    (6.38)
The period T and time for the amplitude to to become either half or double the
initial amplitude are given by
   27r
T--
to,,,~~-
  0.69
     to -. ~r~
(6.39)
(6.40)
Here, to is the time for half amplitude if r is positive, and it is the time for the
amplitude to double if r is negative.
    For the general aviation airplane, we get <1.2 = 0.6997 and co1.2 - 3.6054, cor-
responding to ,/J.2 and C3.4 -.0.0281, and t03.4 = 0.2134, corresponding to A.3.4.
The motion corresponding to A1.2 is heavily damped and is of higher frequency
or a shorter period. The other motion corresponding to A3.4 is lightly damped and
iS of lowPr frequency or longer period. These values of A,1.2 and A,3.4 are typical
of conventional, statically stable airplanes. The high-frequency, heavily damped
oscillator}r motion is called the short-period mode, and the lightly damped, long-
period oscillatory mode is known as phugoid or Iong-period mode. For the general
aviation air)lane, we find that the periods of short-period and phugoid modes are
 1.7424 and 29.4432 s, respectively. The corresponding values of the time for half
 amplitude are 0.2735 and  1J5 s. Because the short-period mode is fast and heavily
damped, it is just felt as a bump by the pilot or the passengers. The pilot does not
have to take any action to kill this mode. The phugoid mode is very lightly damped
and usually persists for a long time. It can be quite annoying if left to die by it-
　
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