NAVWEPS DD-801-80  STABILITY AND CONTROL  (4) The displacement or deflection of the  elevator when the stick-free condition is  considered.  limit is set by the minimum permissible con-  trollability  while the aft c.g. limit is set by  the minimum permissible stability.  All previous considerations of longitudinal  stability have been concerned with the initial  tendency of the airplane to return to equilib-  rium when subjected to a disturbance.  The  considerations of longitudinal  dynamic sta-  bility ate concerned  with time history response  of the airplane to these disturbances, i.e., the  variation of displacement amplitude with time  following a disturbance. From previous deli-  nition, dynamic stability will exist when the  amplitude of motion decreases  with time and  dynamic instability will exist if the amplitude  increases  with time.  The longitudinal  dynamic stability of an  airplane generally consists of three basic modes  (or manners) of oscillation.  While the longi-  tudinal motion of the airplane may consist of a  combination of these modes, the characteristics  of each mode are sufficiently distinct that each  oscillatory tendency may be studied separately.  The first mode of dynamic longitudinal sta-  bility consists of a very long period oscillation  referred  to as the phagoid. The phugoid or long  period oscillation involves noticeable vatia-  tions in pitch attitude, altitude, and airspeed  but nearly constant angle of attack. Such an  oscillation of the airplane could be considered  as a gradual interchange of potential and  kinetic energy about some equilibrium airspeed  and altitude.  Figure 4.20 illustrates the char-  acteristic motion of the phugoid.  Of course, the airplane must demonstrate  positive dynamic stability for the major longi-  tudinal motions.  In addition, the airplane  must demonstrate a certain degree of longitu-  dinal stability by reducing the amplitude of  motion at a certain rate.  The requited degree  of dynamic stability is usually specified by  the time necessary  for the amplitude to reduce  to one-half the original value-the  time to  damp to half-amplitude.  LONGITUDINAL  DYNAMIC  STABILITY.  The period of oscillation in the phugoid is  quite large, typical values being from 20 to 100  seconds. Since the pitching rate is quite low  and only negligible changes in angle of attack  take place, damping of the phugoid is weak and  possibly negative.  However, such weak or  negative damping does not necessarily  have any  great consequence.  Since the period of oscilla-  tion is so great, the pilot is easily able to  counteract the oscillatory tendency by very  slight and unnoticed control movements. In  most cases, the necessary corrections ate so  slight that the pilot may be completely un-  aware of the oscillatory tendency.  The airplane in free flight has six degrees of  freedom: rotation in roll, pitch, and yaw and  translation in the horizontal, vertical, and  lateral directions. In the case of longitudinal  dynamic stability, the degrees of freedom can  be limited  to pitch rotation, vertical and  horizontal translation.  Since the airplane is  usually symmetrical from port to starboard,  there will be no necessity for consideration of  coupling between longitudinal  and lateral-  directional motions. Thus, the principal vari-  ables in the longitudinal motion of an airplane  will be:  Due to the nature of the phugoid, it is not  necessary to make any specific aerodynamic  provisions to contend with  the oscillation.  The inherent long period of the oscillation al-  lows study to be directed to more important  oscillatory tendencies. Similarly, the diffet-  ences between the stick-fixed and stick-free  phugoid are not of great importance.  The second  mode  of longitudinal dynamic sta-  bility is a relatively short period motion that  (3) The flight velocity.  (2) The angle of attack (which will  differ  from the pitch attitude by the inclination of  the flight- path).  (1) The pitch attitude of the airplane. 