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436             PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
    We assulue k - 0.80 and that we have a midwing configuration so that zw = 0.
Substituting in above expressions, we obtain (1 + [aa/af/l)oy = 1.39, Cypjv =
-0.8920/rad, z - 0.12 m, (Clp)V = -0.000742/rad, (Cnp)V = ~0.05615/rad,
(Clr)V = 0.00804/rad, and (Cnr)V = -0.07171/rad.
4.5 Summary
    In this chapter we have studied various axes systems used in airplane dynam~
ics and discussed various methods of transfornung vectors from one coordinate
system into another. We also studied the methods of calculating time history of
Euler angles. The method based on using Euler angle rates has a singularity when
the pitch angle approaches 90 deg. However, the direction cosine method and the
quatenuons do not encounter this problem. We then formulated the problem of
airplane dynamics malang use of a moving coordinate system to avoid the prob-
lem of computing time-varymg moments and products ofinertia but had to deal
with more complex acceleration and angular momentum terms, which render the
equations of motion coupled and nonlinear. We then introduced the concept of
small disturbances, which enabled us to linearize and decouple the equations of
motion into two categories, one set of three equations for longitudinal motion and
 another set of three equations for lateral-directional motion. The advantage of this
 approach is that the two motions can be studied independent of each other. We then
used Bryan's method and assumed that the aerodynamic forces and moments de-
pend linearly on the instantaneous values ofmotion variables. We used the method
ofTaylor series expansion to estimate the aerodynamic forces and moments in the
disturbed state. We also discussed engineering methods to evaluate the stability
and control derivatives appearing in the Taylor series expansions.
     Our next task is to solve the equations of longitudinal and lateral-directional
motions to study the stability and response of the airplane to controlinputs. The
 linearity of the equations enables us to use the powerful methods oflinear control
 systems.ln Chapter  5, we wiU_ discuss basic principles oflinear system theory and
design and then, in Chapter 6, discuss their application for the study of dynamic
 stability and response of the aircraft as well as the design of autopilots and stability
augmentation systems.
References
     I Bryan, G. H., Stability in Avration, McMillan, London,  191 1.
    2Regan, F., Reentry Dynamics, AIAA Education Series, AIAA, New YorK  1984.
  3Rolfe, J. M., and Staples, K. J., Flight Simulamion, Cambridge Uruv. Press, New York,
1986.
  4Robinson, A. C., "On the Use of Quatemions in Simulation of Rigid-Body Motion:'
Wright Air Development Center, WAEDC TR 58-17, Wright-Patterson Air Force Base,
Dec. 1958.
  5MATLAB High Perj'ormcmce Numeric Compzrtation and VisuaLization Software, The
MathWorks, Natick, MA, Oct. 1992.
   6Bate, R. R., Mueller, D. D., and White, J. E., Fundamentals of Astrm~nonucs, Dover,
New York, 1971.
    7Hoak, D. E., et al., "The USAF Stability and Control Datcom:' Air Force Wright Aero-
nautical Laboratories, TR-83-3048, Oct. 1960 (Revised 1978).
EQUATIONS OF MOTION AND ESTIMATION OF STABILITY DERIVATIVES 437
     H Schuler, C. J., Ward, L. K., and Hodapp, A. E., " Techniques for Measurement of Dynamic
Stability Derivatives in Ground rfest Facilities:' AGARDograph 121, Oct. 1967.
  9"Dynamic Stability Parameters:: AGARD, LS-I14, May 1981.
 ioOrlick-Rueckmann, K. J., "Dynamic Stability Parameters:' AGARD, CP-235, 1978.
 'IDayman, B., Jr., "Free Flight Testing in High Speed Wind Tunnels:' AGARDograph
J 13,  1966.
Problems
4.1   At t - 0, a launch vehicle takes off from the surface of the Earth at the
equator with longitude L - 0. At t - 50  s, the vehicle has alongitude of 10  deg,
latitude of0, and an altitude of  50,000 ft.   The velocity components measured with
respect to the Earth-fixed OxEyEzE system are UE = 1500ft/s, VE - 3000ft/s,
and WE -. 0. Assuming that the Earth/fixed system OxE yEzE coincides with the
inertial system at t - O, determine the position and velocity of the vehicle in the
inertial reference system at t -. 50 s.
4.2     An aircraft is undergoing a steady rotation with angular velocity components
in body axes system p - lOdeg/s, q = 2deg/s, and r = 5 deg/s. At a certain
　
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