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0.0007ce deg, sectional lift-curve slope ao = O.l/deg, sectional stall angle arstaii -
 14 deg, and F' = 0.
3.23  An aircraft has a wing loading of 2850 N/m2, a wing span of 27 m, a
maximum lift coefficient of 1.75, and vertical tail lift-cuwe slope of 0.082/deg.
(CnB)fix  =  0.015]deg, vertical tail -volume ratio is 0.2, and the coefficient k  -  0.90.
Assuming that 1 deg of rudder deflection changes the vertical tail sideslip by 0.3
deg and that the maximum rudder deflectionis restricted to :1: 25 deg, determine the
maximum crosswind speed that can be permitted for takeoff at sea level. Assume
that the unstick velocity is 1.2 times the stall velocity. [Answer 7.3598 m/s.]
318             PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
3.24      A twinjet engine aircraft has a thrust of 20,000 N per engine, and the engines
are separated by a spanwise distance of 10 m. The wing area is 60 rn2 and the wing
span is 15 m. Assuming that the rudder effectiveness vanishes beyond +25 deg
deflection, determine the minimum rudder effectiveness to hold zero sideslip with
one engine losing all the thrust at a forward speed of 75 m/s at sea level.  [Answer:
0.0013/deg.]
3.25    For a high-aspect ratio swept-back wing with leading-edge sweep angle A
and dihedral r, show that
(CLp)w =: -
aoc(yh)yh dyh
3.26     For the airplane in Exercise 3.20, determine Cip at M  - 0.3 and an altitude
of 3000 m and atvM = 2.0 and 15,000 m altitude. Also, assuming'that Ca(Y) =
0.3c(y) ancl ailerons extend from y - 0.6s to 0.9s, where s is the semispan,
determine the aileron effectiveness at low subsonic speeds.
3-27     For the flying wing in Exercise 3.21,using strip theory, determine the rolling
moment.
3.28    For the swept-back wing of Exercise 3.22, using strip theory, plot the vari-
ation of Cip with angle of attack.
                              4
             Equations of Motion and
Estimation of Stability Derivatives
4.1 Introduction
     In the preceding chapter, we studied static stability and control of airplanes. We
assumed that the motion following either an external disturbance such as a wind
gust or an intenial disturbance like a control input was so slow that the inertia
and damping forces/moments could be ignored. Thus, we essentially assumed the
airplane to be a static system and studied the stability and control based on the
static forces and moments acting on the airplane following a disturbance.
  In this chapter, we will study the auTlane as a dynamic system and derrve
equations of motiort. We will consider the influence ofits mass and inertia on the
motion. We will also consider aerodynamic damping effects. However, we will not
be considering the aeroelastic effects and,instead, we will assume that the airplane
functions like a rigid body.
   The foundaOons of the airplane dynamic stability and response were laid by
the pioneering work of Bryan.l His'9ormulation was based on two principal as-
sumptions: 1) the instantaneous aerodynamic forces and moments depend only
on instantaneous values of the motionyvariables and 2) the aerodynamic forces
and moments vary linearly with motion variables. This approach ofjBryan,l intro-
duced more than 80 years ago,is used even today in the study of dynamic stability,
control, and response of the airplanc and forms the basis of the subject matter
discussed in this chapter.
       To begin with, we will discuss various axes systems used in the study of airplane
dynamics and present relations for transforming vectors from one coordinate sys-
tem into another system. We will then formulate the problem of airplane dynamics
and derrve equations of motion for six-degree-of-fre9dom analyses. Because these
equations are, in general, coupled and nonlinear, it is difficult to obtain analytical
solutions. In view of this, we will assume that the motion following a disturbance
is one of small amplitudes in all the disturbed variables. With this assumption
and the usual approximation that the airplane has a vertical plane of symmetry, it
is possible to linearize and decouple the equations of motion into two sets, one
for the longitudinal motion and a:other for lateral-directional motion. Then we
use the method of Bryanl and assume that the aerodynamic forces and moments
　
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