)  -8.0 0
           )00
 )
     A          l*5 0 _
                                                          J

Design a full-state feedback controller to place the closed-loop eigenvalues at
 0, -0.6 1:  j1.9079, -0.005, and -8.0.
7.1  Introduction
                     7
Inertia Coupling and Spin
     In the preceding chapters, we have studied the airplane stability, dynamics, and
control with the assumption that the longitudinal and lateral-directional motions
                                           I
ofthe airplane could bevdecoupled and studied separately. With these assumptions,
the problem of airplane dynamics and control was linearized so that the analyses
methods oflinear systems could be used. For example, the response of the airplane
to a 2-deg elevator input would be exactly double that for a'r-deg elevator input.
Also, there was no cross coupling between the longitudinal and lateral-directional
degrees of freedom.ln other words, operating the rudder or the ailerons would not
generate any pitching motion or a change in the forward speed or a change in the
angle of attack. Similarly, moving the elevator would not generate any sideslip.
rolling, or yawing motion. As a consequence, one had to solve the problem for
only one set of control mputs and, using those solutions, the response to any
other combination of control inputs could be quickly deduced. Furthermore, the
response to combined control inputs was the sum of the responses to individual
controlinputs.ln other words, the fiight dynamicist had to solve the problem only
once. TheYnj' he had it solved for all other cases. What could be simpler?
   However, this type of simple approach cannot be used for probt;ems in which
the longitudinal and lateral motions are coupled. Such coupling occurs because of
either inertial cross coupling or aerodynamic nonlinearities. The examples of the
first category are the inertia or roll coupling problems and the spinning motion. We
will study these two problems in this chapter. The cross coupling between longi-
tudinal and lateral motions due to aerodynamic nonlinearities and large amplitude
motions occurs in fiight at high angles of attack close to or exceeding the stalling
angle. We will study these problems in Chapter 8.
7.2   Iner;tia Coupling   '         '.
    The problem of inertia coupling was totally unknown to the aeronautical en-
gineer until the closing years of World War II. During a demonstration flight in
late 1944, the German fighter aircraft Heinkel 162 disintegrated in a fast high-
speed rolling maneuver. The Heinkel 162 was a small single-engine jet fighter. It
so happened that a number of cameras recorded this event, which helped a close
reconstruction of the entire episode. However, with the knowledge and expertise
available then, no one could come up with a convincing explanation as to why
the Heinkel 162 aircraft behaved that way. The second example was that of the
British Fairey Delta aircraft during an air show at Famborough, England. This
aircraft had an impressive roll rate capability of 500 degls. Halfway through the
roll, the pilot lost control, or more appropriately, the airplane took over. Luckily,
this incidence did not result in a fatality or severe damage to the aircraft. The
627
628           PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
Fairey Delta aircraft carried extensive fiight test instrumentation, which recorded
the event. Once again, this incident offered a frustrating puzzle because no calcu-
lated response even with wild assumptions about the misbehaved aerodynamics
　
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