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INERTIA COUPLING AND SPIN 641
aac. <0 (7.65)
where CR iS the resultant force coefficient given by
CR
=,
(7.66)
At high angles of attack exceeding the stalling angle, the resultant force is approx-
imately normal to the chordline so that
CL - CR COSCL
CD - CR sina
(7.67)
(7.68)
Suppose we mount an unswept (rectangular) wing on a single-degree-of-free-
dom, free-to-roll apparatus having frictionless bearings and place it in the test
section of a low-speed wind tunnel at an angle of attack below stall. When it is
disturbed from its eqLulibrium position, the disturbance in roll will quickly die out
because qP < 0, and the wing willimmediately retum to its equilibrium position.
However, if the angle of attack is above the stalling angle, the disturbance in roll
will increase because Cip > 0. In other words, the wing is unstable in roll and
starts autorotating. The rate of roll wiD increase initially but eventually reach a
steady value. The steady-state roll rate is called the autorotational speed.
The autorotational characteristics of a wing depend on the namre of the varia-
tion oflift and drag coefficients with an angle of attack beyond stall. In Fig. 7.5,
schematic variations oflift and dfag coefficients of an airfoil with angle of attack
are shown. Generally, it is possible to identify five regions. In region I, where
ct < ctsem, the damping in roll derivative qP is negatrve, and the airfoilis stable in
roll. Region II, with Cip > 0, is one of spontaneous autorotation because even a
slight disturbance willinitiate autorotation.ln region III, even though the lift-curve
slope i.s negative, the airfoilis stable again because the magnitude of thelift-curve
slope is smaller than the drag coefficient. In other words, the damping effect due
to drag is sufficient to make the airfoil stable in roll. In region IV; once again
the airfoil exhibits autorotative tendency but only to large disturbances in roll so
that the angle of attack of the down-going wing falls in region N and that of the
up-going wing in region II. To distinguish this autorotational tendency from the
spontaneous autorotative tendency of region II, region IV is called one of latent
autorotation. In region V, the airfoilis stable again because oflarge 'values of drag
coefftcient
7-3-2 Autorotation of Fuselages
The autorotation of a fuselage depends on its cross-sectional shape. Generally,
fuselages with circular cross sections or cross sections with round bottoms are
Equation (7.63) is based on the assumption that the angle of attack is small and
is in the linear range. This approximation is satisfactory for most of the airfoils
whose stalling angles are in the range of 10-15 deg. However,if the stalling angles
are higher, then the criterion for instability in roll is given by8
522 PERFORMANCE, STABiLITY, DYNAMICS, AND CONTROL
5.10.12 ObserverDesign
The pole-placement design method requires that all the state variables are accu-
rately measured and are available for feedback.lf this requirement is met and the
system is controllable, then a complete control over all the eigenvalues is possible.
A problem arises if some or all of the states are not actually measured or are not
avaYi:lable for state feedback. An obvious solution would be to add more sensors that
can measure the missing states. However, this approach may not always be feasible
and often can be quite expensive. The other option is to estimate the unavailable
states using a subsystem called a state observer. An observer that estimates all the
states, including t<ose that are actually measured, is called a full-state observer,
and one that estimates only those states that are not measured is called a reduced-
state observer. Here, we will discuss the procedure for the design of a full-state
observer.
The design of an observer is based on the knowledge of a mathematical model
of the plant, input(s), and output(s). The basic idea is to make the estimated states
as close to the actual states as possible, but the problem is that all the actual states
are not available for comparison. However, we do know the output of the given
plant, and we can compare it with the estimated output of the observer. The design
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