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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
642 PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
Fig.T5 Schematic variations of airfoillift and drag coeffiaents with angle of attack.
resistant to autorotation. On the other hand, cross sections with fiat bottoms are
prone to autorotation.
To understand the aerodynamics associated with the autorotation ofthe ffiselage,
consider a noncircular cylindrical model mounted on a single-degree-of-freedom,
free-to-roll apparatus and held at an angle of attack in an airstream of uniform
velocity Uo as shown in Fig. 7.6. We assume that the modelis pivoted atits center
of gravity and is constrained to rotate about the velocity vector Uo. Let the origin
of the body-fixed coordinate system coincide with the center of gravity.
Suppose that the modelis ir;;parted a disturbance that makes it rotate in a clock-
wise direction (viewed from a~"end) with an angular velocity S2. The crossflow
angle 4 at an axial location x is given by
o = t--'(szv7)
(7.69)
We note that, for the cross sections ahead of the center of gravity, the crossflow
angle is positrve and, for the sections aft of the center of gravity, the crossflow
angle is negative.
The local dynamic pressure at the axial location x is given by
q, = -,pUo' [,+ (
S2x sin CL
)2]
(7.70)
For simplicity,-we assume that the side force coefficient Cy of the fuselage cross
sections depends only on the crossflow angle 4.
INERTIA COUPLING AND SPIN
C;
+>o
Section AA
Section BB
Cy
Fig. 7.6 Schematic diagram ofan autorotating fuselage.
643
rfhe moment developed by the section AA about the axis of rotation (velocity
vector) is given by
z\Nn = -,pU- [,+ (
or, in coefficient form,
where
ACfZ = -,, [,+(
C2x sin a
Uo
S2x sin ct
ACcz
)2lCy()bOxsinadx (7.71)
)2],y()xsiDCL-cIX (7.72)
ANrz
21p U02b0/2
(7.73)
Here, bo is the width of the body, and I is the total length of the body.
Given the variation of Cy with crossflow angle 4, an integration of Eq. (7.72)
gives the variation of the moment coefficient Cn as a function of the angular
velocity CZ. The schematic van:ation of Csz is shown in Fig. 7.7. It is possible that
this variation is of two types. For type I variation, the yawing moment coefficient
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