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Ap - BpK =
00
10
Ol
0
0
0
-(ao+ki) -(al+k2) . . -(an_i+kn)
3) Write down the characteristic equation for the plant as follows
(5.261)
}sl - (Ap - Bp K)l = Sn + (an _ I + kn)sn -1 + .. .+ (ai + k2)S
+ (ao + ki) : 0 (5.262)
4) Decide on all the closed-loop pole locations that give the desired system
response, i.e., the desired characteristic equation is given by
Sn + dn _lSn -1 + ... + d] s + do - 0 (5.263)
5) Equate coefficients of the two charactenstic equations
so that
dn -i - an -i + kn, . - ', di -. ai + k2, do - ao + ki (5.264)
ki - do - ao, k2 = di - ai, . .., kn - dn -i - an -1 5.265)
t)
1~
640 PERFORMANCE, STABILIIY, DYNAMICS, AND CONTROL
co.V -
-. 1.5476 rad/s
The slope of the line OP for the given aircraft is given by
(-)2 = 2.2068
As the roll rate increases, we move towards the origin along the line OP. When the
roll rate increases to that corresponding to point Pi, the aircraft experiences yaw
divergence.However, when the rollrate is furtherincreased to that corresponding to
P2, the aircraft becomes stable again (spin stabilized). The roll rates corresponding
to Pi and P2 are, respectively, equal to 1.8344 and 2.3680 rad/s. Thus, the aircraft
experiences yaw divergence due to inertial coupling when
1.8344 rad/s < Po < 2.3680 rad/s
7.3 Autorotation of Wings and Fuselages
Autorotation is an inherent tendency of unswept wings at angles of attack be-
yond the stalling angle and is one of the princiPpal causes for a straight-wing,
propeller-driven light airplane to enter into a spin. The autorotative tendency of
the fuselage also contributes to the development of the spin. However, the auto-
rotative tendencies of the wings and fuselages by themselves are not sufficient
to make an airplane develop a steady-state spin. Whether an airplane develops a
steady-state spin depends on the balance between the aerodyrtamic and inertial
moments as we will discuss Iater in this chapter.
7.3.1 Autorotation of Vfings
Based on strip theor}r, we have the following expression for damping-in-roll
derivative Cip (see Chapter 4) for a rectangular (unswept) wing of high aspect
ratio:
(Clp)W = -g(ao+CD) (7.63)
where ao is the sectionallift-curve slope and CD iS the sectional drag coefficient
For flight at low angles of attack, ao is positive so that Cip < 0. For angles of
attack above stall, ao < O and, if lao} > CD, then the damping-in~roll deriva-
tive changes sign and becomes positive (Cip > 0). When this happens, the wing
becomes unstable in roll. Thus, for instability in roll,
ao + CD < 0 (7.64)
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