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234
5 6 7 8 9 10
t, s .
~g. 6.22 Unit-step response Oateral-directional) of the general a~niation airplane:
yaw rate~
where
Ap.& = Cy8. (Ixllzl - Ix2zl)
(6.237)
Bp.81 = -bi CyBr.(lxl Cnr + Clp Izl + Clr Ixzl + Cnplxzl)
+ bi C),p(lzl CL& + Cn8o Ixz I) - (Fr?l - bi Cyr)(Cl8a Ixzl + Cn8a Ixl)
(6.238)
Cp.8, = b?jCy8a(Clp C,,r - Ctr Cnp) + Cy4(CI8, Izl + Cn8a IxzJ)
- bf C).P (CL8,Cnr - Cn8a C,r) + bi (mi - bi Cyr)(Cn8" C,p - C,Sa Cnp)
(6.239)
Dp.b" = -bi Cy+ (q8,Cnr - Cn8a Clr) (6.240)
Expanding the determinant Aiar, we get a fifth-order polynomial in s, which is of
theYform"F '
Alar = A8 S5 + B8 S4 + CBr"S3 + D8iuS2 + E8rus (6.241)
284 PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
where
a) Cr,p > 0
b) CA6 < 0
Fig.3.88 Sign convention for rudderlunge-moment coefficient
Clp = aac~
C,,8r = aag-
(3.323)
(3.324)
The hinge-moment coefficients Chp and q,ar can be estimated using the methods
given in Section 3.3.14 using the variable p instead of cr and 8r instead of 8e.
Recall that the elevator hinge moment was assumed positive if it rotated the
elevator trailing edge down or in a direction to increase the elevator deflection.
In a similar way, we assume rudder hinge moment to be positive if it rotates the
rudder such that rudder deflection increases,i.e., it makes the rudder deflect to the
left. With this understanding, we observe that a positive sideslip causes a positive
hinge moment so that Chp > 0. Similarly, a positive rudder deflection generates
a negative hinge moment so that Ch8r < O. These concepts are schematically
illustrated in F~g 3.88.
With Chp > 0 and'ChSr < 0 for positive sideslip, we observe from Eq. (3.322)
that the floating angle of the rudder will be positive, i.e., the rudder floats to the
left when the aircraft sideslips to starboard and vice versa.
STATfC STABIUTY AND CONTROL
285
The effective sideslip angle Pv of the vertical tail with rudder free to :float is
given by
Then,
pv N p + Cr + -C28rf
(3.325)
(Cn.V)free=ka,,[p+o- C
. U-T2gt}3]V217u (3.326)
(Cnp.VXtee = ka,[(1+ ~p) - ,2g ] V2?7v (3.327)
Using Eq. (3.317), we get
(Cnp.V)free =(Cnp.v)fix+C,p (g ) (3.328)
and
(Cnp)free = (Cnp)W + (Cnp)B(W) + (Cnp.V)frree (3.329)
For positive rudder-free directional stability, (Cnp)fn:e > O.
3.5.7 PedaIForces
Suppose that the aircraft is required to maintain a given sideslip [3. Then, the
required rudder deflectrion is given by
8r- (Cnp)rixP
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