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prediction of -0.08317/rad is in error by as much as 63%.
Now let us calculate (Clr)W and (Cnr)W. The strip theory gives
CL 0.1*-5~0.16
(Clr)W= 3 = 3 1667lrad
~CDJ 0.0011*5+0.022 (
(Cr)W= 3 =- -0.009/rad
3
Nowlet us estimate (Clr)W and (Cnr)W using Datcod methods.
(Clr)W = c,(C/,).,_ .M + (A2 )~7rad
Num (
(g C~=O,M = D<. (g ),.=o.v=o
EQUATlONS OF MOTION AND ESTIMATION OF STABILITY DERiVATIVES 435
B2) 'AB+2cosAc/4 :an2Ac/4)
Num=1+2B~AABl2B /,)+(7~ :,,)(~ ')
:cosAc/4) ,AB+4~osA 4 8 )
A+2cosAc/, tan2Ac/4'
Den:l+ -.,,,)(- -)
A+4cosA.c/, 8 /
We have r : O, ALE - Ac]4 = O, A - 6, and M -. 0.15. Substituting and
simplifying, we get (Clr)W - 0.1051/rad, Thus, the str:ip theory prediction of
0.1667 differs by as much as 50o/o compared to the Datconl result.
We have
(Cnr)W = (C/: )cl- + (g )CDO
Assuaung g=0 from Fig. 4.29, we get (Cnr/CZ) = -0.02 and (Cnr/CD) =
-0.20. We have CL = 0.3834 and CDO -. 0.022. Substituting, we get (Cnr)W =
-0.0073/rad. In this case, the strip theory result of -0.009/rad differs from the
Datcom result by about 25%.
Example 4.14
Estimate the low-speed vertical tail contributions (Clp)V, (Cnp)V, (Clr)V, and
(Cnr)V at an angle of attack of 5 deg using the following data: lift-curve slope
au - 0.07/deg, leading-edge sweep ALE - 0, vertical tail length /u =' 9.0 m,
wing mean aerodynamic chord c - 3.0 m, wing span b = 15 m, zy = 0.9 m, ratio
of vertical tail area to wing area Sy]S - 0.20, wing aspect ratio A = 6, and wing
taper ratio A, = 0.5.
Solution. We have
(l+?dp T7v=0.724+~06S3S~+2f4 +0.009A
Cyp.v = -kay 1 + ?dp)-7,SS
z N zU COS CL - /v sin a
(qp)V={2(b)( b )ICyp,v
(Cnp)V=-(;)(l,,osa+zusm,)( b )Cyp.v/rad
(C,r)V = b2(lv cos a + zl sinu)(ZU cos a - ly sin ty)Cyp.v
(Cnr)V = b2 (lv cos a + zu SiIICL,)?Cyp.V
436 PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
We assulue k - 0.80 and that we have a midwing configuration so that zw = 0.
Substituting in above expressions, we obtain (1 + [aa/af/l)oy = 1.39, Cypjv =
-0.8920/rad, z - 0.12 m, (Clp)V = -0.000742/rad, (Cnp)V = ~0.05615/rad,
(Clr)V = 0.00804/rad, and (Cnr)V = -0.07171/rad.
4.5 Summary
In this chapter we have studied various axes systems used in airplane dynam~
ics and discussed various methods of transfornung vectors from one coordinate
system into another. We also studied the methods of calculating time history of
Euler angles. The method based on using Euler angle rates has a singularity when
the pitch angle approaches 90 deg. However, the direction cosine method and the
quatenuons do not encounter this problem. We then formulated the problem of
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