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instant of time, the Euler angles are /r : -30 deg, 9 = lO deg, and ~ : 15 deg.
Determine the corresponding Euler angle rates.
4.3 An aircraft executes a velocitjr vector roll at an angle of attack of 40 deg
with an angular velocity of 30 deg/s about the wind axis. Plot the time history of
Euler angles for first 20 s.
4.4 Using the method of direction cosines and quaternions, compute the time
history of Euler angles for the aircraft in Example 4.3 and compare your results.
4.5 An aircraft is flying at an angle of attack of 10 deg, sideslip of 5 deg, and a
bank angle of 10 deg. The onboard accelerometers record art - 10 ft/S2, a,y, =
5 ft/S2, and azb = -5 ft/S2. Determine the components of accelerations in the
wind axes system.
4.6 An aircraft modeJ is tested in a low-speed wind tunnel at an angle of attack
of 25 deg, sideslip of -5 deg, and a bank angle of 5 deg. The internal strain gage
balance records an axial force of 25 lb, a side force of -3 lb, and a normal force
of -75 lb with respect to the model axes system. Determine the effective angle
of attack, effective sideslip, lift, and drag forces acting on the model.
4.7 An airplane is flying at a velocity of 150' ft/s at an angle of attack of 12 deg
and sideslip of 2 deg. It has angular velocities in pitch, roll, and yaw measured by
onboard rate gyros as 10 deg/s, 5 deg/s, and 10 deg/s, respectively. Determine the
angular velocity rates ct, B, and @ associated with the wind axes system.
4.8 Given p = 10 deg/s, q = 5 deg/s, and r = 10 deg/s, determine C2{lt, and g2:b.
4.9 An aircraft is in a spin at an angle of attack of 60 deg. Assuming that the
spin rate about a vertical axis through the center of gravity is 40 deg/s, plot the
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438 PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
time history of Euler angles using (a) method of Euler angles, (b) the method of
direction cosines, and (c) the quaternions.
4.10 Determine the missing elements (marked xx) of the following direction
cosine matrices:
0.1587 xx
(a) DCMi - 0.8595 -0.1218
xx 0.4963
xx 0.5283
(b) DCM2 = -0.5253 xx
0.0888 xx
4.11 An acrobatic aircraft flying at 150 m/s and a 30-deg angle of attack, executes
a body axis roll at a rate of 150 deg/s. Determine the accelerations measured by
onboard accelerometers.
4*12 An airplane fiying at an angle of attack below stall angle is in a steady roll
at a rate po deg/s. Derive the equations of motion for small disturbance motion.
4.13 An aircraft weighs 50,000 N and is in a steady level flight at 150 m/s
at sea level. The drag polar is given by CD = 0.018+0.024Cl.. The lift-curve
slope of the wings is 0.095/deg, and the wing mean aerodynamic chord is 2.5 m.
The lift-curve slope of the horizontal tail is 0.06/deg. As9uming a tail efficiency
of 0.9, estimate the stability derivatives Cxu, Czu, Cxa, Cza, Czq, Cztr, and Cma.
[Answer: Cxu -.-0.03697, Cz" =-0.2844, Cxa -.0.1047, Cza -.-0.11348,
Czq ~ ~3.5373, Cz& = -1.2476, Cmq = -8.5548, and C,na, - -2.9942. All val-
ues are per radian.]
4.14 For the wing-body ofExample 4.12,estimate the stability derivatives CLq,
Cmq, CLa, and CmCr at a Mach number of 0.4.
4.15 For an aircraft wing with Ieading-edge sweep 30 deg, aspect ratio 4, lift-
curve slope O.l/deg, dihedral angle 3 deg, taper ratio 0.5, CDD = 0.021, CDa,l -
0.0012/deg, and span 10 m, estimate the stability derivatives Ctp, Cnp, CLr, and
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