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1. (25 pts.) A sketch of the half-wing planform for a business aviation airplane is given below: 2.75 r 1.34 3.4n 4.8 m 5.2 m a. Assuming that the flow over wing is only altered at the location of the ailerons when the ailerons are deflected, estimate the aileron control power. Hint: the formula for calculating this has been developed in the lecture notes. Control surface area/lifting surface area can be approximated by c/c. Suppose that this wing is incorporated into a low-wing aircraft design with dihedral angle r. b. By neglecting fuselage interference on the wing-dihedral contribution, the change in rolling moment (dL) due to the local change of angle of attack (Aa) at position y is given by dL -c(y)yC, Aa dy, where C1. is constant along the span of the wing. Show that the dihedral effect can be expressed as 2C2 c(y)ydy Sb c. For the wing above, find the wing dihedral angle r to produce a dihedral effect of G. = -0.1 /rad Assume the airfoil used for the wing has a lift slope of 0.1 /°. 2. (25 pts.) Due to lateral-directional control coupling, the steady rolling and yawing moment coefficients (C) and Cn respectively) of an aircraft can be expressed as: C,=-6.3ß+10.25,+0.998, C. = 7.288-0.995 4.285, where B is sideslip angle, & is aileron deflection and & is rudder deflection (all in radians). The maximum aileron and rudder deflections for this aircraft are 8am =+15° and 8. +20°. 1 a. Does the aircraft possess static lateral and directional stability? Explain the reason for your answer. b. When the aircraft performs a turn by using aileron, does this aileron deflection generate adverse or proverse yaw? Explain your answer in terms of the direction of the yawing moment due to aileron deflection. c. During a final approach, it is assumed that lateral-directional moment equilibrium can always be maintained (Ci = Cn = 0). If the approach speed is 50 m/s and the rudder deflection is at its maximum, determine the maximum perpendicular crosswind that the aircraft can withstand to maintain its fuselage parallel to the runway centerline and the necessary aileron deflection to compensate for it. 3. (25 pts.) The wind-axis system of an aircraft can be transformed to body-axis system using two successive rotations through the angles - B and a (positive direction of the rotations is shown in the figure). a. Find the transformation matrix from the wind-axis system to the body-axis system (C.).).). b. If the velocity vector V of the aircraft coincides with the positive x-axis of the wind-axis system (in the opposite direction of the relative wind), find the velocity expression in the body-axis system using the result of part a. X-AXIS (BODY) BODY X-AXIS Y-AXIS (STABILITY) BODY X-AXIS Z-AXIS (WIND) 4. (25 pts.) In the lecture notes, a 3-2-1 sequence of Euler angles, which is commonly used in the aerospace field, is derived. Another popular choice of Euler angles is the 3-2-3 sequence that will bring the XYZ frame to the xyz frame through the following sequence of planar rotations: a rotation of magnitude y, called precession, about axis Z, then a rotation of magnitude 0, called nutation, about axis y' and finally, a rotation of magnitude 0. called spin, about axis z" a. Determine the transformation matrix for this Euler angle sequence. b. Determine the kinematic expressions relating the Euler-angle rates to the rates in the xyz frame (p.q.r). c. Determine the singularity condition with this set of Euler angles.

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Aircraft Stability and Control
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