Question

Consider a fixed-wing airplane with a mass of 3800 kg. At a given time instant, the Euler angles are: ? = 90 deg, ? = 0 deg, ? = 30 deg. a) (10 pts) In which direction does the airplane fly? Find the numerical value of the direction cosine matrix L_{VB}. b) (10 pts) Determine the components of the weight (force) in the vehicle-carried coordinate system (vec{W}_V), and in the body-fixed coordinate system (vec{W}_B). How can you verify that the results are correct? c) (10 pts) At another time instant, the instruments show the following non-zero readings: pitch rate equal to 0.2 rad/s, x-body velocity equal to 90 m/s, angle of attack equal to 4 deg. What is the acceleration of the airplane w.r.t the inertial reference frame (in the body axes)? Useful formula and expressions Elementary transformation matrices: L_1 = egin{bmatrix} 1 & 0 & 0 \ 0 & cos phi & sin phi \ 0 & -sin phi & cos phi end{bmatrix}; L_2 = egin{bmatrix} cos phi & 0 & -sin phi \ 0 & 1 & 0 \ sin phi & 0 & cos phi end{bmatrix}; L_3 = egin{bmatrix} cos phi & sin phi & 0 \ -sin phi & cos phi & 0 \ 0 & 0 & 1 end{bmatrix} Aerodynamic angles: tan ? = w/u, sin ? = v/|V|.

          Consider a fixed-wing airplane with a mass of 3800 kg. At a given time instant, the Euler angles are: ? = 90 deg, ? = 0 deg, ? = 30 deg.

a) (10 pts) In which direction does the airplane fly? Find the numerical value of the direction cosine matrix L_{VB}.

b) (10 pts) Determine the components of the weight (force) in the vehicle-carried coordinate system (vec{W}_V), and in the body-fixed coordinate system (vec{W}_B). How can you verify that the results are correct?

c) (10 pts) At another time instant, the instruments show the following non-zero readings: pitch rate equal to 0.2 rad/s, x-body velocity equal to 90 m/s, angle of attack equal to 4 deg. What is the acceleration of the airplane w.r.t the inertial reference frame (in the body axes)?

Useful formula and expressions

Elementary transformation matrices:

L_1 = egin{bmatrix} 1 & 0 & 0 \ 0 & cos phi & sin phi \ 0 & -sin phi & cos phi end{bmatrix}; L_2 = egin{bmatrix} cos phi & 0 & -sin phi \ 0 & 1 & 0 \ sin phi & 0 & cos phi end{bmatrix}; L_3 = egin{bmatrix} cos phi & sin phi & 0 \ -sin phi & cos phi & 0 \ 0 & 0 & 1 end{bmatrix}

Aerodynamic angles: tan ? = w/u, sin ? = v/|V|.

Consider a fixed-wing airplane with a mass of 3800 kg. At a given time instant, the Euler angles are: ? = 90 deg, ? = 0 deg, ? = 30 deg.

a) (10 pts) In which direction does the airplane fly? Find the numerical value of the direction cosine matrix LVB.

b) (10 pts) Determine the components of the weight (force) in the vehicle-carried coordinate system (vecWV), and in the body-fixed coordinate system (vecWB). How can you verify that the results are correct?

c) (10 pts) At another time instant, the instruments show the following non-zero readings: pitch rate equal to 0.2 rad/s, x-body velocity equal to 90 m/s, angle of attack equal to 4 deg. What is the acceleration of the airplane w.r.t the inertial reference frame (in the body axes)?

Useful formula and expressions

Elementary transformation matrices:

L1 = eginbmatrix 1 0 0 0 cos phi sin phi 0 -sin phi cos phi endbmatrix; L2 = eginbmatrix cos phi 0 -sin phi 0 1 0 sin phi 0 cos phi endbmatrix; L3 = eginbmatrix cos phi sin phi 0 -sin phi cos phi 0 0 0 1 endbmatrix

Aerodynamic angles: tan ? = w/u, sin ? = v/|V|.

Added by Vanessa O.

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