4. An electric vertical take-off and landing (eVTOL) aircraft uses electric power to hover, take off,...

 

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4. An electric vertical take-off and landing (eVTOL) aircraft uses electric power to hover, take off, and land vertically. The eVTOL aircraft is modelled as shown with a total mass of M = 40m and radius of gyration about the centre of mass axis OX of d and includes four rotor blade systems, each of mass m and diameter d. The rotor blade systems can provide thrust in the +z directions that increases with the magnitude of the spin rate p. The direction of the spin axis, z, can be tilted to be aligned with the vertical, Z, during takeoff as shown. Assume no friction between the aircraft body and each rotor. Assume OXYZ is aligned with the principal axes and the rotor blade (note there are two blades per rotor) may be modeled as a uniform thin rod with mass moment of inertia about its centre of mass Icon m, d/12. The rotors are spun up to speeds of p, p+4p and p-4p (in rad/s) relative to the aircraft body in the directions as shown above using motor torques acting between the rotors and body. The aircraft body is initially on the ground during spinup. P Back Rotor blades P Z Y. Y X, x Mg Front p-4p p+4p (a) Predict and explain the direction of the aircraft tilting and displacement motion after it lifts off the ground. [3mks] (b) Calculate the angular momentum components of the aircraft after it is spun up to the steady speeds shown, and has tilting angular rate of 2 rad/s and velocity v m/s. [5 marks] (c) Calculate the total kinetic energy of the aircraft after it is spun up to the steady speeds shown, and has tilting angular rate of 2 rad/s and velocity v m/s. [5 marks] [2 marks] (d) Predict and explain any ground reactions on the aircraft while the rotors are being spun up. (e) Assuming steady rotor speeds and tilting angular rate of 2 rad/s, calculate the gyroscopic moment on each of the wings and the total on the aircraft. (f) Explain why the choice of rotor speeds and directions is desirable. [8 marks] [2 marks] (g) Predict the free precession rates and directions of the rotor blades if they break free from the eVTOL as a ratio of their spin rates. Assume the nutation angle is small. Explain whether this motion is stable about the spin axis or not? [5 marks] 4. An electric vertical take-off and landing (eVTOL) aircraft uses electric power to hover, take off, and land vertically. The eVTOL aircraft is modelled as shown with a total mass of M = 40m and radius of gyration about the centre of mass axis OX of d and includes four rotor blade systems, each of mass m and diameter d. The rotor blade systems can provide thrust in the +z directions that increases with the magnitude of the spin rate p. The direction of the spin axis, z, can be tilted to be aligned with the vertical, Z, during takeoff as shown. Assume no friction between the aircraft body and each rotor. Assume OXYZ is aligned with the principal axes and the rotor blade (note there are two blades per rotor) may be modeled as a uniform thin rod with mass moment of inertia about its centre of mass Icon m, d/12. The rotors are spun up to speeds of p, p+4p and p-4p (in rad/s) relative to the aircraft body in the directions as shown above using motor torques acting between the rotors and body. The aircraft body is initially on the ground during spinup. P Back Rotor blades P Z Y. Y X, x Mg Front p-4p p+4p (a) Predict and explain the direction of the aircraft tilting and displacement motion after it lifts off the ground. [3mks] (b) Calculate the angular momentum components of the aircraft after it is spun up to the steady speeds shown, and has tilting angular rate of 2 rad/s and velocity v m/s. [5 marks] (c) Calculate the total kinetic energy of the aircraft after it is spun up to the steady speeds shown, and has tilting angular rate of 2 rad/s and velocity v m/s. [5 marks] [2 marks] (d) Predict and explain any ground reactions on the aircraft while the rotors are being spun up. (e) Assuming steady rotor speeds and tilting angular rate of 2 rad/s, calculate the gyroscopic moment on each of the wings and the total on the aircraft. (f) Explain why the choice of rotor speeds and directions is desirable. [8 marks] [2 marks] (g) Predict the free precession rates and directions of the rotor blades if they break free from the eVTOL as a ratio of their spin rates. Assume the nutation angle is small. Explain whether this motion is stable about the spin axis or not? [5 marks]